Riemann-Hilbert approach for the nonlocal modified Korteweg-de Vries equation with a step-like oscillating background

TL;DR

This paper develops a Riemann-Hilbert framework for solving the nonlocal modified Korteweg-de Vries equation with step-like oscillating boundary conditions, introducing new two-soliton solutions for specific parameter regimes.

Contribution

It introduces a novel Riemann-Hilbert approach for the nonlocal mKdV equation with oscillating backgrounds and derives three new families of two-soliton solutions.

Findings

Established Riemann-Hilbert formalism for the problem.

Derived three new two-soliton solution families.

Analyzed solutions for different parameter regimes.

Abstract

This work focuses on the Cauchy problem for the nonlocal modified Korteweg-de Vries equation $$ u_t(x,t)+6u(x,t)u(-x,-t)u_x(x,t)+u_{xxx}(x,t)=0, $$ with the oscillating step-like boundary conditions: $u(x,t)\to 0$ as $x\to-\infty$ and $u(x,t)\backsimeq A\cos(2Bx+8B^3t)$ as $x\to\infty$, where $A,B>0$ are arbitrary constants. The main goal is to develop the Riemann-Hilbert formalism for this problem, paying a particular attention to the case of the ``pure oscillating step'' initial data, that is $u(x,0)=0$ for $x<0$ and $u(x,0)=A\cos(2Bx)$ for $x\geq0$. Also, we derive three new families of two-soliton solutions, which correspond to the values of $A$ and $B$ satisfying $B<\frac{A}{4}$, $B>\frac{A}{4}$, and $B=\frac{A}{4}$.