Sobre la estabilidad de ciertos tipos de ecuaciones no lineales de Klein-Gordon

Facultad de Matem´aticas Universidad de Sevilla Trabajo Fin de Grado Doble grado en F´ısica y Matem´aticas Sobre la estabilidad de ciertos tipos de ecuaciones no lineales de Klein-Gordon Presentado por Pablo Rab´an Mond´ejar Dirigido por los Drs. Renato ´ Alvarez Nodarse y Niurka Rodr´ıguez Quintero. Sevilla, junio de 2021 Faculty of Mathematics University of Seville Final Degree Project Double Degree in Physics and Mathematics On the stability of certain kind of nonlinear Klein-Gordon equations Presented by Pablo Rab´an Mond´ejar Directed by Drs. Renato ´ Alvarez Nodarse and Niurka Rodr´ıguez Quintero. Seville, June, 2021 Resumen En el presente trabajo, nuestro objetivo es estudiar la estabilidad de ciertas soluciones estacionarias de una determinada familia de ecuaciones no lineales de Klein-Gordon. El trabajo se encuentra dividido en tres cap´ıtulos. El primero de ellos tiene como fin el servir de motivaci´on sobre la singularidad y la importancia de las soluciones de tipo “solit´on” y de tipo “kink”, dado que las soluciones objeto de nuestro estudio de estabilidad siempre pertenecen a esta ´ultima clase de soluciones. En el segundo cap´ıtulo desarrollamos el m´etodo del pseudopotencial, con objeto de obtener soluciones tipo “kink” de ecuaciones no lineales de Klein-Gordon, y lo aplicamos a dos casos particulares. Adem´as, se realiza un estudio de estabilidad de las soluciones tipo “kink” de ecuaciones no lineales de Klein-Gordon y se sientan las bases y las definiciones de nuestro estudio de estabilidad lineal hasta llegar a una ecuaci´on diferencial ordinaria de tipo hipergeom´etrico. Finalmente, en el tercer cap´ıtulo, resolveremos dicha ecuaci´on para toda una familia de ecuaciones no lineales de Klein-Gordon, haciendo uso de funciones especiales. Abstract In the present work, our aim is to study the stability of certain static solutions of a particular family of nonlinear Klein-Gordon equations. The work is divided in three chapters. The purpose of the first chapter is to motivate about the singularity and the importance of the “soliton” kind and “kink” kind solutions, due to the fact that the solutions object of our stability study always belong to this last class of solutions. In the second chapter, we develop the pseudopotential method in order to obtain “kink” kind solutions of nonlinear Klein-Gordon equations, and we apply it to two particular cases. Furthermore, a stability study of “kink” kind solutions of nonlinear Klein-Gordon equations is performed, and we establish the foundations and definitions of our linear stability study until we reach a hypergeometric kind ordinary differential equation. Finally, in the third chapter, we solve such equation for an entire family of nonlinear Klein-Gordon equations, making use of special functions. Agradecimientos Quer´ıa agradecer a todos aquellos que me han acompa˜nado en esta etapa y han permitido que llegar hasta aqu´ı sea posible. En primer lugar a mi familia, especialmente a mi padre Miguel y a mi madre Teresa, por soportarme y apoyarme d´ıa a d´ıa, y por compartir miles de momentos conmigo sin los que todo esto no habr´ıa sido posible. Tambi´en a mi hermano Miguel, por estar siempre cuando lo necesito y aportar alegr´ıa tanto en los mejores como en los peores momentos, y a mi t´ıa Pilar y a mi abuela Carmen, por traer mucha diversi´on y buenos ratos cada noche. De igual forma, especial menci´on merecen mis tutores de TFG, Renato y Niurka, por su enorme atenci´on y apoyo, y por ense˜narme maravillosamente a investigar, siempre de la forma m´as cercana y amena posible. Finalmente, quer´ıa mostrar mi agradecimiento a las personas que me han acompa- ˜nado directamente en esta etapa acad´emica. Entre ellos cabe destacar a algunos compa˜neros de clase que se han convertido en verdaderos amigos, sobre todo a Paula, por acompa˜narme siempre y ayudarme a recorrer este camino cada d´ıa, aportando luz en los momentos m´as oscuros, y haciendo que los m´as luminosos sean inmejorables. Tambi´en a todos aquellos profesores que me han hecho aprender y evolucionar, y que han despertado mi curiosidad en mayor o menor medida, y que de igual forma me han transmitido nuevos conocimientos y formas de pensar. Todos ellos han contribuido de distinta manera a que este trabajo se haya realizado, y algunos de ellos son fundamentales para que sea quien soy hoy. De nuevo, gracias a todos. Contents 1 Nonlinear waves: solitons 1 1.1 A brief history of solitons ........................ 1 1.2 Effect of dispersion and nonlinearity on waves ............. 5 1.3 Topological solitons and kinks ...................... 8 2 Linear stability of the nonlinear Klein-Gordon equation 13 2.1 The sine-Gordon equation ........................ 13 2.2 Obtaining kink solutions. Pseudopotential method ........... 17 2.3 Linear stability of kink solutions ..................... 20 2.4 Some examples .............................. 25 2.4.1 Sine-Gordon equation: soliton solutions and their stability . . 25 2.4.2 ϕ4equation: kink solutions and their stability ......... 28 3 The stability problem for a family of NLKG equations 31 3.1 The stability of a class of potentials ................... 33 3.1.1 Solution of the Sturm-Liouville problem: connection with Jacobi polynomials ......................... 34 3.2 Boundedness of the solutions of the Sturm-Liouville problem ..... 35 3.2.1 Case I: νpure imaginary ..................... 37 3.2.2 Case II: νreal ........................... 39 4 Conclusions and further problems 49 A Klein-Gordon equation 53 B The intrinsic energy and pseudopotential 57 C Jacobi polynomials 59 Bibliography .................................. 63 6Chapter 1. Nonlinear waves: solitons ∂2ϕ ∂x2=1 r2 ∂2ϕ ∂t2,(1.2) where ris a real constant. The general solution is the superposition of two travellingwaves, both with speed r, one travelling to the right and the other to the left on the xaxis: ϕ(x, t) = f(x−rt) + g(x+rt). Another example, maybe the simplest one, is obtained from KdV equation (1.1) setting =γ= 0, in other words, dispersive and nonlinear effects are not considered. In this case, the equation is ∂u ∂t +d∂u ∂x = 0,(1.3) which solution now is ϕ(x, t) = h(x−dt), a travelling-wave too. But, what do both equations (1.2) and (1.3), and other linear and non dispersive equations have in common? First of all, they are linear, and we know that this is a very interesting property because if we have two solutions of the equation, any linear combination of them is also a solution, so the general problem can be divided into simpler ones. Nonlinear equations do not verify this property, i.e. “superposition principle” does not take place. This means that nonlinear wave equations are more difficult to analyse mathematically and that a general method for their solution does not exist. Moreover, the behaviour of their solutions depends on the specific equation. On the other hand, both equations are also non dispersive. Indeed, by inserting the plane wave solution ϕ(x, t) = ei(kx−ωt)in any non dispersive and linear equation, one can obtain the following relation between kand ω: ω=±ck, where cis known as phase velocity vp, generally defined as the quotient of the frequency ωand the wave number k. Notice that in the two examples that we have considered, in effect that is what happens. In general, the relation between kand ω, usually considered as a function ω(k), is called dispersion relation, and the particular case ω=±ck is the one which characterises the non dispersive wave equations. When a wave equation is both linear and non dispersive, the harmonic wave solutions are f+(x−ct) = eik(x−ct)and f−(x+ct) = eik(x+ct), and because of the linearity, the general solution is the sum of two travelling-waves ϕ(x, t) = f+(x−ct) + f−(x+ct). That is, the general solution of a linear and non dispersive wave equation is 1.2. Effect of dispersion and nonlinearity on waves 7 similar to the solitary waves that Russell observed in nonlinear and dispersive media, a function ϕ(x, t) = f(x−ct). Nevertheless, a linear and dispersive equation does not have either these solutions in general. As an example, we can consider a particular case of the KdV equation ∂u ∂t +d∂u ∂x +∂3u ∂x3= 0,(1.4) where of course, we have made the nonlinear constant γ= 0. If we look for harmonic wave solutions, we obtain the dispersion relation ω(k) = dk(1 −k2/d), where the phase velocity now is vp=d(1 −k2/d). The equation is linear, so if we consider an arbitrary solution of the equation and we decompose it in harmonic waves (through Fourier analysis), each of these waves would have certain ωand kleading to a particular phase velocity ϕk(x, t) = ei(kx−ωt)=eik(x−vp(k)t)=f(x−vp(k)t). As a consequence, each of these waves will travel at its own speed, and as a result the arbitrary solution will not be a travelling-wave. The solution does not have a fixed shape, it changes with time until it disappears, and it is said that it is a dispersive wave. We can see in figures 1.4 and 1.5 this effect for a certain solution of equation (1.4). Figure 1.4: Evolution in time for a solution of linear and dispersive KdV equation. Notice how the initial wave shape changes with time and “secondary waves” appear. And as we have previously commented, nonlinear partial differential equations do not allow decomposition of the solution into other solutions, and in many cases they even do not hold harmonic wave solutions. What is extremely remarkable 8Chapter 1. Nonlinear waves: solitons about solitary waves in nonlinear and dispersive media is that in these media both effects compensate for certain solutions and as a result a travelling-wave solution arises. That is why Russell and many other scientifics were very impressed by the existence of these solutions, not only because they are analytical solutions of certain partial differential equations, but also due to the fact that they are real observable waves. Figure 1.5: Solution of linear and dispersive KdV equation for initial time and after a certain time. Notice how the initial wave shape changes and “secondary waves” appear as time goes by. As a conclusion, dispersive equations need nonlinear effects so that a travellingwave is a possible solution, as well as nonlinearity tends to make solutions “steepen” (so a travelling-wave solution is not possible either in most of the cases) and also need dispersion effect in order to have a travelling-wave solution. It is quite impressive that two effects which avoid the existence of these solutions can compensate each other in certain cases and make it possible that the medium hold a “solitary wave” solution. 1.3 Topological solitons and kinks Now, we will introduce the concepts of topological soliton and kink solutions. In this project, our soliton definition (which is almost the most commonly accepted) is restricted to one dimension and it will be: Definition 1.1 A soliton is a travelling-wave solution of a nonlinear and dispersive partial differential equation which verifies that, after a collision with other soliton, it emerges with the same shape and speed. Remark We will understand that a collision between two travelling-waves happens when two travelling-waves are moving in opposite directions until they overlap. 1.3. Topological solitons and kinks 9 It is important to highlight that the wave that Russell saw 150 years ago satisfies this definition, as well as solitary wave solutions to the KdV equation, because Korteweg and de Vries showed that those solutions “reappear virtually unaffected in size or shape”. Those solitons solutions are known as “nontopological”. But there is a second type of solitons that we want to briefly study in this section: topological solitons. The name arises from the fact that those solutions are the result of geometric restrictions of certain problems. Topological solitons are usually the solutions of field equations, understanding that a field ψis a scalar or vectorial function (the arrival is Ror RNwith N > 1, respectively) of the position and time. From this point on, we will suppose a scalar field. In these conditions, the field has an energy density ρ(associated scalar magnitude, dependent on the field at each position and time) according with different physics hypotheses, which follows the following expression for each position and time: ρ(ψ(x, t)) = 1 2∂ψ ∂t (x, t)2 +1 2∂ψ ∂x (x, t)2 +g(ψ(x, t)),(1.5) where gis a continuous function that can be defined as non negative, and also1 g∈C2(R) and ψ∈C2(R2) is verified. The first term is known as kinetic energy, the second one as potential energy, and the last one as intrinsic energy (see [8]). If we think in a wave on a pond, the first two terms are intuitive: a wave which varies more quickly on time or on space (more up and downs) carries a greater energy than the one that a wave with a lower variation on time or space would carry. On the other hand, the intrinsic energy can be thought of in that same example as the increase of the energy of the wave in terms of having a higher height. However, in general intrinsic energy strongly depends on the particular field we are studying. In order to understand the concept of topological soliton, we must focus on the intrinsic energy first. In general, as we have previously said, gcan be defined as non negative, and this means that it has a global minimum, which can be reached in one or more points. The necessary condition so that a topological soliton (or more generally, a solitary wave) solution is possible is that greaches the global minimum in at least two different points. Now, we are going to focus on this necessary condition that has just been men- 1Remember the general definitions for an arbitrary set A⊂Rkand an open set Ω ⊂Rk, with k∈N C0(A) = {f:A→R|fis continuous }, L∞(Ω) = {f: Ω →R| ∃ M > 0\ |f(x)| ≤ M∀x∈Ω}. In addition, f∈Cn(Ω) for n≥1 if f∈C0(Ω) and all its partial derivatives up to order ninclusive belong to C0(Ω) as well. 10 Chapter 1. Nonlinear waves: solitons tioned. Under normal conditions, the field would be in one of the points where the minimum is reached, for every position and time, but let us suppose a different field solution, a wave which has a different value in some positions at a certain time. Due to the fact that physics systems tend to go to the lowest energy state, the wave will spread and will disperse while the energy is redistributed and the density of energy decays to zero at each point. Figure 1.6: Three examples of intrinsic energy EI=g(ψ). Take into account that in the first case the minimun is reached in only one point, in two in the second one, and in a infinite number of points in the last one. Therefore, only in the second and in the third cases topological soliton solutions could be possible. Usually, minima are set to zero, that is why gcan be defined as non negative [8]. Nevertheless, there is another possible configuration of the system, if it extends to ±∞ in space, where a solution whose density of energy does not decay to zero may be possible, leading to a wave that persists on time, a necessary condition for a soliton wave. Let us consider a solution of the field equation with an intrinsic energy which has at least two points where the minimum is reached (from now on, they will be known as vacuum states). Let us also suppose that at a given time, the solution is on a vacuum state in every position (not the same vacuum state for all the positions) but a finite region where the wave changes its values from one vacuum state to another one different. Under these conditions, the wave cannot disperse since a continuous change between the two vacuum states (which extend one to +∞and the other one to −∞) is mandatory. That is why the existence of at least two vacuum states is necessary for the existence of a soliton solution, because if there were only one vacuum state, every wave solution would disperse. In the third picture of figure 1.6 it appears the intrinsic energy characteristic of the sine-Gordon equation, and as one can see, it has infinite minima. Indeed, we will see that solitary wave solutions (in fact, soliton solutions) are possible, and we will 1.3. Topological solitons and kinks 11 study their stability properties. In addition, the wave that we have supposed in the previous paragraph would have a kinetic and a potential energies associated, and because of the sake of minimum energetic configurations of the system, the wave would change to an optimal shape and “constant speed”, leading to a soliton solution candidate (because it needs to verify the collision condition as well). In case that solutions would be a soliton, physicist would refer to it as a “topological soliton”, since it is a consequence of the geometry and topology of the problem. Looking the figure 1.7 may be clarifying as a global view of the qualitative concepts we have discussed. Figure 1.7: Different possible solutions to the field equation (sine-Gordon equation which we will analyse in a later section and that will be one of the motivating problems of the project), with their intrinsic energy associated. Topological solitons connect different vacuum states: ϕ= 0, ϕ= 180, ϕ= 360... [8]. As another result of the discussion, topological solitons (if they exist) of a given physical system will always connect different vacuum states of the intrinsic energy. This allows to define the concept of topological charge: Definition 1.2 Given a solution ψ(x, t)of a field equation, we can define the topological charge as Q= lim x→+∞ψ(x, t)−lim x→−∞ ψ(x, t). Take into account a couple of things. Firstly, as topological solitons connect two different vacuum states, topological charge is well defined and it is the difference between the values of the field in those vacuum states. Secondly, topological charge 12 Chapter 1. Nonlinear waves: solitons is different than zero for every topological soliton and is also time independent. Nevertheless, non topological solitons (as KdV solitons) have a null topological charge, this definition makes the difference between those two types of solitons. Additionally, we want to comment that there is a third type of soliton, called envelop soliton (for