Riemann–Hilbert problems and N-soliton solutions for a coupled mKdV system

Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions

A class of Riemann–Hilbert problems on the real axis is formulated for solving the multicomponent AKNS integrable hierarchies associated with a kind of bock matrix spectral problems. Through special Riemann–Hilbert problems where a jump matrix is taken to be the identity matrix, soliton solutions to all integrable equations in each hierarchy are explicitly computed. A class of specific reductions of the presented integrable hierarchies is also made, together with its reduced Lax pairs and soliton solutions.

Application of the Riemann–Hilbert approach to the multicomponent AKNS integrable hierarchies

We present an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4 × 4 matrix spectral problem. An integrable coupled mKdV hierarchy with three potentials is first generated. Based on the corresponding oscillatory Riemann-Hilbert problem, the leading asympototics of the three-component mKdV system is then evaluated by using the nonlinear steepest descent method.

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Long-Time Asymptotics of a Three-Component Coupled mKdV System

The BO equation depicts the evolution of algebraic Rossby solitary waves in ocean and atmosphere. But, despite its overt fame, the BO equation is restricted as a two-dimensional model. On the basis of the great success in the soliton theory, a lot of works have recently been directed to three-dimensional models and investigations of solitary waves properties in three-dimensional systems. In this paper, a new ZK–BO equation for three-dimensional algebraic Rossby solitary waves is derived by employing perturbation expansions and stretching transformations of time and space. By virtue of the trial function method, the exact solution of the ZK–BO equation is presented. By comparing the solution with the solution of two-dimensional algebraic Rossby solitary waves, we can find that the wave length of three-dimensional algebraic Rossby solitary waves is shorter while the wave amplitude is higher. Based on the exact solution of ZK–BO equation, the dissipation effect is studied. The results reveal the influence of dissipation effect on the three-dimensional algebraic solitary waves. Further, after theoretical analysis, the conservation laws of three-dimensional algebraic Rossby solitary waves are discussed and the fission property is studied.

A new ZK–BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property

In the paper, based on the modified Riemann–Liouville fractional derivative and Tu scheme, the fractional super NLS–MKdV hierarchy is derived, especially the self-consistent sources term is considered. Meanwhile, the generalized fractional supertrace identity is proposed, which is a beneficial supplement to the existing literature on integrable system. As an application, the super Hamiltonian structure of fractional super NLS–MKdV hierarchy is obtained.

Bao Shu Yin

Papers

Generalized fractional supertrace identity for Hamiltonian structure of NLS–MKdV hierarchy with self-consistent sources