Symmetry analysis, conservation laws and exact solutions of ill-posed Boussinesq equation

 Corresponding author: Gabriel Onwudiwe Omaba Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0. Symmetry analysis, conservation laws and exact solutions of ill-posed Boussinesq equation Gabriel Onwudiwe Omaba *, Justina Ebele Okeke and Ifeanyi Enuma Ezenekwe Department of Mathematics, faculty of physical sciences Chukwuemeka Odumegwu University, Uli Anambra, Nigeria. World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 Publication history: Received on 31 March 2025; revised on 14 May 2025; accepted on 16 May 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.26.2.1800 Abstract This thesis presents a comprehensive study on the construction of conservation laws and the application of double reduction techniques to obtain exact solutions for the ill-posed boussinesq equation Conservation laws are utilized to perform a double reduction of the original PDE. This reduction process involves two key stages: firstly, the application of the derived conservation laws to convert the PDE to ODE then reduction of the order of the ODE and secondly, the further simplification of the reduced equation by exploiting additional symmetries to obtain the exact solutions. The exact solutions are analyzed and graphically demonstrated to gain insight into the underlying physical and mathematical properties of the original PDE. The dissertation contributes to the field of applied mathematics by providing a rigorous framework for constructing conservation laws and applying reduction techniques to nonlinear PDEs. The exact solutions obtained not only advance the theoretical understanding of the equation but also offer potential applications in areas such as fluid dynamics, nonlinear optics, and other fields where similar equations arise. Keywords: Boussineq Equation; Lie Point Symmetry; Conserved Vectors; Double Reductions; Exact Solutions 1. Introduction Partial Differential Equations (PDEs) play a pivotal role in the modeling and analysis of various physical phenomena across disciplines such as physics, engineering, and applied mathematics [2,11,12]. These equations describe how physical quantities such as heat, sound, fluid flow, and electromagnetic fields change over space and time. The study of PDEs is fundamental to understanding the behavior of systems governed by the laws of