Comment on “Soliton ratchets induced by excitation of internal modes”

Comment on “Soliton ratchets induced by excitation of internal modes” Niurka R. Quintero* Departamento de Física Aplicada I, E. U. P., Universidad de Sevilla, Virgen de África 7, 41011 Sevilla, Spain and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain Bernardo Sánchez-Rey Nonlinear Physics Group, University of Seville, Spain Jesús Casado-Pascual Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, Sevilla 41080, Spain 共Received 23 July 2004; published 27 May 2005兲 Very recently Willis et al. 关Phys. Rev. E 69, 056612 共2004兲兴 have used a collective variable theory to explain the appearance of a nonzero energy current in an ac-driven, damped sine-Gordon equation. In this Comment, we prove rigorously that the time-averaged energy current in an ac-driven nonlinear Klein-Gordon system is strictly zero. DOI: 10.1103/PhysRevE.71.058601 PACS number共s兲: 05.45.Yv, 05.60.Cd, 63.20.Pw Recently several papers have been published trying to understand soliton ratchets 共see, for example, Refs. 关1–6兴and for a recent review, Chap. 9 in Ref. 关7兴, pp. 343–364兲. This phenomenon is a generalization of the ratchet effect 关8兴to spatially extended systems, and manifests as a unidirectional motion of a soliton induced by zero-average forces. A paradigmatic example is the driven, damped nonlinear KleinGordon equation ␾ ,tt共x,t兲− ␾ ,xx共x,t兲=−U⬘关 ␾ 共x,t兲兴 +f共t兲− ␤␾ ,t共x,t兲,共1兲 where g,z= ⳵ g/ ⳵ z,f共t兲is a periodic field with period Tand zero time average 关i.e., 1/T兰0 Tdtf共t兲=0兴, ␤ ⬎0 is the dissipation parameter determining the inverse relaxation time in the system, and U⬘共z兲is the derivative with respect to zof the potential U共z兲. In this Comment, we will assume that the potential U共z兲is periodic with period ␭, and presents minima at zj=z0+j␭, with j苸Z. The ac-driven, damped sine-Gordon equation considered in Ref. 关6兴is a particular case of this more general problem, with U共z兲=1−cos共z兲and f共t兲=−关 ⑀ 1cos共 ␻ t兲+ ⑀ 2cos共2 ␻ t+ ␪ 兲兴.共2兲 To fully specify the mathematical problem, the partial differential equation 共1兲must be amended by both initial conditions for ␾ 共x,0兲and ␾ ,t共x,0兲, and boundary conditions for limx→±⬁ ␾ 共x,t兲. Several boundary conditions can be imposed to have a well-posed boundary value problem. For instance, in the absence of the periodic field f共t兲, it is possible to choose the fixed boundary conditions: limx→+⬁ ␾ 共x,t兲=zland limx→−⬁ ␾ 共x,t兲=zm. In the presence of f共t兲, the fixed boundary conditions become incompatible with Eq. 共1兲, and they are usually replaced by the aperiodic boundary conditions: lim x→+⬁ ␾ 共x,t兲= lim x→−⬁ ␾ 共x,t兲+␭Q,共3兲 lim x→+⬁ ␾ ,x共x,t兲= lim x→−⬁ ␾ ,x共x,t兲,共4兲 where Q苸Zis the so-called topological charge. The discrete version of these aperiodic boundary conditions are also the most used in the numerical solution of Eq. 共1兲共see, for example, Refs. 关1,5兴兲. It can be derived from the continuity equation that the energy current density generated by the field ␾ 共x,t兲in the absence of damping and external forcing is given by j共x,t兲 =− ␾ ,t共x,t兲 ␾ ,x共x,t兲and, consequently, the energy current reads J共t兲=− 冕 −⬁ +⬁ dx ␾ ,x共x,t兲 ␾ ,t共x,t兲.共5兲 The time-averaged energy current 具J典is defined as the limit 具J典= lim ␶ →+⬁ 1 ␶ 冕 0 ␶ dtJ共t兲.共6兲 In Ref. 关1兴, it has been proved by symmetry considerations that a necessary condition for the appearance of a nonvanishing time-averaged energy current is that either the potential presents broken spatial symmetry, or the field f共t兲violates the symmetry property f 冉 t+T 2 冊 =−f共t兲,共7兲 or both simultaneously. Following this idea, a collective variable approach has been developed in Ref. 