N-soliton solutions of an integrable equation studied by Qiao

Abstract

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 ( 2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.

Dark solitons of the Qiao's hierarchy

“…The time-dependence of the scattering data is given by (31). The N-soliton solution then is given in parametric form by (17) and (18) for the eigenfunction (32). The condition 0 < κ n < m −1 0 is sufficient to ensure smoothness of the solitons.…”

“…The (white) soliton solutions of (1) and (5) have been found previously [15,17]. These studies rely on the fact that the spectral problem for (1) is gauge-equivalent to the one for the mKdV equation.…”

Multi-soliton solutions for the coupled modified nonlinear Schrödinger equations via Riemann–Hilbert approach

“…Due to the profound theoretical and practical significance, the study of multi-soliton solutions of nonlinear partial differ-ential equations has always been a hot topic. To date, a variety of approaches have been well established to investigate the multi-soliton solutions of nonlinear partial differential equations, such as the inverse scattering method, [9] the Hirota's bilinear method, [10] the Darboux transformation method, [11][12][13] the Riemann-Hilbert approach, [14] the KP hierarchy reduction method, [15][16][17] the generalized unified method, [18,19] and so forth. The Riemann-Hilbert approach as a very powerful approach in dealing with nonlinear partial differential equations with initial-boundary value conditions has attracted more and more attention in recent years.…”

Some properties for a new integrable soliton equation

“…In fact, under some appropriate transformations of coordinates, equations (1.2)-(1.4) are equivalent. In recent years, equation (1.4) has attracted much attention in soliton theory [14][15][16][17][18][19].…”