nature, and their solutions provide insights into the dynamics of these systems. Among the vast array of PDEs, nonlinear equations, in particular, pose significant challenges due to their complexity and the rich variety of behaviors they can exhibit. Nonlinear PDEs often describe processes where the effects of interactions cannot be simply added together, leading to phenomena such as shock waves, solitons, and turbulence [12]. Finding exact solutions to these equations is crucial, as they can serve as benchmarks for numerical simulations and offer deep insights into the underlying physical processes. The Boussinesq equation, first derived by Joseph Boussinesq [1] is a nonlinear partial differential equation (PDE) that has far-reaching implications in various fields, including physics, engineering, and mathematics. Boussinesq[1], a French mathematician, presented his work in a paper titled "Théorie des ondes et des remous qui se propagent le long World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3380 d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond" (Theory of waves and ripples that propagate along a horizontal rectangular channel, communicating to the liquid contained in this channel speeds that are roughly equal from the surface to the bottom). Lord Rayleigh [8], an English physicist, independently derived a similar equation in 1876, now known as the RayleighBoussinesq equation. The renowned work done by Korteweg and de Vries [3] led to the derivation of the KdV equation, a simplified version of the Boussinesq equation, to model shallow water waves. The Boussinesq equation has found applications in various fields, including shallow water waves and coastal engineering, nonlinear optics and fiber optics, and plasma physics and ion-acoustic waves.The generalized Boussinesq (GB) equation with a damping term is given by 𝑢𝑡𝑡=2𝑘𝑢𝑥𝑥𝑡+𝑞𝑢𝑥𝑥𝑥𝑥+𝑐 𝑢𝑛𝑥𝑥, where k, q, c are constants and n is a nonzero real number[11]. This equation is widely used as a model to describe natural phenomena in many scientific fields, such as plasma waves, solid physics, and fluid mechanics. A special case of the Boussinesq equation is the modified Boussinesq equation, which is obtained when k = 0, q = 1, c = -1, and n = 3. This equation is used to model the temporal evolution of nonlinear finite amplitude waves on a density front in a rotating fluid. Exact traveling wave solutions for the