instance, the one that appears as a solution of the nonlinear Schr¨odinger equation). However, the analysis of nontopological and envelop solitons is outside the current study. In order to finish this section, notice that connecting two vacuum states is a necessary condition for a topological soliton solution to exist, but it is not sufficient, even if the wave moves with constant shape and speed, because the collision property is needed as well. The solutions which connect different vacuum states are known as “kinks”, because they are like knots on an infinite rope, they cannot be removed. Notice that Definition 1.2 can also be applied to kinks, since the collision property is not needed in order to define the topological charge. In fact, that name is given when the wave travels in the direction of the bigger of the two vacuum states it connects (for that reason, it has a positive topological charge), whereas the term “antikink” is used when the opposite happens (and therefore, the wave has an opposite topological charge than the associated kink, i.e. same in absolute value but negative). When a kink and an antikink which connect the same states collide, in some cases (of course not when they are solitons) they can annihilate, leading to an additional property of the “kinks” which is quite similar to particles in physics, in the sense that they always have an antiparticle and if they collide an annihilation can be produced. We will analyse both, a topological soliton solution (sine-Gordon kink), and a non soliton kink solution (ϕ4kink, it does not verify collision property as far as it is known) whose stability study motivated the mathematical problems we have dealt with in this work. Chapter 2 Linear stability study of the nonlinear Klein-Gordon equation In this second chapter, we are going to discuss the problems which caused this report: the study of stability of the kink solutions for the sine-Gordon equation and ϕ4equation. For that purpose, we will show a method for obtaining solitary wave solutions (actually, kink solutions) of nonlinear Klein-Gordon equations. Then, under certain hypotheses, we will study their stability (the so-called linear stability); and finally, we will focus on the stability of the two particular cases we have just mentioned: sine-Gordon equation and ϕ4equation. As a consequence, we will obtain certain ordinary differential equations which will be generalized and studied, and it is the central point of this work. But before that, let us introduce the general equation where we will restrict our study in the next two sections, the nonlinear Klein-Gordon equation1 ∂2ϕ ∂x2(x, t)−∂2ϕ ∂t2(x, t) + ∂V ∂ϕ (ϕ(x, t)) = ∂2ϕ ∂x2−∂2ϕ ∂t2+∂V ∂ϕ (ϕ) = 0.(2.1) In the next section, we will obtain a particular case of (2.1), the sine-Gordon equation, for a classical mechanical system, and at the end of the section, we will discuss the importance of that equation. 2.1 The sine-Gordon equation Sine-Gordon equation is the result of considering V(ϕ) = 1 + cos ϕin (2.1) ∂2ϕ ∂x2−∂2ϕ ∂t2= sin ϕ. (2.2) As we have commented, using a certain method (the pseudopotential method) 1See Appendix A for a brief section about the linear version of this equation. 13 14 Chapter 2. Linear stability of the nonlinear Klein-Gordon equation we can obtain kink solutions and even more in this case: soliton solutions (and therefore, topological solitons). Before doing that, and for the sake of completeness, we will show how the sine-Gordon equation (2.2) can model a classical mechanical system. Let us consider a chain of coupled pendula connected by torsional springs on a common axis (see [3]) as we can see in the following figure: Figure 2.1: Chain of coupled pendula joint by torsional springs on a common axis [3]. We denote by θn(t) (observe that it is time-dependent) the angular rotation of the nth pendulum with respect its equilibrium position, the vertical axis. In classical mechanics, a variational principle is supposed, which states that a magnitude called “action”, a functional of all possible trajectories (i.e. the different possible timedependent functions involved in the mechanical problem), must be minimum for the solution (stationary action principle). There are several formulations of classical mechanics, but we are going to make use of the Hamiltonian one. Hamiltonian mechanics is a reformulation from Lagrangian mechanics, and both are equivalent to Newton’s laws of motion. In Hamiltonian mechanics, a classical physical system is described by some “canonical” coordinates, in this case (θn, ωn= ∂θn ∂t )nwith n= 1, ..., N, and a function of the coordinates and time over a trajectory H(θn(t), ωn(t), t), which represents the energy of the system in many cases. In this particular system, that function turns out to be H(θn(t), ωn(t), t) = N−1 X n=1 I 2ω2 n+C 2(θn+1 −θn)2+mgl(1 −cos θn)+I 2ω2 N+mgl(1 −cos θN),(2.3) where Iis the moment of inertia of an individual pendulum respect the common axis, which we will denote as xaxis, Cis the torsion constant of the torsional springs, 2.1. The sine-Gordon equation 15 lis the length of the pendula, mthe mass of a pendulum and gthe gravitational acceleration. The first term of the sum is the classical kinetic energy associated to the rotation motion of a pendulum, the second one makes reference to the coupling energy of two pendula connected by a spring (notice that if two consecutive pendula rotate the same angle, the spring stays in its equilibrium position, so the contribution to the energy is zero), and the last contribution is from the gravitational potential energy of an individual pendulum. The solution of this mechanical system is given by the solution of the Hamilton equations ∂H ∂ωn =Iωn,∂H ∂θn =−I∂ωn ∂t .(2.4) The first equation is a tautology, but the second one, for n6∈ {1, N}leads to ∂H ∂θn =−I∂ωn ∂t ⇒C(2θn−θn−1−θn+1) + mgl sin θn=−I∂2θn ∂t2 ⇒I∂2θn ∂t2−C(θn+1 +θn−1−2θn) + mgl sin θn= 0.(2.5) The solution to this coupled partial differential system would provide us the variation of the coordinates in time for each pendulum, the solution to the problem. Notice that, if we suppose that the chain is infinite, every pendulum coordinate will satisfy equation (2.5). Now we will obtain the sine-Gordon equation from equation (2.5), making certain hypotheses. Assume that every couple of pendula is separated by a fixed distance a that is small enough (in our case, this means that the terms with a4can be neglected in comparison with those with a2). Suppose as well that there exists a function of space and time θ∈C3(R2)∩L∞(R2) which verifies that θ(x=na, t) = θn(t), ∀n∈Z(the last condition is equivalent to supposing that the chain is infinite in both directions of the xaxis). Then, equation (2.5) becomes the following one: I∂2θ ∂t2(x, t)−C(θ(x+a, t) + θ(x−a, t)−2θ(x, t)) + mgl sin(θ(x, t)) = 0.(2.6) In fact, solving (2.6) would provide us the solution to our problem just by substituting the solution on x=na. Due to our hypothesis about the distance a, we will be able to neglect a4terms in comparison with a2ones. If we make a Taylor expansion of θ(x+a, t) and θ(x−a, t) respect xvariable, then θ(x+a, t) = θ(x, t) + a∂θ ∂x(x, t) + a2 2 ∂2θ ∂x2(x, t) + a3 6 ∂3θ ∂x3(x, t) + Oa4, 22 Chapter 2. Linear stability of the nonlinear Klein-Gordon equation The static solution ϕ0(x)is stable if, for all  > 0there exists δ > 0such that for all ˜ϕ(x, t)solution of the partial differential equation which verifies k∆(x, 0)kL∞(Ω) = k˜ϕ(x, 0) −ϕ0(x)kL∞(Ω) < δ, then k∆(x, t)kL∞(Ω) =k˜ϕ(x, t)−ϕ0(x)kL∞(Ω) < , for all t > 0. Furthermore, in case that ϕ0(x)does not verify the previous condition, we will say that ϕ0(x)is unstable. Notice that the condition is set for t > 0, that is why we will work in C2 ∞(R× [0,+∞)). Additionally, we will take Ω = R. In this project we are interested in the study of a different and weaker concept of stability: linear stability. Our aim will be to study the linear stability of the static kinks for certain nonlinear Klein-Gordon equations. In the next pages we will discuss this concept. Let us now focus our attention in the so-called linear stability [11, § 2.5.2]. The idea is to reach a linear equation that the function ∆(x, t) satisfies and then to obtain its bounded solutions (in case they exist). Therefore, to discuss the linear stability of the static solution ϕ0(x) of equation (2.10) we will linearize it around the kink static solution ϕ0(x). For our purpose, we will consider a “small” perturbation Ψ(x, t) to the static kink solution ϕ0(x)∈C2 ∞(R) ϕ(x, t) = ϕ0(x) + Ψ(x, t),(2.18) where Ψ(x, t)∈C2 ∞(R×[0,+∞)) and is a real number which is small in two ways: first it verifies that7kΨk∞ kϕ0k∞for all t > 0, and second ||  1 in the way that its powers greater that one are negligible in comparison with . In addition, we want (2.18) to satisfy8equation (2.10). So if we introduce the function ϕ(x, t) on the original equation (2.10) we obtain 0 = ∂2ϕ0 ∂x2+∂2Ψ ∂x2−∂2ϕ0 ∂t2−∂2Ψ ∂t2+∂V ∂ϕ (ϕ0+Ψ) = ∂2ϕ0 ∂x2−∂2ϕ0 ∂t2+∂V ∂ϕ (ϕ0) + ∂2Ψ ∂x2−∂2Ψ ∂t2+∂2V ∂ϕ2(ϕ0)Ψ + O(2),(2.19) where a Taylor expansion of second degree has been applied to ∂V ∂ϕ (ϕ0+Ψ) around ϕ0. As we have mentioned, we are supposing that powers greater than one are negligible, and as a result, we obtain the following linear wave equation with a source term for Ψ, due to the fact that the first three terms in (2.19) cancel because ϕ0is 7From this point on, when we write k.k∞we will refer to the norm k.kL∞(R) 8Equation (2.10) is Lorentz invariant as well, that is why (2.18) must satisfy that equation even when we make a Lorentz transformation to the wave frame of reference x0=x−vt √1−v2,t0=t−vx √1−v2 (in the natural system of units, where the speed of light c= 1). We maintain the same notation for simplicity. Again, more details are given in Appendix A. 2.3. Linear stability of kink solutions 23 a solution of equation (2.10) ∂2Ψ ∂x2−∂2Ψ ∂t2+V00(ϕ0)Ψ = 0,(2.20) where the prime denotes the derivative of a one variable function with respect to its variable. In order to solve equation (2.20), we are going to use the following ansatz Ψ(x, t) = c1eiωt +c2e−iωtψ(x),(2.21) where ω2∈R, and c1and c2are complex constants9. As a consequence of introducing (2.21) in equation (2.20), ψ(x) is a solution of the following Sturm-Liouville problem which will provide us the possible values10 of ω: ψ00 +ω2+V00(ϕ0)ψ= 0.