相关论文: A new matrix modified Korteweg-de Vries equation: …

We focus on the semi-discrete complex modified Korteweg-de Vries (DcmKdV) equation in this paper. The direct and inverse scattering theory is developed with zero and non-zero boundary conditions (BCs) of the potential. For direct problem,…

We study the plus and minus type discrete mKdV equation. Some different symmetry conditions associated with two Lax pairs are introduced to derive the matrix Riemann-Hilbert problem with zero. By virtue of regularization of the…

We delve into the inverse scattering transform of the real-valued vector modified Korteweg--de Vries equation, emphasizing the challenges posed by $N$ pairs of higher-order poles in the transmission coefficient and the enhanced spectral…

In this work, we mainly study the general $N$-soliton solutions of the nonlocal modified Korteweg-de Vries (mKdV) equation by utilizing the Riemann-Hilbert (RH) method. For the initial value belonging to Schwarz space, we firstly obtain the…

The theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2m\times2m$ $(m\geq 1)$ Lax pairs under the nonzero boundary conditions at infinity. In…

New reductions for the multicomponent modified Korteveg-de Vries (MMKdV) equations on the symmetric spaces of {\bf DIII}-type are derived using the approach based on the reduction group introduced by A.V. Mikhailov. The relevant inverse…

The soliton resolution for the focusing modified Korteweg-de vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation…

We take a closer look at the Riemann-Hilbert problem associated to one-gap solutions of the Korteweg-de Vries equation. To gain more insight, we reformulate it as a scalar Riemann-Hilbert problem on the torus. This enables us to derive…

Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Baecklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified…

In this work, we extend the Riemann-Hilbert (RH) method in order to study the coupled modified Korteweg-de Vries equation (cmKdV) under nonzero boundary conditions (NZBCs), and successfully find its solutions with their various dynamic…

In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit $\lambda_{j}$ $\rightarrow$ $\lambda_{1}$ of the Lax pair…

We explore systematically a rigorous theory of the inverse scattering transforms with matrix Riemann-Hilbert problems for both focusing and defocusing modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at…

An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in Ref. 7 (S. Carillo, M. Lo Schiavo, and C. Schiebold. Matrix solitons solutions of the modified Korteweg-de Vries…

We formulate the inverse spectral theory of infinite gap Hill's operators with bounded periodic potential as a Riemann--Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for…

The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include…

The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo\-di\-fied Kor\-te\-weg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the $n$-soliton solutions of the mKdV…

We consider perturbations of the special pole-free joint solution $U(x,t)$ of the Korteweg--de Vries equation $u_t+uu_x+\frac{1}{12}u_{xxx}=0$ and $P_I^2$ equation $u_{xxxx}+10u_x^2+20uu_{xx}+40(u^3-6tu+6x)=0$ under the action of the KdV…

We demonstrate the existence of complex solitary wave and periodic solutions of the Kortweg de-vries (KdV) and modified Kortweg de-Vries (mKdV) equations. The solutions of the KdV (mKdV) equation appear in complex-conjugate pairs and are…

A formula for certain exact solutions to the modified Korteweg-de Vries (mKdV) equation is obtained via the inverse scattering transform method. The kernel of the relevant Marchenko integral equation is written with the help of matrix…

We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg--de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to $-\infty$. This…