generalized Boussinesq equation have been studied using various methods, including the extended tanh method and the direct method. The Boussinesq equation appears in different forms, depending on the values of the constants k, q, c, and n. For example, if 𝑢𝑥𝑥𝑡 is replaced by 𝑢𝑥𝑥 , the general form of the Boussinesq equation becomes𝑢𝑡𝑡=𝑢𝑥𝑥+𝑞𝑢𝑥𝑥𝑥𝑥+ 𝑢2𝑥𝑥 for n = 2, k = 1, and c = 1. This equation is known as the good Boussinesq or well-posed equation when q = -1, and the bad or ill-posed Boussinesq equation when q = 1. Therefore, the equation under investigation in this dissertation is the ill-posed nonlinear PDE 𝑢𝑡𝑡=𝑢𝑥𝑥+𝑢2𝑥𝑥+𝑢𝑥𝑥𝑥𝑥 ………….. (1.0) This equation is characterized by its combination of second-order time derivatives and a mix of second and fourth-order spatial derivatives, along with a nonlinear term involving the square of the dependent variable. Such equations often arise in the study of wave propagation in nonlinear media, including the analysis of elastic waves, fluid dynamics, and other areas where higher-order dispersion effects and non-linearities are significant. Despite the challenges encountered in finding solutions to the (1.0), researchers have made significant progress in solving the ill-posed Boussinesq equation using various numerical methods. Recent advances in symmetry analysis and conservation laws through the multiplier method have provided new insights into solving the ill-posed Boussinesq equation[11,12]. This method has been successfully applied to other nonlinear PDEs, and researchers are hopeful that it will provide a breakthrough in solving the ill-posed Boussinesq equation. 1.1. Statement of problem The ill-posed Boussinesq equation (1.0) poses significant challenges due to its non-integrable nature, which makes it difficult to find exact solutions. Furthermore, the equation's ill-posedness leads to numerical instability, making it challenging to obtain accurate numerical solutions. Due to the crucial role played by this equation, finding its exact solutions becomes significant as it throws more light into the physical features and intricate behavior of the system 1.2. Significance of the study The significance of this study lies in its contribution to the theory and application of nonlinear PDEs. By constructing conservation laws and performing double reductions, this research advances the understanding of complex nonlinear systems and provides a methodology that can be applied to other PDEs with similar structures. The exact solutions obtained in this study offer valuable insights into the behavior of the equation and can serve as benchmarks for future analytical and numerical studies. 