(2.22) Now, we are in condition for defining a suitable, for our purposes, concept of linear stability: Definition 2.3 The static solution ϕ0(x)∈C2 ∞(Ω) of equation (2.10)is linearly stable in Ωif there exists a set of functions Ψ(x, t)∈C2 ∞(Ω ×[0,+∞)) solution of equation (2.20)of the form (2.21), where ψ(x)and ωare the solution of the Sturm- Liouville problem (2.22). Moreover, in the case Ω = R, we will say that ϕ0(x)is linearly stable. Again, we will work with Ω = R. Notice that in the case when =ω6= 0, the function (2.21) blows up exponentially when t→ ∞, so our linear stability definition will demand ω∈Rsince Ψ(x, t)∈L∞(Ω ×(0,+∞)) is necessary. Similarly, if ψ(x) is not a bounded function in its domain, then it will be impossible to guarantee that the term Ψ in (2.18) really constitutes a small perturbation. For this reason in the Literature (see e.g. [9, § 5]) is said that the static solution is stable for the perturbation if the function ψ(x), that is the solution of the Sturm-Liouville problem (2.22), is bounded, and also its first derivative. This is the reason of Definition 2.3. Take also into account that, if there exists M > 0 such that kΨ(x, t)kL∞(Ω) ≤M for all t≥0, then kϕ(x, t)−ϕ0(x)kL∞(Ω) ≤M for all t≥0, so in some sense the Definition 2.3 is similar to the standard one, Definition 2.2. The linear stability is 9Although the arrival of the solutions we want to obtain is R, we will work with functions whose arrival is Cbecause it will simplify some calculations. Once we have the general solution, either imposing the initial conditions or taking real or imaginary parts, the arrival of that solution will be R. We will repeat this artifice several times in this work, but keep in mind that it is an equivalent treatment which simplifies the different steps we will follow. Equation (2.21) is an example of it. 10The final Sturm-Liouville problem we will study will be formalised later on. 24 Chapter 2. Linear stability of the nonlinear Klein-Gordon equation much weaker, because we do not demand it for all the solutions of (2.19) within a given margin (see Definition 2.2), but for the existence of a function ϕ(x, t) = ϕ0(x) + Ψ(x, t) such that Ψ is a solution of (2.20) of the form (2.21) and it verifies kϕ(x, t)−ϕ0(x)kL∞(Ω) ≤M for all t≥0. Now, we will make an additional assumption lim x→+∞V00(ϕ0(x)) = lim x→−∞ V00(ϕ0(x)) = c, (2.23) where cis a real constant. Then, we look for solutions of the form11 ψ(x) = eikx F(x),(2.25) where k2=ω2+c. Introducing (2.25) in equation (2.22), F(x) satisfies the next second order differential equation: d2F dx2(x)+2i k dF dx (x)+[V00(ϕ0(x)) −c]F(x) = 0.(2.26) Finally, we can do a change of variable s= tanh(x). Applying the chain rule ∂ ∂x =∂ ∂s ∂tanh(x) ∂x = (1 −tanh2(x)) ∂ ∂s = (1 −s2)∂ ∂s, and introducing it on (2.26), we obtain (1−s2)d2F ds2(s)−2(s−i k)dF ds (s)+ 1 1−s2[V00(ϕ0[arctanh(s)])−c]F(s) = 0,(2.27) which is the ordinary differential equation that will play an important role in this work. Summing up, once we have solved the previous equation, the perturbation Ψ will verify Ψ(x, t) = c1eiωt +c2e−iωteikxF(tanh(x)),(2.28) where k2=ω2+cand limx→±∞ V00(ϕ0(x)) = c. 11An usual way to try to solve equations similar to (2.22) in Physics, is to analyse what happens to the equation for large values of x(associated “asymptotic” equation), and then look for solutions of the original equation as product of the “asymptotic” solution and an unknown function that needs to be determined. We have done a similar procedure. In (2.22), if we take into account the condition (2.23), for high enough values of |x|we have that ψ00 + [ω2+c]ψ+h(x)ψ= 0,(2.24) where h(x) = V00(ϕ0(x)) −c. If khψk∞would be negligible in comparison with the norm of the other terms of the equation (2.24), and we would neglect the khψk∞term, the solution of the equation would be ψ(x) = e±ikx (with k2=ω2+c). Under the hypothesis (2.23), the “asymptotic solution” of (2.22) would be the previous one. This is why we are going to look for solutions of the form showed in (2.25). Notice that we can make this regardless of whether khψk∞is negligible or not. At first, we only consider one of the solutions e±ikx, we will discuss this with more details in Chapter 3. 2.4. Some examples 25 Keep in mind that (2.27) and (2.28) give us the bounded linear perturbations of a kink solution of the nonlinear Klein-Gordon equation (obtained by the pseudopotential method) under certain approaches we have done. In the next section, we will study the particular cases of sine-Gordon and ϕ4equations, where we will apply the pseudopotential method, for obtaining the kink solutions, and we will particularize the linear stability study as well. 2.4 Some examples 2.4.1 Sine-Gordon equation: soliton solutions and their stability In this section we want to calculate two kink solutions (kink and antikink12) of the sine-Gordon equation (2.2), provided by the pseudopotential method, and then to obtain their corresponding equation coming from (2.27). In this case V(ϕ) = 1 + cos ϕ, and all their local maxima verify V(ϕ0) = 2. If we introduce it on (2.16) 2 = 1 2∂ϕ ∂z 2 + 1 + cos ϕ⇒∂ϕ ∂z =±p2(1 −cos ϕ),(2.29) where integrating and imposing that ϕ(z)∈C2 ∞(R), which is equivalent to impose to the solution the initial condition ϕ(0) ∈(2nπ, 2(n+ 1)π), because we know that it must connect ϕ= 2nπ and ϕ= 2(n+ 1)π, and choosing ϕ(0) ∈(2n+ 1)π, we get ϕ(z) = 4 arctan[exp(±z)] + 2πn , n ∈Z,(2.30) which are kink and antikink solutions13 (+ and −signs, respectively) of the sine- Gordon equation. Even more, it can be shown that after a collision where two of these solutions are involved, they emerge unaffected in their mathematical expression (alternatively, in their shape and speed): they are soliton solutions, according to Definition 1.1. The kinks connect the stationary solutions ϕ= 2nπ and ϕ= 2(n+1)π. In figure 2.3 one can observe representations for n= 0 of these solutions and of the pseudopotential. In addition, we can calculate their topological charge (see Definition 1.2). For 12In fact, we will get infinite solutions, but every kink (antikink) solution is related with other kink (antikink) by a translation of 2nπ. 13If we go to the original variables, the solutions are ϕ(x, t) = 4 arctan exp ±x−vt √1−v2+ 2πn , n ∈Z. 26 Chapter 2. Linear stability of the nonlinear Klein-Gordon equation the kink solutions Q= lim x→+∞ϕ(x, t)−lim x→−∞ ϕ(x, t) = 2π, whereas antikink solutions, as we commented at the end of the Section 1.3, have the opposite topological charge, Q=−2π. (a) Kink (blue) and antikink (red) solutions (equation (2.30) for n= 0). They connect the stationary solutions ϕ= 0 and ϕ= 2π. (b) For n= 0, kink and antinkink solutions connect the stationary solutions ϕ= 0 and ϕ= 2π, which correspond with the red dots, maxima of the pseudopotential. Figure 2.3: Kink and antikink solutions of sine-Gordon equation. Now, we can analyse what happens with the stability of these kink solutions. Remember that (2.27) gives us the unknown function Ffrom the linear stability perturbation Ψ of a kink solution (1 −s2)d2F ds2(s)−2(s−i k)dF ds (s) + 1 1−s2[V00(ϕ0[arctanh(s)]) −c]F(s)=0, where limx→±∞ V00(ϕ0(x)) = c,k2=ω2+c,ωis an undetermined constant (Sturm- Liouville problem), and ϕ0is the kink solution (after performing the corresponding Lorentz transformation to its frame of reference) whose stability is being studied. 2.4. Some examples 27 Thus, we need to calculate cand V00(ϕ0[arctanh(s)]). We start with c c= lim x→±∞ V00(ϕ(x)) = lim x→±∞ −cos(ϕ(x)) = lim x→±∞ −cos(4 arctan[exp(±x)] + 2πn) = lim x→±∞ −cos(4 arctan[exp(±x)]) = −1.(2.31) Notice that (2.23) holds. In order to calculate V00(ϕ0[arctanh(s)]), we use (2.29), that yields 2 = 1 2∂ϕ ∂z 2 + 1 + cos ϕ⇒cos ϕ= 1 −1 2∂ϕ ∂z 2 .(2.32) On the other hand, taking derivatives of the kink and antikink solutions ϕ(z) = 4 arctan[exp(±z)] + 2πn ⇒ ∂ϕ ∂z = 4 ±1 1 + e±2ze±z=±4 ez+e−z=±2 cosh z.