The multi-triple-pole solitons for the focusing mKdV hierarchy with nonzero boundary conditions

In this paper, the general triple-pole multi-soliton solutions are proposed for the focusing modified Korteweg–de Vries (mKdV) equation with both nonzero boundary conditions (NZBCs) and triple zeros of analytical scattering coefficients by means of the inverse scattering transform. Furthermore, we also give the corresponding trace formulae and theta conditions. Particularly, we analyze some representative reflectionless potentials containing the triple-pole multi-dark-anti-dark solitons and breathers. The idea can also be extended to the whole mKdV hierarchy (e.g. the fifth-order mKdV equation, and third-fifth-order mKdV equation) with NZBCs and triple zeros of analytical scattering coefficients. Moreover, these obtained triple-pole solutions can also be degenerated to the triple-pole soliton solutions with zero boundary conditions.

A Integrable Generalized Super-NLS-mKdV Hierarchy, Its Self-Consistent Sources, and Conservation Laws

A generalized super-NLS-mKdV hierarchy is proposed related to Lie superalgebra B(0,1); the resulting supersoliton hierarchy is put into super bi-Hamiltonian form with the aid of supertrace identity. Then, the super-NLS-mKdV hierarchy with self-consistent sources is set up. Finally, the infinitely many conservation laws of integrable super-NLS-mKdV hierarchy are presented.

On Type-II Bäcklund Transformation for the MKdV Hierarchy

The study of new integrable defects leads to new type of Bäcklund transformations named as the type-II Bäcklund transformations. In this article, we show, for the MKdV hierarchy, that the type-II Bäcklund transformation is the compound type-I Bäcklund transformation.

Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients

In this paper, Hirota’s bilinear method is extended to a new modified Kortweg–de Vries (mKdV) hierarchy with time-dependent coefficients. To begin with, we give a bilinear form of the mKdV hierarchy. Based on the bilinear form, we then obtain one-soliton, two-soliton and three-soliton solutions of the mKdV hierarchy. Finally, a uniform formula for the explicit N-soliton solution of the mKdV hierarchy is summarized. It is graphically shown that the obtained soliton solutions with time-dependent functions possess time-varying velocities in the process of propagation.