1.3. Scope of the study This study is focused on the mathematical analysis of (1.0). The research is divided into four main components: the construction of conservation laws and Lie point symmetries, the reduction of the PDE using these laws, solving the reduced equations to get exact solutions and analysis of the solutions obtained World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3381 2. Methodology lie point symmetry method were used for the derivation of the Lie symmetries while the conservation laws which serve as integral in-variants under the dynamics described by the PDE.associated are systematically constructed using multiplier method. We chose to use the multipliers method to construct the conservation laws of the equation (1.0) for the first time because of its numerous advantages over other methods. The method of multipliers helps to find conserved integrals and local continuity equations of PDES. The conserved vectors and lie symmetry vectors derived in turn serves as tools or approach to perform double reduction of the PDE under consideration 2.1. Lie point symmetry method Lie point symmetries admitted by equation (1.0) are generated by a vector field of the form V=𝜉( 𝑥,𝑡,𝑢)𝜕 𝜕𝑥+ 𝜏(𝑥,𝑡,𝑢)𝜕 𝜕𝑡 +𝜙( 𝑥,𝑡,𝑢)𝜕 𝜕𝑢 …………….. (2.0) and we need to solve for the coefficient functions 𝜉(x,t,u), 𝜏(x,t,u), 𝜙 (x,t,u). V must satisfy Lie’s symmetry condition (1. 3), that is V[4] [𝑢𝑡𝑡 −𝑢𝑥𝑥−𝑢2𝑥𝑥−𝑢𝑥𝑥𝑥𝑥 =0]|(1.0) =0, ………………….. (2.1) where V[4] is the fourth prolongation of the operator V defined by V[4] =V+ ҁ𝑥𝜕 𝜕𝑢𝑥 +ҁ𝑡 𝜕 𝜕𝑢𝑡＋ ҁ𝑥𝑥 𝛿 𝜕𝑢𝑥𝑥 + ҁ𝑡𝑡 𝜕 𝜕𝑢𝑡𝑡 + ҁ𝑥𝑥 𝜕 𝜕𝑢𝑥＋ ҁ𝑥𝑥𝑥 𝜕 𝜕𝑢𝑥𝑥𝑥＋ҁ𝑥𝑥𝑥𝑥 𝜕 𝜕𝑢𝑥𝑥𝑥𝑥 and the coefficients ҁ𝑥, ҁ𝑡 , ҁ𝑥𝑥 , ҁ𝑡𝑡 , ҁ𝑥𝑥𝑥 and ҁ𝑥𝑥𝑥𝑥 are given by ҁ𝑥 = Dx(ϕ) − utDx(ξ) − uxDx(τ), ҁ𝑡 = Dt(ϕ) −utDt(ξ) − uxDt(τ), ҁ𝑥𝑥 = Dx (ζx) − uxtDx(ξ) − uxx Dx(τ), ҁ𝑡𝑡 = Dt (ζt) – uttDx(ξ) – utx Dx((τ), ҁ𝑥𝑥𝑥 = Dx(ζxx) − uxxt Dx(ξ) − uxxx Dx((τ), ҁ𝑥𝑥𝑥𝑥 = Dx(ζxxx) − Uxxxt Dx(ξ) − Uxxxx Dx((τ). Here Dx , Dt denote the total derivative operators defined by Dx = ∂ ∂x＋𝑢x ∂ ∂u＋𝑢tx ∂ ∂ut＋..., Dt = ∂ ∂t ＋𝑢t ∂ ∂u＋𝑢xt ∂ ∂ux＋… ………… (2.2) Expansion and separation of (2.1) with respect to the powers of different derivatives of u