(2.33) Now, if we introduce (2.33) in (2.32), and we make the change of variable s= tanh z, we get cos (ϕ(arctanh(s))) = cos (ϕ(z)) = 1 −1 2±2 cosh z2 = 1−2 cosh2z= 1 −2(1 −tanh2z) = 1 −2(1 −s2).(2.34) Therefore, V00(ϕ(arctanh(s))) = −cos (ϕ(arctanh(s))) = 2(1 −s2)−1.(2.35) Finally, introducing (2.31) and (2.35) in (2.27), we obtain (1 −s2)d2F ds2(s)−2(s−i k)dF ds (s) + 1 1−s2[V00(ϕ0[arctanh(s)]) −c]F(s)=0 ⇒(1 −s2)d2F ds2(s)−2(s−i k)dF ds (s) + 1 1−s2[2(1 −s2)−1 + 1]F(s)=0 ⇒(1 −s2)d2F ds2(s)−2(s−i k)dF ds (s)+2F(s) = 0.(2.36) We just wanted to add that a particular perturbation, solution of the problem, is obtained for ω= 0 (k=±√ω2+c=±i), known as Goldstone mode, whose solutions of (2.36) are14 F(s) = s+ 1 (k=i), F(s) = s−1 (k=−i). 14As we will see in the next chapter, these two solutions are linearly dependent. 28 Chapter 2. Linear stability of the nonlinear Klein-Gordon equation 2.4.2 ϕ4equation: kink solutions and their stability In this section we will do a similar study to that performed in the previous section, but with ϕ4equation: we will obtain kink (not soliton) solutions from the pseudopotential method, as well as the corresponding equation coming from (2.27). In other words, we will repeat the scheme of the previous section, but considering the pseudopotential V(ϕ) = −1 2(ϕ2−1)2in the nonlinear Klein-Gordon equation (2.10) ∂2ϕ ∂x2−∂2ϕ ∂t2= 2ϕϕ2−1.(2.37) Again, we will apply the pseudopotential method in order to get kink and antikink solutions. The pseudopotential V(ϕ) = −1 2(ϕ2−1)2has only two maxima, for ϕ0=±1, where V(ϕ0) = 0. Thus, if we use (2.16) 0 = 1 2∂ϕ ∂z 2 −1 2ϕ2−12⇒∂ϕ ∂z =±(ϕ2−1),(2.38) which is a Riccati’s equation, and whose general solution is ϕ(z) = ±1−e2(C+z) e2(C+z)+ 1, where Cis an undetermined constant which requires an initial condition. But due to the fact that our kink solution must connect ϕ=−1 and ϕ= 1, we have to impose ϕ(z0)∈(−1,1), for an arbitrary z0∈R. This condition also assures that ϕ(z)∈L∞(R). For the seek of simplicity, we impose that ϕ(z0= 0) = 0, and we get the solutions ϕ(z) = ±tanh z. (2.39) In this case we only have two solutions, a kink (+ sign) and its corresponding antikink (−sign), because the pseudopotential only has two maxima likely to be connected by a kink solution. As far as it is known, they are not solitons, due to the fact that after a collision they may be distorted (see [7] where a broader soliton definition is given and where this property is not required). In this case, the topological charge can be easily calculated either, obtaining for the kink Q= lim x→+∞ϕ(x, t)−lim x→−∞ ϕ(x, t) = 2, and, as a consequence, for the antikink Q=−2. As we did in the previous section, now we want to obtain the equation (2.27) for the kink solutions of this case. Remember (2.27): (1 −s2)d2F ds2(s)−2(s−i k)dF ds (s) + 1 1−s2[V00(ϕ0[arctanh(s)]) −c]F(s)=0, 2.4. Some examples 29 (a) Kink (blue) and antikink (red) solutions. They connect the stationary solutions ϕ=−1 and ϕ= 1. (b) Kink and antinkink solutions connect the stationary solutions ϕ= −1 and ϕ= 1, which correspond with the red dots, maxima of the pseudopotential. Figure 2.4: Kink and antikink solutions of ϕ4equation (equation (2.39)). where again limx→±∞ V00(ϕ0(x)) = c,k2=ω2+c,ωis an undetermined constant (Sturm-Liouville problem), and ϕ0is the kink solution (after performing the corresponding Lorentz transformation to its frame of reference) given by (2.39). In this case, cand V00(ϕ0[arctanh(s)]) are c= lim x→±∞ V00(ϕ(x)) = lim x→±∞ 2−6 (ϕ(x))2=−4,(2.40) V00(ϕ(arctanh(s))) = 2 −6 (ϕ(arctanh(s)))2= 2−6 (±tanh(arctanh(s)))2= 2 −6s2.(2.41) Therefore, if we introduce (2.40) and (2.41) in (2.27) we obtain (1 −s2)d2F ds2(s)−2(s−i k)dF ds (s) + 1 1−s2[V00(ϕ0[arctanh(s)]) −c]F(s)=0 ⇒(1 −s2)d2F ds2(s)−2(s−i k)dF ds (s) + 1 1−s2[2 −6s2+ 4]F(s)=0 30 Chapter 2. Linear stability of the nonlinear Klein-Gordon equation ⇒(1 −s2)d2F ds2(s)−2(s−i k)dF ds (s)+6F(s) = 0,(2.42) Goldstone mode (remember, the case ω= 0, which implies k=±√ω2+c=±2i) can be easily obtained as well, and the functions solution of the previous equation associated to it are15 F(s) = (s+ 1)2(k= 2i), F(s) = (s−1)2(k=−2i). We hope that all the analysis we have just done has been motivating for the problem we will formulate in the next section. In the two cases we have just studied, the obtained equations (2.36) and (2.42) show a similar structure: (1 −s2)d2F ds2(s)−2(s−i k)dF ds (s)+2F(s)=0, (1 −s2)d2F ds2(s)−2(s−i k)dF ds (s)+6F(s)=0. Notice that, if we further suppose V00(ϕ0[arctanh(s)]) −c=l(l+ 1)(1 −s2), l ∈N, then the equation (2.27) becomes (1 −s2)d2F ds2−2(s−i k)dF ds +l(l+ 1) F(s)=0, l ∈N.(2.43) Observe that if we set l= 1 and l= 2 in (2.43) we recover the cases of sine-Gordon and ϕ4, respectively. The equation (2.43) will be key for solving our stability problem as we will see in the next chapter. 15Again, we will see in the next chapter their linear dependence. Chapter 3 The stability problem for a family of nonlinear Klein-Gordon equations In this chapter, we are going to formulate and solve the stability problem for a given family of potentials from the nonlinear Klein-Gordon equation. With that purpose, and in order that this chapter is self-contained, first we will recall what we did in Section 2.3. Our aim was studying the stability of kink solutions of nonlinear Klein-Gordon equation. We introduced a general stability definition (Definition 2.2), but we mentioned that we were going to work with a weaker concept of stability: linear stability (given in Definition 2.3). Our starting point was the nonlinear Klein-Gordon equation ∂2ϕ ∂x2(x, t)−∂2ϕ ∂t2(x, t) + ∂V ∂ϕ (ϕ(x, t)) = ∂2ϕ ∂x2−∂2ϕ ∂t2+∂V ∂ϕ (ϕ) = 0.(3.1) We supposed that the static kink solution of (3.1), ϕ0(x), was known and we considered a small perturbation to the static kink solution ϕ0(x)∈C2 ∞(R) ϕ(x, t) = ϕ0(x) + Ψ(x, t),(3.2) where Ψ(x, t)∈C2 ∞(R×[0,+∞)) and was a real number, small in two ways: first it verified that kΨk∞ kϕ0k∞, and second, ||  1 in the way that its powers greater that one were negligible in comparison with . We wanted (3.2) to satisfy (3.1), what, at first order in , leads to ∂2Ψ ∂x2−∂2Ψ ∂t2+V00(ϕ0)Ψ = 0,(3.3) where the prime denotes the derivative of a one variable function respect its variable. 31 38 Chapter 3. The stability problem for a family of NLKG equations Proof: To know if the solutions given in (3.22) are linearly independent we can use the results of the previous section. We can compute de Wronskian of both solutions W[ψk(x), ψ−k(x)] = det  ψk(x)ψ−k(x) ψ0 k(x)ψ0 −k(x) = det eikxP(−ik,ik) l(s)e−ikxP(ik,−ik) l(s) eikx ikP (−ik,ik) l(s) + 1 cosh2(x)P(−ik,ik) l0 (s)e−ikx −ikP (ik,−ik) l(s) + 1 cosh2(x)P(ik,−ik) l0 (s) , where s= tanh x. From above W[ψk(x), ψ−k(x)] = −2ikP (−ik,ik) l(s)P(ik,−ik) l(s) + 1 cosh2(x)W[P(−ik,ik) l(s), P(ik,−ik) l(s)]. Taking the limit x→ ∞, and taking into account that s= tanh x→1, cosh x→ ∞, that P(−ν,ν) lis a polynomial on the bounded interval [−1,1], and therefore bounded; and the values of P(−ν,ν) l(1) given by (3.17), we obtain that lim x→+∞W[ψk(x), ψ−k(x)] = −2ikP(−ik,ik) l(1)P(ik,−ik) l(1) = −2ik (l!)2 l Y m=1 (k2+m2)6= 0, thus by Corollary 3.6 they are linearly independent.  Due to Theorem 3.3, the general solution of (3.4) is a linear combination of the solutions given in (3.22). Notice that we can assume that k > 0, since ψk(x) transforms into ψ−k(x) if k→ −k. Take into account that the solutions belong to C2 ∞(R), because the exponential is bounded and8C∞(R), the Jacobi polynomials are bounded and C∞((−1,1)), and tanh(x) is also C∞(R), so ψkis bounded (it belongs to C∞(R)∩L∞(R)). Similarly, as its first derivative is a combination of the same polynomial and its first derivative, the same exponential and 1/cosh2(x) is also C∞(R), the first derivative of the solutions is also bounded. As a consequence, the static kink solution is linearly stable according to our Definition 2.3. Moreover, we have found that for every k∈Rthere exist two independent bounded solutions of (3.3), the linear perturbation of the kink solution of the nonlinear Klein-Gordon equation, associated to the same ω. Each kis linked to a ω value by ω2=k2−c, so for every ω∈R\(−√−c, √−c) we have bounded solutions of the original equation, which are a continuous and infinite spectrum of solutions in this case. 8A function belongs to C∞(Ω) if it belongs to Cn(Ω) for all n∈N. 3.2. Boundedness of the solutions of the Sturm-Liouville problem 39 3.2.2 Case II: νreal Now, let us consider the case ν∈R, or equivalently, k=iκ,κ∈R\{0}. The necessary condition −c≥ν2is not trivially verified since ν2≥0. So, this condition restricts the νspectrum to ν∈[−√−c, √−c]\{0}(the same spectrum holds for κ). As a consequence, the spectrum of ωlikely to be part of the solution in this case is the complementary of Case I: ω∈(−√−c, √−c). Just for the spectrums of these two cases, the time dependent part of the solution associated to the Sturm-Liouville problem (3.4) is bounded. As in the previous section, we must require eikxFk(tanh(x)) ∈C2 ∞(R), where (k, Fk(s)) is a solution of equation (3.13), and the spectrum κ∈(−√−c, √−c) might be reduced. As before, we are going to look for the solutions given by Jacobi polynomials P(−ν,ν) l(s) = P(κ,−κ) l(s). Now, we only have a solution of (3.4) given by Jacobi polynomials for each κ∈[−√−c, √−c]\{0}: ψκ(x) = e−κxP(κ,−κ) l(tanh x).