Bell Polynomial Approach and N -Soliton Solutions for a Coupled KdV-mKdV System

Abstract

In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.

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“…Gepreel, Omran and Elegan (2011) employed the modified truncated expansion method to obtain its traveling wave solutions. Qin et al (2012) derive its N-soliton solutions by means of the Bell polynomials. Rui and Qi (2016) employed the Hirota bilinear method to construct its quasi-periodic wave solution.…”

Binary Bell polynomials, Hirota bilinear approach to Levi equation

“…So both the bilinear forms and bilinear Bäcklund transformations can be obtained directly. Lots of studies have devoted to the application and extension of this method [11][12][13][14][15][16][17][18][19][20][21].…”

Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid

“…[21][22][23] Burgers' equation can be reduced to a linear equation under the Cole-Hopf transformation, then the analytical solutions can be derived, and some Burgers-type NLEEs which cannot be linearized directly have been solved via the Hirota method. [33][34][35] Coupled NLEEs have been considered, such as the generalized variable-coefficient Drinfeld-Sokolov-Satsuma-Hirota system, [36] Downloaded by [University of Nebraska, Lincoln] at 17:03 31 May 2016 coupled KdV-mKdV system, [37] and (2+1)-dimensional Boiti-Leon-Pempinelli system for water waves. [38] To our knowledge, analytic properties such as the soliton solutions and bilinear BT for Equation (3) have not been studied via the binary Bell polynomials.…”