yields an over determined system in the unknown coefficients 𝜉, 𝜏 and 𝜙. However the over determined system cannot be presented here due to its lengthy calculations. We present only the result and refer the reader to [8] for details. Solving the over determined system for arbitrary parameters we obtain thee coefficients 𝜉(𝑥,𝑡,𝑢)=1 2𝑐1𝑥+𝑐3,𝜏(𝑥,𝑡,𝑢)=𝑐1𝑡+𝑐2,𝜙(𝑥,𝑡,𝑢)=−𝑐1(𝑢+1 2), where 𝑐1, 𝑐2 and 𝑐3 are constants. Without loss of generality, World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3382 taking𝑐3=1,𝑐1=0,𝑐2=0, we obtain the symmetry vector 𝑉1=𝜕 𝜕𝑥. Similarly, we obtain the rest as 𝑉2=𝜕 𝜕𝑡,𝑉3=1 2𝑥𝜕 𝜕𝑥+𝑡𝜕 𝜕𝑡−(𝑢+1 2)𝜕 𝜕𝑢 These 3 symmetry vectors, 𝑣1,𝑣2 and 𝑣3 will be used to reduce equation (1.0) to simpler and solvable form. 2.2. Conservation laws via multiplier approach A conserved vector corresponding to a conservation law of the equation (1.0) is a 2−tuple (𝑇𝑡,𝑇𝑥) such that 𝐷𝑡 𝑇𝑡+𝐷𝑥𝑇𝑥=0 along the solutions of the equation. We derive the conservation laws using the multiplier approach [9,10] Consider the multiplier 𝛬 of order up to two, viz. 𝛬=(𝑡,𝑥,𝑢,𝑢𝑥,𝑢𝑡,𝑢𝑥𝑡,𝑢𝑥𝑥,𝑢𝑡𝑡,𝑢𝑥𝑥𝑥𝑥) for eqn.(1.0). The conserved vector (𝑇𝑡,𝑇𝑥) of eqn.(1.0) satisfies the divergence relation 𝐷𝑡 𝑇𝑡+𝐷𝑥𝑇𝑥=𝛬(𝑢𝑡𝑡 = 𝑢𝑥𝑥+ (𝑢2)𝑥𝑥 + 𝑢𝑥𝑥𝑥𝑥)=0. Moreover, we have 𝛿 𝛿𝑢(𝛬)(𝑢𝑡𝑡 = 𝑢𝑥𝑥+ (𝑢2)𝑥𝑥 + 𝑢𝑥𝑥𝑥𝑥)=0. (2.3) After a lengthy calculation with the help of maple software we obtain the following conserved vectors with their corresponding multipliers: 𝛬 1=1 𝑇1𝑡=−𝑢𝑡,𝑇1𝑥=2𝑢𝑢𝑥+𝑢𝑥+𝑢𝑥𝑥𝑥 𝛬 2=𝑥 𝑇2𝑡=−𝑥𝑢𝑡,𝑇2𝑥=2𝑥𝑢𝑢𝑥−𝑢2+𝑥𝑢𝑥+𝑥𝑢𝑥𝑥𝑥−𝑢−𝑢𝑥𝑥 …………… (2.4) 𝛬 3=𝑡 𝑇3𝑡=−𝑢𝑡𝑡+𝑢,𝑇3𝑥=2𝑡𝑢𝑢𝑥+𝑡𝑢𝑥+𝑡𝑢𝑥𝑥𝑥 𝛬 4=𝑥𝑡 𝑇4𝑡=−𝑥𝑡𝑢𝑡+𝑥𝑢,𝑇4𝑥=2𝑥𝑡𝑢𝑢𝑥−𝑡𝑢2+𝑥𝑡𝑢𝑥+𝑥𝑡𝑢𝑥𝑥𝑥−𝑡𝑢−𝑡𝑢𝑥𝑥 2.3. Definition Consider a scalar PDE F = 0 with n = 2,(x1,x2) = (t,x) which admits a symmetry X associated with a conserved vector (TT, TX ). In terms of the canonical variables r,s obtained by mapping X to Y = 𝜕 𝜕𝑠 the conservation laws can be expressed as[5] DrTr + DsTs = 0, with Tr and Ts given as , . World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3383 This now allows for a double reduction of the PDE. 2.4. Double reduction of equation (1.0) We firstly show that the lie vector symmetries are associated with the conserved vectors. This happens if Vi[3](Tit Tix)+(Dt𝜉+Dxτ)(Tit Tix)−(Tit Tix)(Dt𝜉 Dx𝜉 Dt𝜏 Dxτ)=(00),i=1,2,3 Tix and Tit i=1,2,3 and 4 are conserved vectors and Vi[3] is the third prolongation of the Vi , i=1,2 and 3 and Vi[3] = Vi + ҁ𝑥 𝜕 𝜕𝑢𝑥 +ҁ𝑡 𝜕 𝜕𝑢𝑡＋ ҁ𝑥𝑥 