(3.23) We need to discuss the boundedness of these functions and their first derivatives in order to find solutions for our problem, since again we have that they belong to C∞(R). First, we will show a general result. Proposition 3.9 (Necessary condition for boundedness) Consider the function ψκgiven in (3.23). If it is bounded, then κ=±1,±2,...,±l. Proof: We will consider κ > 0, the case κ < 0 is analogous. If the function e−κxP(κ,−κ) l(tanh x)∈C2 ∞(R), then it belongs to L∞(R). As a consequence, it must happen lim x→−∞ e−κxP(κ,−κ) l(tanh x)∈R. But due to the fact that limx→−∞ e−κx = +∞, it must be verified 0 = lim x→−∞ P(κ,−κ) l(tanh x) = P(κ,−κ) l(−1). If we make use of the second equation of (3.17) with ν=−κ, then κ= 1,2, . . . , l needs to be verified. Proceeding in a similar way, the case κ < 0 provides the necessary condition κ=−1,−2,...,−l. A consequence of the proposition is that this case cannot provide a continuous spectrum: it is discrete. We will refer to each solution from this case as a “discrete mode”. Now, we will prove that the solutions corresponding to κ=±1,±2,...,±land their first derivatives are bounded, so the spatial part of the corresponding pertur- 40 Chapter 3. The stability problem for a family of NLKG equations bation is bounded9. Notice that since the functions are continuous as in the previous case, it would be enough to prove that their limits to ±∞ exist and are finite. Theorem 3.10 The function given in (3.23)for κ=±1,±2,...,±land its first derivative are bounded, i.e. ψκ(x),(ψκ)0(x)∈L∞(R), and therefore ψκ(x)belongs to C2 ∞(R). Moreover, it verifies that lim x→±∞ ψκ(x) = lim x→±∞ e−κxP(κ,−κ) l(tanh x)=0, lim x→±∞ (ψκ)0(x) = lim x→±∞ −κψκ(x) + e−κx 1 cosh2xP(κ,−κ) l0(tanh x)= 0. Proof: Notice that proving the limits given in the theorem, then the function and its first derivative are bounded and we have the result, since the function belongs to C2 ∞(R). Again, we will consider the case κ > 0. The case κ < 0 is similar and a bit easier since it is not necessary the use of the symmetry property (3.18). For κ > 0 is trivial that the limit to +∞is zero. Consider the first limit lim x→−∞ e−κxP(κ,−κ) l(tanh x) First we make the change x→ −x(we maintain the same notation for simplicity) and use the symmetry property of the Jacobi polynomials (3.18), that allows us to write the limit as follows lim x→∞(−1)leκxP(−κ,κ) l(tanh x). Next we set t=e2x. As a consequence, s= tanh x=t−1 t+1 , so that when x→ ∞ then t→+∞. Then, using the explicit expression of the Jacobi polynomials (3.16), which reads as P(−κ,κ) l(s) = 1 l! l X m=κ (−l)m(l+ 1)m(1 −κ+m)l−m m!1 t+ 1m , we find lim t→+∞(−1)ltκ/2P(−κ,κ) l(s) = lim t→+∞ (−1)l l! l X m=κ (−l)m(l+ 1)m(1 −κ+m)l−m m! tκ/2m t+ 1 !m = 0, since, m≥κ, and therefore κ/2m≤1/2. Now, for the second limit in Theorem 3.10, we need to prove lim x→−∞ (ψκ)0(x) = lim x→−∞ −κψκ(x) + e−κx 1 cosh2xP(κ,−κ) l0(tanh x) = 9Of course, we must additionally require the boundedness of the time dependent part that, as we commented, in this case forces to κ∈[−√−c, √−c]\{0}. 3.2. Boundedness of the solutions of the Sturm-Liouville problem 41 lim x→−∞ e−κx 1 cosh2xP(κ,−κ) l0(tanh x) = 0. Firstly, applying the change of variable s=−tanh x, and secondly making use of the identity e2x=1+tanh x 1−tanh x, as well as the symmetry property of Jacobi polynomials (3.18), then one obtains lim x→−∞ e−κx 1 cosh2xP(κ,−κ) l0(tanh x) = lim s→11 + s 1−sκ 2 (1 −s2) (−1)l l! l X m=κ (−l)m(l+ 1)m(1 −κ+m)l−m m!1−s 2m!0 = lim s→1−(1 + s)κ 2+1 (1 −s)κ 2−1 (−1)l l! l X m=κ (−l)m(l+ 1)m(1 −κ+m)l−m (m−1)!2m(1 −s)m−1 = lim s→1−(1 + s)κ 2+1 (−1)l l! l X m=κ (−l)m(l+ 1)m(1 −κ+m)l−m (m−1)!2m(1 −s)m−κ 2= 0, since m≥κ≥1, we have that m−κ 2≥m 2≥1 2, and we have proven the result.  From the Theorem 3.10 we conclude that ψκis a bounded solution of equation (3.13) for κ=±1,±2,...,±l, and so does its first derivative. Nevertheless, for each κthere is a second linearly independent solution of that equation that we do not know its analytic form neither its boundedness. In the following theorem we will prove that this solution does not belong to C2 ∞(R), and therefore is not a solution of our stability problem. Theorem 3.11 Consider the equation (3.4)and its solution ψκ(x)given in (3.22) for κ=±1,±2,...,±l. Then, any other linearly independent solution of (3.4)for the same value of ω=√−κ2−cverifies that either it is unbounded or its first derivative is unbounded. As a consequence, it does not belong neither to L∞ 1(R)nor to C2 ∞(R). Proof: Again, we will prove it for κ > 0, the other case is similar. We can prove it by contrapositive. Suppose that there exists a bounded solution χof the Sturm- Liouville problem (3.4) for the same eigenvalue, ω=√−κ2−c, and that its first derivative is also bounded. It is enough to prove that under these hypotheses, ψκ and χare linearly dependent. If we use Theorem 3.10 and the boundedness of χ and its first derivative, then lim x→−∞ W[ψκ(x), χ(x)] = lim x→−∞ ψκ(x)χ0(x)−(ψκ)0(x)χ(x)= 0. Therefore, by Proposition 3.7 (in our case P(x) = 0 in I=R), it follows that ψκ and χare linearly dependent. Thus, if such functions are linearly independent, then 42 Chapter 3. The stability problem for a family of NLKG equations χis unbounded or its first derivative is unbounded.  Remark: Notice that if we consider ψκ(x) for κ=±1,±2,...,±l, then for the corresponding eigenvalue of (3.4)ω=√−κ2−c,ψ−κ(x) is a solution as well, and it is bounded due to Theorem 3.10. Actually, Theorem 3.10 proves that both solutions are linearly dependent. One can easily check it, taking into account the Wronskian we computed in the proof of Theorem 3.8 and taking k=iκ: lim x→+∞W[ψκ(x), ψ−κ(x)] = 2κP(κ,−κ) l(1)P(−κ,κ) l(1) = 2κ (l!)2 l Y m=1 (m2−κ2)=0, where the first equation of (3.17) has been used. Proposition 3.7 assures their linearly dependence. Theorem 3.11 assures that the solution of (3.4) linearly independent of ψκ(x), for κ=±1,±2,...,±l, does not belong to C2 ∞(R), so in this case, we have only one (up to a constant factor) solution to our problem. Let us put together all the results we have obtained in this case. We considered the real νcase, which is equivalent to the case k=iκ with κ∈R\{0}. The boundedness of the time dependent part of the solution associated with problem (3.4) implied that κ∈[−√−c, √−c]\{0}and therefore ω∈(−√−c, √−c). On the other hand, the spatial part of the solution ψκ(x) = e−κxP(κ,−κ) l(tanh x) associated with problem (3.4) only belongs to C2 ∞(R) if κ=±1,±2,...,±l. Thus, the number of discrete modes depends on land c. Since ±κis associated with the same ω, at the most we will have ldiscrete modes with only one function associated to each one. But if √−c < l, we will only have b√−ccdiscrete modes. Therefore, for a general land c, in this second case we have n= min{l, b√−cc} solutions of the Problem 3.2, and those solutions are given by the pair (ω, ψκ), where ψκ(x) = e−κxP(κ,−κ) l(tanh x), ω=√−c−κ2and κ= 1,2, . . . , n. We have proven that there exists only one bounded solution (its first derivative is bounded as well) of (3.3), the linearization around the kink of the nonlinear Klein-Gordon equation, for each κ= 1,2, . . . , n. So, we have a bounded solution of our problem only for ω=√−c−κ2∈(−√−c, √−c) and κ= 1,2, . . . , n, in other words, a discrete spectrum of solutions, or discrete modes as we have refered to them. Notice the difference with Case I, where there exist two linearly independent solutions of our problem for every ω∈R\(−√−c, √−c), i.e. a continuous and infinite spectrum of solutions of Problem 3.2. As we will see soon, for the particular cases of sine-Gordon and ϕ4equations, there exist discrete modes, concretely one and two discrete modes, respectively. Notice it can happen that these solutions do not exist, what in fact occurs for 3.2. Boundedness of the solutions of the Sturm-Liouville problem 43 0≥c > −1. Moreover, we will see that some of the discrete modes of those two particular cases of our family of potentials, given by (3.11), are related with the so-called Goldstone modes we introduced in Section 2.4. In the next theorem we collect the main results of our work: Theorem 3.12 Let be the nonlinear Klein-Gordon equation ∂2ϕ ∂x2−∂2ϕ ∂t2+∂V ∂ϕ (ϕ)=0,(3.24) and assume that it has a static solution ϕ0(x). Assume also that lim x→±∞ V00(ϕ0(x)) = c≤0, V 00(ϕ0[arctanh(s)])−c=l(l+1)(1−s2), l ∈N.(3.25) Consider the small perturbations of the solutions of (3.24)in the sense we discussed ϕ(x, t) = ϕ0(x) + Ψ(x, t),(3.26) where Ψ(x, t) = (c1e−iωt +c2eiωt)ψ(x).(3.27) Then, the static solution ϕ0(x)is linearly stable in the sense of Definition 2.3, i.e. the associated Sturm-Liouville problem ψ00 +ω2+V00(ϕ0)ψ= 0,(3.28) has a set of eigenfunctions that belong to L∞ 1(R)and such that ω∈R. In fact, we have the following situations:  If ω∈R\(−√−c, √−c), then, the real and imaginary parts of ψk(x) = eikxP(−ik,ik) l(tanh x), k ∈R,(3.29) constitute the basis of solutions for each value ω2=k2−c > 0. In this case it is said that the Sturm-Liouville problem has a continuous spectrum of solutions ω∈R\(−√−c, √−c)where for each ωthere are two linearly independent solutions of the differential equation (3.28).  