Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation

Abstract The inverse scattering transform of the coupled modified Korteweg–de Vries equation is studied by the Riemann–Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann–Hilbert problem is established for the equation. In the inverse scattering process, by solving Riemann–Hilbert problems corresponding to the reflectionless cases, three types of multi-soliton solutions are obtained. The multi-soliton classification is based on the zero structures of the Riemann–Hilbert problem. In addition, some figures are given to illustrate the soliton characteristics of the coupled modified Korteweg–de Vries equation.

[1]  A. Fordy,et al.  Generalised KdV and MKdV equations associated with symmetric spaces , 1987 .

[2]  Deng-Shan Wang,et al.  Prolongation structures and matter-wave solitons in F=1 spinor Bose-Einstein condensate with time-dependent atomic scattering lengths in an expulsive harmonic potential , 2014, Commun. Nonlinear Sci. Numer. Simul..

[3]  Ryogo Hirota,et al.  Soliton Solutions of a Coupled Modified KdV Equations , 1997 .

[4]  Ye Tian,et al.  Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects , 2014, Appl. Math. Comput..

[5]  Bo Xue,et al.  Explicit solutions and conservation laws of the coupled modified Korteweg–de Vries equation , 2015 .

[6]  M. Wadati,et al.  The Coupled Modified Korteweg-de Vries Equations , 1998, solv-int/9812003.

[7]  Ryogo Hirota,et al.  Exact Solution of the Modified Korteweg-de Vries Equation for Multiple Collisions of Solitons , 1972 .

[8]  D. J. Kaup,et al.  Squared Eigenfunctions for the Sasa-Satsuma Equation , 2009, 0902.1210.

[9]  Deng-Shan Wang,et al.  Integrable properties of the general coupled nonlinear Schrödinger equations , 2010 .

[10]  M. Wadati,et al.  The Modified Korteweg-de Vries Equation , 1973 .

[11]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[12]  Da‐jun Zhang,et al.  Solutions to the modified Korteweg-de Vries equation , 2012, 1203.5851.

[13]  Xianguo Geng,et al.  Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy , 2014 .

[14]  Lixin Tian,et al.  Orbital stability and asymptotic stability of mKdV breather-type soliton solutions , 2017, Commun. Nonlinear Sci. Numer. Simul..

[15]  X. Geng,et al.  Riemann–Hilbert approach and N-soliton solutions for a generalized Sasa–Satsuma equation , 2016 .

[16]  Jianke Yang,et al.  Nonlinear Waves in Integrable and Nonintegrable Systems , 2010, Mathematical modeling and computation.

[17]  N. Yajima,et al.  A Class of Exactly Solvable Nonlinear Evolution Equations , 1975 .