The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

Abstract

Abstract. The general KdV equation (gKdV) derived by T. Chou is one of the famous (1 + 1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painlevé test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained.

Application of Lie-Group Theory for solving CalogeroBogoyavlenskii- Schiff Equation

“…A similarity solution has been found explicitly in terms of , and for CBS equation which is different from previous findings [9][10][11][12][13][14]. The Eq.…”

“…The solutions reflect elastic soliton behaviour of the wave which can be used to test accuracy, comparison and analysis of numerical results in the field. This appears to be more suitable than the previous findings [9][10][11][12][13][14] as it provides physical analysis of an exact solution of the (2+1)-CBS equation. Again the STM used here can be extended to other exact solutions of NLEEs which are arising in theoretical and applied Physics, Engineering and the like.…”

“…The researches [7][8][9][10][11][12][13][14] motivate the author to solve Eq. (1.1) analytically by using the similarity transformations method (STM).…”

“…The CBS equation was initially constructed by Bogoyavlenskii and Schiff in different ways [7][8][9][10]. Recent history of some past researches show that several effective methods for obtaining exact solutions of the CBS equation are contributed by a diverse group of researchers across the globe [9][10][11][12][13][14], for example, the periodic and soliton solutions of the CBS equation were obtained by Gandarias et al [11]. Its integrability has been proved by Zhang et al [12] and derived also the symmetry reductions of the equation.…”

A modified Calogero-Bogoyavlenskii-Schiff equation with variable coefficients and its non-isospectral Lax pair

“…Therefore, equations with variable coefficients may provide various models for real physical phenomena, for example, in the propagation of small-amplitude surface waves, which run on straits or large channels of slowly varying depth and width. On one hand, the variable-coefficient generalizations of nonlinear integrable equations are a currently exciting subject [19][20][21][22] (and also [21,]). Many researchers have mainly investigated (1 + 1) dimensional nonlinear integrable systems with constant coefficients for discovery of new nonlinear integrable systems.…”