𝜕 𝜕𝑢𝑥𝑥 + ҁ𝑡𝑡 𝜕 𝜕𝑢𝑡𝑡 + ҁ𝑥𝑥 𝜕 𝜕𝑢𝑥＋ ҁ𝑥𝑥𝑥 𝜕 𝜕𝑢𝑥𝑥𝑥 since V1 and V2 are trivial symmetries, they are associated with T1= (T1t T1x) with its multiplier Λ1. Next, we verify if V3 is associated with (T1t T1x) 𝑉3=12𝑥𝜕 𝜕𝑥+𝑡𝜕 𝜕𝑡−(𝑢+12)𝜕 𝜕𝑢, 𝜉(𝑥,𝑡,𝑢)=1 2𝑥,𝜏(𝑥,𝑡,𝑢)=𝑡,𝜙 (𝑥,𝑡,𝑢)= −(𝑢+ 1 2 ). Dt(𝜙)=−ut,,Dt(𝜏)=1,Dt(𝜉)=0, Dx(𝜙)= – ux , Dx(𝜉) = 1 2, Dx(𝜏) = 0 ζx = Dx(ϕ) − utDt(ξ) − uxDx(τ) = – ux – ut .0– ux .0= -ux ζt = ζt =Dt(ϕ) − utDt(ξ) − uxDt(τ)= -ut -ut.0 – ux .1= -ut - ux ζtt = Dt (ζt ) - utt Dx (ξ ) - utx Dx (τ)= -utt - utx - 1 2utt - utx. 0= - 3 2 utt -utx ζxx = Dx (ζx) – uxt Dx (ξ ) – uxx Dx (τ) = -uxx1 2uxt - uxx . 0 = -uxx1 2uxt ζxxx = Dx (ζxx) – uxxt Dx (ξ ) – uxxx Dx (τ) = -uxxx1 2uxxt1 2uxxt1 2uxxx.0= -uxxx-uxxt Substituting (1.) we obtain the third prolongation of 𝑉3 as V3(3)= 1 2x∂ ∂x+t∂ ∂t−(u+1 2)∂ ∂u -ux∂ ∂ux+(-ut - ux )∂ ∂ut＋(-uxx1 2uxt)δ ∂uxx +(−3 2𝑢𝑡𝑡−𝑢𝑥𝑡 ) ∂ ∂utt + (-uxx1 2uxt)∂ ∂ux＋( - uxxxuxxt)∂ ∂uxxx Also substituting into ( 12x∂ ∂x+t∂ ∂t−(u+12)∂ ∂u −ux ∂ ∂ux+(−ut − ux ) ∂ ∂ut＋(−uxx− 12uxt) δ ∂uxx +(−32𝑢𝑡𝑡−𝑢𝑥𝑡 ) ∂ ∂utt + (−uxx − 12uxt) ∂ ∂ux＋ ( − uxxx− uxxt)∂ ∂uxxx )(−ut 2uux+ux+uxxx)+( 12 + 1)(−ut 2uux+ux+uxxx) −(1 0 012)(−ut 2uux+ux+uxxx) ≠(00) Therefore 𝑉3 is not associated with (T1t T1x) with multiplier Λ1. The table below shows the relationship between the 3 vector symmetries and the conserved vectors with their multipliers. World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3384 Table 1 Relationship between the point symmetry vectors and conserved vectors Multipliers Conserved Vectors Symmetry Vectors Association Λ1=1 𝑇1𝑡, 𝑇1𝑥 V1 , V2 , V3 x Λ2=𝑥 𝑇2𝑡, 𝑇2𝑥 V1 , V2 , V3 x Λ3=𝑡 𝑇3𝑡, 𝑇3𝑥 V1 , V2 , V3 x Λ4=𝑥𝑡 𝑇4𝑡, 𝑇4𝑥 V1 , V2, V₃ x We now use the lie point symmetries which are associated with the conserved vector (T1t T1x) to transform the variables of the Boussinesq equation (1.0) into new similarity variables. we consider the linear combination V2 + cV1 where c is a non - zero constant. solving the characteristic equation. 𝑑𝑡 1= 𝑑𝑥 𝑐= 𝑑𝑢 0= 𝑑𝑟 0= 𝑑𝑠 1= 𝑑𝑤 0 we obtain the canonical coordinates s =𝑡, 𝑟=𝑥−𝑐𝑡, 𝑤 (𝑟)=𝑢 In the new canonical coordinates, the conservation law 𝐷𝑡 𝑇1𝑡+𝐷𝑥 𝑇1𝑠=0 is rewritten as 𝐷𝑟 𝑇1𝑟+𝐷𝑠 𝑇1𝑠=0 We can find 𝑇1𝑟 and 𝑇1𝑠 by T1r= 𝑇1𝑡𝐷𝑡 (𝑟)+ 𝑇1𝑥𝐷𝑥 (𝑟) 𝐷𝑡 (𝑟) 𝐷𝑥 (𝑠)−𝐷𝑥 (𝑟) 𝐷𝑡 (𝑠) …………….. (2.5) T1s= 𝑇1𝑡𝐷𝑡 (𝑠)+ 𝑇1𝑥𝐷𝑥 (𝑠) 𝐷𝑡 (𝑟) 𝐷𝑥 (𝑠)−𝐷𝑥 (𝑟) 𝐷𝑡 (𝑠) Substituting the canonical variables, their derivatives and the conserved vectors into (2.5) we obtain T1r=−𝑐𝑢𝑡−2𝑢𝑢𝑥−𝑢𝑥−𝑢𝑥𝑥𝑥 …………. (2.6) T1s=−𝑢𝑡 Now, 𝑢𝑡= 𝑑𝑢 𝑑𝑡= 𝑑𝑤 𝑑𝑡 = 𝜕𝑤 𝜕𝑟. 