If ω∈(−√−c, √−c), then there exists only a discrete set of linearly independent functions solution of our problem ψκ(x) = eκxP(−κ,κ) l(tanh x), κ = 1,2,...,min{l, b√−cc}.(3.30) In this case it is said that the Sturm-Liouville problem has a discrete spectrum ω2 κ=−κ2−c,κ= 1,2, ..., min{l, b√−cc}, with just one solution of the 44 Chapter 3. The stability problem for a family of NLKG equations differential equation (3.28)associated to each value. For ω= 0, if the solution exists it is known as Goldstone mode. Now we will apply this theorem to the two particular cases we have studied: sine-Gordon equation ans ϕ4equation. Corollary 3.13 (Particular case l= 1 and c=−1: sine-Gordon equation) Let be the sine-Gordon equation ∂2ϕ ∂x2−∂2ϕ ∂t2= sin ϕ, (3.31) and consider the following static solutions ϕ0(x) = 4 arctan[exp(±x)] + 2πn , n ∈Z.(3.32) Then, the static solutions ϕ0(x)are linearly stable in the sense of Definition 2.3. In other words, the associated Sturm-Liouville problem ψ00 +ω2−cos(ϕ0)ψ= 0,(3.33) has a set of eigenfunctions that belong to L∞ 1(R)and such that ω∈R. In fact, we have the following situations:  If ω∈R\(−1,1), then, the real and imaginary parts of ψk(x) = eikxP(−ik,ik) 1(tanh x) = eikx(tanh x−ik),(3.34) where k∈R, constitute the basis of solutions for each value ω2=k2+ 1 >0. Those solutions are, for k≥0 ψk,1(x) = cos kx tanh x+ksin kx, ψk,2(x) = sin kx tanh x−kcos kx. (3.35) Therefore, the Sturm-Liouville problem has a continuous spectrum of solutions ω∈R\(−1,1) where for each ωthere are two linearly independent solutions of the differential equation (3.33).  For ω∈(−1,1), in the discrete spectrum there exists only one function solution of our problem, for ω= 0, the Goldstone mode ψ1(x) = exP(−1,1) 1(tanh x) = ex(tanh x−1).(3.36) Notice that, as we remarked in the previous section, ψ1(x) = ex(tanh x−1) and ψ−1(x) = e−x(tanh x+1) are linearly dependent solutions of (3.33) for the same value 3.2. Boundedness of the solutions of the Sturm-Liouville problem 45 ω= 0, and that in fact they are the Goldstone modes ((s±1)e∓x, with s= tanh x) that we introduced in Section 2.4.1. Therefore, they are linearly dependent, as we advanced. In the following figures we show the discrete mode and the solutions of the continuous spectrum for k= 1 (i.e. ω=±√2). Notice the difference between the behaviour of the solutions. The discrete mode does not oscillate and is spatially localized (figure 3.1), in contrast to the solutions of the continuous spectrum. The real part of (3.34) (it is, the first solution of (3.35), figure 3.2) is odd, while the imaginary part (the second solution of (3.35), figure 3.3) is even. Additionally, the Goldstone mode is even. All the solution are, naturally, bounded. Figure 3.1: Golstone mode (κ= 1, equation (3.36)) of the kink and antikink solutions of sine-Gordon equation. Figure 3.2: Real part of the solution (3.34) for k= 1. Bounded linear perturbation of the kink and antikink solutions of sine-Gordon equation. 46 Chapter 3. The stability problem for a family of NLKG equations Figure 3.3: Imaginary part of the solution (3.34) for k= 1. Bounded linear perturbation of the kink and antikink solutions of sine-Gordon equation. Corollary 3.14 (Particular case l= 2 and c=−4:ϕ4equation) Let be the ϕ4 equation ∂2ϕ ∂x2−∂2ϕ ∂t2= 2ϕ(ϕ2−1),(3.37) and consider the following static solutions ϕ0(x) = ±tanh x. (3.38) Then, the static solutions ϕ0(x)are linearly stable in the sense of Definition 2.3. In other words, the associated Sturm-Liouville problem ψ00 +ω2−6(ϕ0)2+ 2ψ= 0,(3.39) has a set of eigenfunctions that belong to L∞ 1(R)and such that ω∈R. In fact, we have the following situations:  If ω∈R\(−2,2), then, the real and imaginary parts of ψk(x) = eikxP(−ik,ik) 2(tanh x) = 1 2eikx(3 tanh2x−3ik tanh x−k2−1),(3.40) where k∈R, constitute the basis of solutions for each value ω2=k2+ 2 >0. Those solutions are, for k≥0 ψk,1(x) =1 2cos kx(3 tanh2x−k2−1) + 3 2ksin kx tanh x, ψk,2(x) =1 2sin kx(3 tanh2x−k2−1) −3 2kcos kx tanh x. (3.41) Therefore, the Sturm-Liouville problem has a continuous spectrum of solutions ω∈R\(−2,2) where for each ωthere are two linearly independent solutions 3.2. Boundedness of the solutions of the Sturm-Liouville problem 47 of the differential equation (3.39).  For ω∈(−2,2), in the discrete spectrum there exist only two solutions of our problem ψ1(x) =exP(−1,1) 2(tanh x) = 3 2extanh x(tanh x−1), ψ2(x) =e2xP(−2,2) 2(tanh x) = 3 2e2x(tanh x−1)2, (3.42) for ω=√3and ω= 0, respectively. Therefore, the second solution is the Goldstone mode. Notice again that, ψ2(x) = 3 2e2x(tanh x−1)2and ψ−2(x) = 3 2e−2x(tanh x+ 1)2 are linearly dependent solutions of (3.39) for the same value ω= 0, and that in fact they are proportional to the Goldstone modes ((s±1)2e∓2x, with s= tanh x) that we introduced in Section 2.4.2. Therefore, they are linearly dependent, as we advanced. In the following figures we show the discrete modes and the solutions of the continuous spectrum for k= 1 (i.e. ω=±√5). Again, notice the difference between the behaviour of the solutions. The discrete modes do not oscillate (figures 3.4 and 3.5), in contrast to the solutions of the continuous spectrum. The real part of (3.40) (it is, the first solution of (3.41), figure 3.6) is even, while the imaginary part (the second solution of (3.41), figure 3.7) is odd. The Goldstone mode is even and the other discrete mode is odd. All the solutions are, naturally, bounded. Figure 3.4: Discrete mode (κ= 1, first solution from equation (3.42)) of the kink and antikink solutions of ϕ4equation. 54 Appendix A. Klein-Gordon equation (A.1) and (A.2) we obtain, respectively i∂ϕ(~r, t) ∂t =−1 2m∇2ϕ(~r, t), i∂ϕ(~r, t) ∂t =−1 2m∇2ϕ(~r, t) + V(~r)ϕ(~r, t),(A.4) where ∇2=∂2 ∂x2+∂2 ∂y2+∂2 ∂z2. In other words, Schr¨odinger equation is obtained and ϕ(~r, t), the solution of (A.4), gives us information about the system. An analogous procedure can be made in order to obtain the relativistic quantum equation, from the relativistic energy equation E=pp2+m2,(A.5) or equivalently E2=p2+m2.(A.6) If we do the same with equation (A.6) as we did with equations (A.1) and (A.2), then we obtain −∂2ϕ(~r, t) ∂t2=−∇2ϕ(~r, t) + m2ϕ(~r, t).(A.7) If we introduce the D’Alembert operator =∂2 ∂t2−∇2, then one can rewrite (A.7) in a more compact way +m2ϕ(~r, t)=0,(A.8) which is the linear Klein-Gordon equation. In the equation, the quantum spin does not appear, that is why this equation can only be applied to 0-spin particles such as pions. In relativistic classical mechanics, one of the hypotheses is that the equations which rule the systems have to be Lorentz invariants, in other words, they have to be invariant when a Lorentz transformation is applied. For the particular case of a Lorentz transformation in the x axis (which is the case that we have dealt with in Sections 2.2 and 2.3), its expression is t0=t−vx √1−v2, x0=x−vt √1−v2, y0=y , z0=z, (A.9) where |v|< c = 1 is supposed. Notice that D’Alembert operator is invariant to this transformation ∂ ∂t =1 √1−v2 ∂ ∂t0−1 √1−v2~v ·~ ∇0,~ ∇=1 √1−v2 ~ ∇0−~v √1−v2 ∂ ∂t0, the differential operators are transformed by this way, where ~ ∇0=∂ ∂x0~ux+∂ ∂y0~uy+ 55 ∂ ∂z0~uzand ~v =v~ux. Therefore =∂2 ∂t2−∇2=1 √1−v2 ∂ ∂t0−1 √1−v2~v ·~ ∇0 1 √1−v2 ∂ ∂t0−1 √1−v2~v ·~ ∇0− 1 √1−v2 ~ ∇0−~v √1−v2 ∂ ∂t0 1 √1−v2 ~ ∇0−~v √1−v2 ∂ ∂t0 ⇒=1 1−v2∂2 ∂t02+v2∇02−∇02−v2∂2 ∂t02=∂2 ∂t02−∇02=0,(A.10) which is a property that was used in Section 2.3 when we maintained the form of the equation even though we moved to another frame of reference (through a Lorentz transform in a one dimensional space, which is a particular case of what we have just proven). In addition, it can be demonstrated that is invariant when a Lorentz transformation in any direction is applied. As a consequence, if we want equation (A.8) to be Lorentz invariant, due to the fact that mand are Lorentz invariants, then the solution has to be it as well1 ϕ(~r0, t0) = ϕ(~r, t),(A.11) this is one of the properties of a solution to nonlinear Klein-Gordon equation that we have used (see Section 2.3), because if we have a kink solution with speed ~v of this equation, then we can make a Lorentz transformation of speed ~v and as a result ϕ(~r0, t0) does not depend on t0(see the beginning of Section 2.2 where the transformation was performed). Furthermore, equation (A.8) can be generalized by adding a potential dependent on the solution and removing the mass term ϕ(~r, t) = ∂V ∂ϕ (ϕ(~r, t)) ⇒ ∇2ϕ(~r, t)−∂2ϕ ∂t2(~r, t) + ∂V ∂ϕ (ϕ(~r, t)) = 0,(A.12) which is the three dimensional nonlinear Klein-Gordon equation. If we restrict to one spatial dimension, then we obtain the equation we have studied in this project, and which has two specific cases we have focused in: sine-Gordon equation and ϕ4 equation. 1In fact, the condition is ϕ(~r0, t0) = ±ϕ(~r, t), but the −sign does not hold in the cases we have studied. 