关6兴for the acdriven, damped sine-Gordon equation with a field of the form 共2兲that leads to a nonvanishing time-averaged energy current. The purpose of this Comment is to prove that the time-averaged energy current, 具J典, of a driven, damped nonlinear Klein-Gordon equation of the form 共1兲is necessarily zero. To prove that 具J典=0, firstly we will obtain an ordinary differential equation for the energy current J共t兲. In order to *Electronic address: [email protected] PHYSICAL REVIEW E 71, 058601 共2005兲 1539-3755/2005/71共5兲/058601共2兲/$23.00 ©2005 The American Physical Society058601-1 do that, we differentiate with respect to time Eq. 共5兲, resulting J ˙共t兲=− 冕 −⬁ +⬁ dx关 ␾ ,xt共x,t兲 ␾ ,t共x,t兲+ ␾ ,x共x,t兲 ␾ ,tt共x,t兲兴.共8兲 By making use of Eq. 共1兲in the second term on the righthand side of the above expression, it is straightforward to write it in the form J ˙共t兲=−1 2 兵 lim x→+⬁ 关 ␾ ,t共x,t兲兴2− lim x→−⬁ 关 ␾ ,t共x,t兲兴2 其 −1 2 兵 lim x→+⬁ 关 ␾ ,x共x,t兲兴2− lim x→−⬁ 关 ␾ ,x共x,t兲兴2 其 + lim x→+⬁U关 ␾ 共x,t兲兴 − lim x→−⬁U关 ␾ 共x,t兲兴 − ␤ J共t兲 − 关 lim x→+⬁ ␾ 共x,t兲− lim x→−⬁ ␾ 共x,t兲 兴 f共t兲.共9兲 Differentiating Eq. 共3兲with respect to t, it is easy to see that the first term between the brace brackets on the right-hand side of Eq. 共9兲is equal to zero. From Eq. 共4兲, it follows that the second term between the brace brackets on the right-hand side of Eq. 共9兲is also equal to zero. The two terms of Eq. 共9兲 containing U关 ␾ 共x,t兲兴 also cancel each other due to the boundary condition 共3兲and the periodicity of U共z兲. Thus, from Eq. 共3兲we finally obtain J ˙共t兲=− ␤ J共t兲−␭Qf共t兲.共10兲 Notice that Eq. 共10兲is a direct consequence of Eq. 共1兲and the boundary conditions 共3兲and 共4兲. Therefore, it is an exact result valid for any periodic potential U共z兲of the type described in the paragraph below Eq. 共1兲, and any external field f共t兲. Equation 共10兲appears in Ref. 关6兴as an approximate result obtained after neglecting the dressing due to phonons. The general solution of Eq. 共10兲is J共t兲=J共0兲e− ␤ t−␭Q 冕 0 t dt⬘e− ␤ 共t−t⬘兲f共t⬘兲,共11兲 and making use of the definition of the time-averaged energy current in Eq. 共6兲, it results 具J典= lim ␶ →+⬁ 再 J共0兲 ␤␶ 共1−e− ␤␶ 兲−␭Q ␤␶ 冕 0 ␶ dt f共t兲 +␭Q ␤␶ 冕 0 ␶ dt f共t兲e− ␤ 共 ␶ −t兲 冎 .共12兲 The first limit in the above expression is obviously equal to zero. The second one is also equal to zero as the external field is periodic with zero time average. The last integral appearing in Eq. 共12兲can be bounded using the fact that 冏 冕 0 ␶ dt f共t兲e− ␤ 共 ␶ −t兲冏艋 冕 0 ␶ dt兩f共t兲兩e− ␤ 共 ␶ −t兲艋fm ␤ 共1−e− ␤␶ 兲, 共13兲 where fmis an upper bound of 兩f共t兲兩 and, thus, the third limit in Eq. 共12兲is also equal to zero. We conclude that 具J典=0 for any periodic potential U共z兲of the type described in the paragraph below Eq. 共1兲, and any bounded, zero time-averaged periodic field f共t兲. It is important to emphasize that the result in this Comment is not applicable when the external field not only depends on tbut also on x. In that case Eq. 共10兲cannot be obtained and, in principle, it is possible to observe a nonvanishing time-averaged energy current. A field of this kind has been considered in Ref. 关1兴, where f共x,t兲=E共t兲+ ␰ 共x,t兲, with E共t兲being an ac field with zero mean and ␰ 共x,t兲a Gaussian white noise. We acknowledge financial support from the Ministerio de Ciencia y Tecnología of Spain under Grant Nos. BFM20013878 共N.R.Q兲, BFM2003-03015 共B.S-R兲, and BFM200203822 共J.C.-P.兲, and from la Junta de Andalucía. 关1兴S. Flach, Y. Zolotaryuk, A. E. Miroshnichenko, and M. V. Fistul, Phys. Rev. Lett. 88, 184101 共2002兲. 关2兴M. Salerno and Y. Zolotaryuk, Phys. Rev. E 65, 056603 共2002兲. 关3兴M. Salerno and N. R. Quintero, Phys. Rev. E 65 025602共R兲 共2002兲; N. R. Quintero, B. Sánchez-Rey, and M. Salerno, e-print nlin.SI/0405023, Phys. Rev. E 共to be published兲. 关4兴G. Costantini, F. Marchesoni, and M. Borromeo, Phys. Rev. E 65, 051103 共2002兲. 关5兴L. Morales-Molina, N. R. Quintero, F. G. Mertens, and A. Sánchez, Phys. Rev. Lett. 91, 234102 共2003兲. 关6兴C. R. Willis and M. Farzaneh, Phys. Rev. E 69, 056612 共2004兲. 关7兴Oleg M. Braun and Yuri S. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods and Applications 共Springer, Berlin, 2004兲. 关8兴P. Reimann, Phys. Rep. 361,57共2002兲. COMMENTS PHYSICAL REVIEW E 71, 058601 共2005兲 058601-2