𝜕𝑟 𝜕𝑡 = − 𝑐𝑤𝑟 𝑢𝑥= 𝑑𝑢 𝑑𝑥= 𝑑𝑤 𝑑𝑥 = 𝜕𝑤 𝜕𝑟 .𝜕𝑟 𝜕𝑥 = 1.𝑤𝑟= 𝑤𝑟 Similarly, 𝑢𝑥𝑥= 𝑑𝑢𝑥 𝑑𝑥 𝑑𝑤𝑟 𝑑𝑥 = 𝜕𝑤𝑟 𝜕𝑟 .𝜕𝑟 𝜕𝑥= 𝑤𝑟𝑟 𝑢𝑥𝑥𝑥= 𝑤𝑟𝑟𝑟 using (2.6) we obtain T1r=(1−𝑐2)𝑤𝑟+ 2𝑤𝑤𝑟+ 𝑤𝑟𝑟𝑟 …………………. (2.7) T1s= 𝑐𝑤𝑟 ……………… (2.8) World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3385 Where 𝑤𝑟 is total derivative with respect to 𝑟. Since (2.8) does not depend on s, we deduce from that 𝐷𝑟 𝑇1𝑟=0 This means that 𝑇1𝑟 is a constant. That is m (1−𝑐2)𝑤𝑟+ 2𝑤𝑤𝑟+ 𝑤𝑟𝑟𝑟 = 𝑘1 …………….. (2.9) where 𝑘1 is a constant. Equation (2.9) is a third order ODE which is a double reduction of the fourth order ill-posed boussinesq equation (1.0). 3. Results and Discussions By integrating (2.9) once with respect to r, while setting the constant of integration to zero gives rise to (1−𝑐2)𝑤+ 𝑤2+ 𝑤𝑟𝑟 = 0 Further integration results to (1-𝑐2) 𝑤2 2 + 𝑤3 3 + 𝑤𝑟2 2 = 𝑘3w where 𝑘3 is a constant. This implies 𝑤𝑟2 =(𝑐2−1)𝑤2 - 2 𝑤3 3 + 2w𝑘3 𝑤𝑟= √(𝑐2−1)𝑤2 − 2 𝑤3 3 + 2w𝑘3 𝑑𝑤 𝑑𝑟 = √(𝑐2−1)𝑤2 − 2 𝑤3 3 + 2w𝑘3 𝑑𝑟 𝑑𝑤 = 1 √(𝑐2−1)𝑤2 − 2 𝑤3 3 + 2w𝑘3 𝑑𝑤 √(𝑐2−1)𝑤2 − 2 𝑤3 3 + 2w𝑘3 = dr ∫ 𝑑𝑤 √(𝑐2−1)𝑤2 − 2 𝑤3 3 + 2w𝑘3 =r+ 𝑘4 where 𝑘4 is constant. In terms of the original variables, we obtain ∫ 𝑑𝑢 √(𝑐2−1)𝑢2 − 2 3𝑢3 + 2u𝑘3 =𝑥−𝑐𝑡+𝑘4 ………….. (3.0) Equation (1.12) is the integral solution of the ill-posed Boussinesq equation (1.0). 4. Exact solutions by improved generalised Riccati equation mapping method Here, we solve the reduced equation (2.9) using improved generalized Riccati equation mapping method [4]. Our main aim is to obtain exact or at least approximate solutions if possible for the reduced equation (2.9). We express the solution, w(r) of equation (2.9) in the finite series w(r)= ∑𝑎𝑖𝜓𝑖, 𝑚 𝑖=−𝑚 ……………… (3.1) World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3386 where 𝑎𝑖 are constants to be determined and 𝜓 satisfies the Riccatti equation 𝜓𝐼=𝜇+𝛽𝜓+(𝜈−1)𝜓2 …………… (3.2) We determine the positive integer 𝑚 in equation (3.1) by balancing the highest order derivative, 𝑤𝑟𝑟 and the nonlinear term, 𝑤2 by solving 𝑚+2=2𝑚 ⇒ 𝑚=2, so that the solution of equation (4.36) can be written as w(r)= 𝑎−2𝜓−2+𝑎−1𝜓−1+𝑎0+𝑎1𝜓+𝑎2𝜓2 ………… (3.3) After substituting and collecting all the terms of the same power, 𝜓𝑖,𝑖=−2,−1,0,1, 2 and equating them to zero, we obtain a system of an algebraic equations ( Due to the size of the equations we decided not to display the equation