56 Appendix A. Klein-Gordon equation Appendix B Relation between intrinsic energy and pseudopotential In this section, we are focused on proving equation (2.17). For this purpose, we must consider, as we did in Section 2.2 within the study of the chain of coupled pendula, Hamiltonian mechanics. In this case, we are dealing with continuum media, and instead of a function of the coordinates over a trajectory, now we have a functional H H[ϕ](t) = Z+∞ −∞ ρ(ψ(x, t))dx , where ρ(ϕ) is the energy density which was introduced in equation (1.5): ρ(ψ(x, t)) = 1 2∂ψ ∂t (x, t)2 +1 2∂ψ ∂x (x, t)2 +g(ψ(x, t)), where we were supposing that g∈C2(R) and ψ∈C2(R2) is verified, and additionally, ρ(ψ(·, t0)) ∈L1(R) is supposed for each t0∈R. Under this hypotheses, the hamiltonian H(t) is well-defined and the equations which rule this system are given by ∂ψ ∂t =δH δφ =∂ρ ∂φ =φ , ∂φ ∂t =−δH δψ =−∂ρ ∂ψ −∂ ∂x ∂ρ ∂ψx=∂2ψ ∂x2−∂ρ ∂ψ, where we have denoted φ=∂ψ ∂t and ψx=∂ψ ∂x (notice that the first equation is a tautology), and δF δg is the functional derivative of F[g]. The second equation leads to ∂2ψ ∂t2=∂2ψ ∂x2−∂ρ ∂ψ =∂2ψ ∂x2−∂EI ∂ψ ⇒∂2ψ ∂x2−∂2ψ ∂t2=∂EI ∂ψ .(B.1) If we compare equation (B.1) with equation (2.10), the nonlinear Klein-Gordon equation 57 58 Appendix B. The intrinsic energy and pseudopotential ∂2ψ ∂x2−∂2ψ ∂t2+∂V ∂ψ (ψ)=0 ⇒∂2ψ ∂x2−∂2ψ ∂t2=−∂V ∂ψ (ψ), due to the fact that both equations give the possible solutions ψto the physical system, we have proven the relation (2.17): ∂EI ∂ψ =−∂V ∂ψ (ψ)⇒EI(ψ) = −V(ψ) + C, where Cdepends on the specific EIand Vthat are being considered. Therefore, we have accomplished the goal of this section. Appendix C Hypergeometric differential equation and Jacobi polynomials The hypergeometric differential equation is the following σ(z)y00 +τ(z)y0+λy = 0,(C.1) where σis a polynomial of degree at most two, τis a polynomial of degree at most one and λis a constant. A very well known example of (C.1) is z(1 −z)y00 + (c−(a+b+ 1)z)y0−ab y = 0, which was introduced by Euler and detailed studied by Gauss. One solution of the above equation is the Gauss hypergeometric function 2F1(a, b, c;x) (for c6= 0,−1,−2, . . .) 2F1(a, b, c;x) := 2F1a, b c x,(C.2) where the generalized hypergeometric function pFqis defined as pFqa1, a2, . . . , ap b1, b2, . . . , bq x=∞ X k=0 (a1)k(a2)k···(ap)k (b1)k(b2)k···(bq)k xk k!,(C.3) being (a)kthe so-called Pochhammer symbol (a)0= 1,(a)k=a(a+ 1)(a+ 2) ···(a+k−1), k = 1,2,3, . . . . Notice that the Pochhammer symbol is related with the classical Gamma function of Euler by the expression (a)k=Γ(a+k) Γ(a). Another important instances of the equation (C.1) are the so-called classical 59 60 Appendix C. Jacobi polynomials polynomials of Hermite Hn(x), Laguerre Lα n(x), and Jacobi P(α,β) n(x). The polynomials σand τof their respective hypergeometric differential equations can be found below, as well as the eigenvalues λnassociated with the solutions. Pn(x)Hn(x)Lα n(x)P(α,β) n(x) σ(x) 1 x1−x2 τ(x)−2x−x+α+ 1 −(α+β+ 2)x+β−α λn2n n n(n+α+β+ 1) For the purpose of this work, we will include several properties of the Jacobi polynomials. In particular, we need some properties of the Jacobi polynomials when the parameters are, in general, complex numbers. In the following we will discuss some of them. Let us introduce the Jacobi differential equation (1 −z2)y00 + [−(α+β+ 2)z+β−α]y0+n(n+α+β+ 1)y= 0,(C.4) where n= 0,1,2, . . . . Let be αand βcomplex numbers such that1<α > −1 and <β > −1. Therefore, we start discussing some properties of the Jacobi polynomials. In the following it is assumed that 1/n! = 0 if nis a negative integer. The explicit formula for the Jacobi polynomials is the following [13, § 4.3] P(α,β) n(x) =(α+ 1)n n! n X m=0 n+α mn+β n−m(x−1)n−m(x+ 1)m =(n+α+β+ 1)n 2nn!xn+n(α−β) 2n+α+βxn−1+···, (C.5) that can be obtained by using the Rodrigues formula P(α,β) n(x) = (−1)n 2nn!(1 −x)α(1 + x)β dn dxn(1 −x)α+n(1 + x)β+n. The Jacobi polynomials can also be written as a terminating hypergeometric series P(α,β) n(x) =(α+ 1)n n!2F1−n, n +α+β+ 1 α+ 1  1−x 2 =(β+ 1)n n!2F1−n, n +α+β+ 1 β+ 1  1 + x 2, (C.6) 1For detailed properties of the solutions of Jacobi equation (C.4) as well as the corresponding proofs we refer to the books [6,13] for the case when all the parameters are real, and for the complex case to the book [10]. 61 from where it follows their values at x=±1 P(α,β) n(1) = (α+ 1)n n!, P(α,β) n(−1) = (−1)n(β+ 1)n n!.(C.7) Let us point out that the first formula in (C.6) is well defined if α+1 6= 0,−1,−2, . . . , whereas the second one is well defined for β+ 1 6= 0,−1,−2, . . . . Notice that, if α+β+ 2 6= 0,−1,−2, . . . then at least one of the numbers α+ 1 and β+ 1 is different from 0,−1,−2, . . . , therefore for all values of the parameters such that α+β+ 2 6= 0,−1,−2, . . . , one of the representations (C.6) is always valid. Before continuing let us point out that the representations (C.6) allows us to write down the solution of the differential equation (C.4) for an arbitrary choice of the parameters αand βin Cthat satisfy α+β+ 2 6= 0,−1,−2, . . . . In fact, from (C.6) we find, respectively, that P(α,β) n(x) = 1 n! n X m=0 (−n)m(n+α+β+ 1)m(α+m+ 1)n−m m!1−x 2m =(−1)n n! n X m=0 (−n)m(n+α+β+ 1)m(β+m+ 1)n−m m!1 + x 2m . (C.8) From the above equations it is clear that the polynomials P(α,β) n(x) exists for any complex values of the parameters αand βsuch that α+β+ 2 6= 0,−1,−2, . . . , but it is important to notice that in some special cases the degree of P(α,β) n(x) is less than n. In fact, it can be checked that the order of the polynomial P(α,β) n(x) is reduced if and only if n+α+β+k= 0 for a certain integer k, 1 ≤k≤n. Moreover, P(α,β) n(x) vanished identically if n+α+β+k= 0, α=−l, 1 ≤k≤l≤n,l∈N. The differential equation (C.4) has another linearly independent solution. If α+β+n6=−1,−2, . . . , then it can be expressed in terms of the hypergeometric function [13, § 4.23] P(α,β) n(x) =(1 −x)−α−β−n−12F1n+β+ 1, n +α+β+ 1 2n+α+β+ 2  2 1−x,(C.9) and, therefore, for any choice of parameters such that α+β+n6=−1,−2, . . . , the general solution of the equation (C.4) can be represented as y(x) = AP(α,β) n(x) + BP(α,β) n(x),(C.10) where Aand Bare arbitrary constants. It is also important to point out that the second solution P(α,β) n(x) is unbounded in x= 1. Moreover, if y(x) is any other solution of (C.4) that is linearly independent of the Jacobi polynomials, then it can be shown (see [13, § 4.2, page 62]) that y(x) is unbounded on x= 1 or x=−1. 62 Appendix C. Jacobi polynomials Let us now discuss the very important, for our work, case when α=−νand β=ν. Since for this case α+β+ 2 = 2 >0, then the polynomials P(−ν,ν) l(x) is always a polynomial of degree exactly equal l= 0,1,2,··· and all the formulas hold. We will particularize some of them here. The Jacobi polynomials P(−ν,ν) l(s), is a solution of the differential equation (1 −s2)d2y ds2−2(s−ν)dy ds +l(l+ 1) y(s) = 0, l = 0,1,2,.... (C.11) The explicit expression can be obtained from (C.8) P(−ν,ν) l(s) = 1 l! l X m=0 (−l)m(l+ 1)m(−ν+m+ 1)l−m m!1−s 2m , which, for the special cases when ν= 1,2, . . . , l, becomes P(−ν,ν) l(s) = 1 l! l X m=ν (−l)m(l+ 1)m(−ν+m+ 1)l−m m!1−s 2m .(C.12) From (C.7) it follows that P(−ν,ν) l(1) =(−1)l(ν−1)(ν−2) ···(ν−l) l!, P(−ν,ν) l(−1) =(−1)l(ν+ 1)(ν+ 2) ···(ν+l) l!. (C.13) Notice that, if ν= 1,2, . . . , l, then P(−ν,ν) l(1) = 0, whereas if ν=−1,−2,...,−l, P(−ν,ν) l(−1) = 0. To conclude this appendix, we include a list of the first polynomials P(−ν,ν) l, because they are used at the end of the Chapter 3 P(−ν,ν) 0(s)=1, P(−ν,ν) 1(s) = s−ν, P(−ν,ν) 2(s) = 3 2s2−νs +(ν−1) (ν+ 1) 3. Bibliography [1] Al-Gwaiz, M.: Sturm-Liouville Theory and its Applications. Springer-Verlag, London, 2008. [2] Dauxois, T.: Fermi, Pasta, Ulam, and a mysterious lady. Physics Today, American Institute of Physics, 2008. [3] Dauxois, T., Peyrard, M.: Physics of soliton. Cambridge University Press, number 47, page 42, 2010. [4] Eckaus, W.: Studies in Non-Linear Stability Theory. Springer-Verlag New York Inc., 1965. [5] Hansen, P.J., Nicholson, D.R.: Simple soliton solutions. American Journal of Physics, number 47, page 769, 1979. [6] Nikiforov, A. F., Uvarov, V. B.: Special Functions of Mathematical Physics. Birkh¨auser Verlag, Basilea, 1988. [7] Parsa, Z.: Topological solitons in physics. American Journal of Physics, number 47, page 56, 1979. [8] Rebbi, C.: Solitons. Scientific American, 1979. [9] Rubinstein, J.: Sine-Gordon Equation, Journal of Mathematical Physics, volume 11, number 1, page 258, 1970. [10] Rusev, P.: Classical Orthogonal Polynomials and Their Associated Functions in a Complex Domain, Bulgarian Mathematical Monographs, number 3, Publishing House of Bulgarian Academy of Sciences, Sofia, Bulgaria, 1983. [11] Scott, A.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford University Press, 2003. [12] Simmons, G.F.: Differential Equations With Applications and Historical Notes. McGraw-Hill Education, 2 º Edition. [13] Szeg¨o, G.: Orthogonal Polynomials, American Mathematical Society, Colloquium Publications XXIII, New York, 1959. 63