for simplicity). Solving the system of the algebraic equations for 𝑎−2,𝑎−1,𝑎0,𝑎1,𝑎2,𝑐, using symbolic computation software, Mathematica 9, we obtain 𝑎1=𝑎2=0,𝑎0= −6𝜇(𝜈−1),𝑎−1=−6𝛽𝜇,𝑎−2=−6𝜇2 ……………. (3.4) Substituting equation (3.4) into the solution formula (3.3), we obtain w(r)= −6𝜇2𝜓−2−6𝛽𝜇𝜓−1−6𝜇(𝜈−1) …………….. (3.5) Substituting the known solutions, 𝜓(𝑟) of the Riccati equation (3.2) into equation (3.5) and simplifying the resulting equation in terms of the original variable, 𝑢(𝑥,𝑡), we obtained the following new types of solutions: TYPE 1: Ω= 𝛽2−4𝜇(𝜈−1)>0,𝛽(𝜈−1)≠0,(𝑜𝑟 𝜇(𝜈−1)≠0), Soliton like solutions 𝑢1(𝑥,𝑡)= −6𝜇(𝜈−1)+ 12𝛽𝜇(𝜈−1)[𝛽+√Ω 𝑇𝑎𝑛ℎ(√Ω 2(𝑥−𝜆𝑡)) ]−1 −24(𝜇(𝜈−1))2[𝛽+√Ω 𝑇𝑎𝑛ℎ(√Ω 2(𝑥−𝜆𝑡)) ]−2 𝑢2(𝑥,𝑡)= −6𝜇(𝜈−1)+ 12𝛽𝜇(𝜈−1)[𝛽+√Ω 𝐶𝑜𝑡ℎ(√Ω 2(𝑥−𝜆𝑡)) ]−1 −24(𝜇(𝜈−1))2[𝛽+√Ω 𝐶𝑜𝑡ℎ(√Ω 2(𝑥−𝜆𝑡)) ]−2 𝑢3(𝑥,𝑡)=−6𝜇(𝜈−1)+ 12𝛽𝜇(𝜈−1)[𝛽+√Ω (𝑇𝑎𝑛ℎ(√Ω (𝑥−𝜆𝑡))±𝑖𝑆𝑒𝑐ℎ(√Ω (𝑥−𝜆𝑡))) ]−1 −24(𝜇(𝜈− 1))2[𝛽+√Ω (𝑇𝑎𝑛ℎ(√Ω (𝑥−𝜆𝑡))±𝑖𝑆𝑒𝑐ℎ(√Ω (𝑥−𝜆𝑡))) ]−2 𝑢4(𝑥,𝑡)= −6𝜇(𝜈−1)+ 12𝛽𝜇(𝜈−1)[𝛽+√Ω (𝐶𝑜𝑡ℎ(√Ω (𝑥−𝜆𝑡))±𝐶𝑠𝑐ℎ(√Ω (𝑥−𝜆𝑡))) ]−1 −24(𝜇(𝜈−1))2[𝛽+ √Ω (𝐶𝑜𝑡ℎ(√Ω (𝑥−𝜆𝑡))±𝐶𝑠𝑐ℎ(√Ω (𝑥−𝜆𝑡)))]−2 𝑢5(𝑥,𝑡)= −6𝜇(𝜈−1)+ 24𝛽𝜇(𝜈−1)[2𝛽+√Ω (𝑇𝑎𝑛ℎ(√Ω 4(𝑥− 𝜆𝑡))±𝐶𝑜𝑡ℎ(√Ω 4(𝑥−𝜆𝑡)))]−1 −96(𝜇(𝜈−1))2[2𝛽+√Ω (𝑇𝑎𝑛ℎ(√Ω 4(𝑥−𝜆𝑡))±𝐶𝑜𝑡ℎ(√Ω 4(𝑥−𝜆𝑡)))]−2 World Journal of Advanced Research and Reviews, 2025, 26(02), 3379-3393 3387 𝑢6(𝑥,𝑡)= −6𝜇(𝜈−1)−12𝛽𝜇(𝜈−1)[−𝛽+√(𝐴2+𝐵2)Ω−𝐴√Ω 𝐶𝑜𝑠ℎ(√Ω (𝑥−𝜆𝑡)) 𝐴 𝑆𝑖𝑛ℎ(√Ω (𝑥−𝜆𝑡))+𝐵 ]−1 −24(𝜇(𝜈−1))2[−𝛽+√(𝐴2+𝐵2)Ω−𝐴√Ω 𝐶𝑜𝑠ℎ(√Ω (𝑥−𝜆𝑡)) 𝐴 𝑆𝑖𝑛ℎ(√Ω (𝑥−𝜆𝑡))+𝐵 ]−2 𝑢7(𝑥,𝑡)= −6𝜇(𝜈−1)−12𝛽𝜇(𝜈−1)[−𝛽−√(𝐴2+𝐵2)Ω+𝐴√Ω 𝐶𝑜𝑠ℎ(√Ω (𝑥−𝜆𝑡)) 𝐴 𝑆𝑖𝑛ℎ(√Ω (𝑥−𝜆𝑡))+𝐵 ]−1 −24(𝜇(𝜈−1))2[−𝛽−√(𝐴2+𝐵2)Ω+𝐴√Ω 𝐶𝑜𝑠ℎ(√Ω (𝑥−𝜆𝑡)) 𝐴 𝑆𝑖𝑛ℎ(√Ω (𝑥−𝜆𝑡))+𝐵 ]−2 where A and B are two non-zero constants and satisfies 𝐵2− 𝐴2>0. 𝑢8(𝑥,𝑡)= −6𝜇(𝜈−1)−3𝛽 [ 𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) √Ω 𝑆𝑖𝑛ℎ(√Ω 2(𝑥−𝜆𝑡))−𝛽𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) ] −1 −32 [ 𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) √Ω 𝑆𝑖𝑛ℎ(√Ω 2(𝑥−𝜆𝑡))−𝛽𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) ] −2 𝑢9(𝑥,𝑡)= −6𝜇(𝜈−1)+3𝛽 [ 𝑆𝑖𝑛ℎ(√Ω 2(𝑥−𝜆𝑡)) 𝛽 𝑆𝑖𝑛ℎ(√Ω 2(𝑥−𝜆𝑡))−√Ω 𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) ] −1 −32 [ 𝑆𝑖𝑛ℎ(√Ω 2(𝑥−𝜆𝑡)) 𝛽 𝑆𝑖𝑛ℎ(√Ω 2(𝑥−𝜆𝑡))−√Ω 𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) ] −2 𝑢10(𝑥,𝑡)= −6𝜇(𝜈−1)−3𝛽 [ 𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) √Ω 𝑆𝑖𝑛ℎ(√Ω 2(𝑥−𝜆𝑡))−𝛽𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡))±𝑖√Ω ] −1 −32 [ 𝐶𝑜𝑠ℎ(√Ω 2(𝑥−𝜆𝑡)) √Ω 𝑆𝑖𝑛ℎ(√Ω (𝑥−𝜆𝑡))−𝛽𝐶𝑜𝑠ℎ(√Ω (𝑥−𝜆𝑡))±𝑖√Ω ] −2