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

Yan Rybalko

TL;DR

This work addresses the Cauchy problem for the nonlocal mKdV equation $u_t+6u(x,t)u(-x,-t)u_x+u_{xxx}=0$ with oscillating step-like boundaries, and develops a rigorous Riemann-Hilbert formulation via the inverse scattering transform. The authors carefully characterize the spectral data associated to step-like backgrounds, including zeros of $a_1$ and $a_2$ and their behavior near $k=\pm B$, and show that, for small perturbations of the pure step, the data are determined by a single function $b(k)$ with determinant constraints. They classify spectral scenarios into several cases and provide trace formulas that express $a_1$ and $a_2$ in terms of $b(k)$, enabling a complete basic RH problem from which the solution $u(x,t)$ can be reconstructed. In the reflectionless setting, they construct three new families of two-soliton solutions with explicit formulas and demonstrate blow-up along certain curves, accompanied by detailed large-time asymptotics featuring decaying, transition, and periodic regimes. The results extend the Riemann-Hilbert approach to a nonlocal, two-place integrable system with oscillating backgrounds, revealing rich soliton dynamics with PT-symmetric structure.

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 $B=\frac{A}{4}$.

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

TL;DR

This work addresses the Cauchy problem for the nonlocal mKdV equation with oscillating step-like boundaries, and develops a rigorous Riemann-Hilbert formulation via the inverse scattering transform. The authors carefully characterize the spectral data associated to step-like backgrounds, including zeros of and and their behavior near , and show that, for small perturbations of the pure step, the data are determined by a single function with determinant constraints. They classify spectral scenarios into several cases and provide trace formulas that express and in terms of , enabling a complete basic RH problem from which the solution can be reconstructed. In the reflectionless setting, they construct three new families of two-soliton solutions with explicit formulas and demonstrate blow-up along certain curves, accompanied by detailed large-time asymptotics featuring decaying, transition, and periodic regimes. The results extend the Riemann-Hilbert approach to a nonlocal, two-place integrable system with oscillating backgrounds, revealing rich soliton dynamics with PT-symmetric structure.

Abstract

This work focuses on the Cauchy problem for the nonlocal modified Korteweg-de Vries equation with the oscillating step-like boundary conditions: as and as , where 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 for and for . Also, we derive three new families of two-soliton solutions, which correspond to the values of and satisfying , , and .
Paper Structure (8 sections, 14 theorems, 203 equations, 3 figures)

This paper contains 8 sections, 14 theorems, 203 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Inverse scattering transform and the Riemann-Hilbert problem
  3. Eigenfunctions
  4. Scattering data
  5. Pure step initial data
  6. Assumptions on the spectral data
  7. Basic Riemann-Hilbert problem
  8. Two-soliton solutions

Key Result

Proposition 2.1

Assume that $xu(x,t)\in L^1(-\infty,a)$ and $\left( u(x,t)-A\cos(2Bx+8B^3t) \right)\in L^1(a,\infty)$ with respect to the spatial variable $x$, for all fixed $t,a\in\mathbb{R}$. Then the matrices $\Psi_j(x,t,k)$, $j=1,2$, given in Psi1--Psi2, have the following properties:

Figures (3)

  • Figure 1: The graphs of the two-soliton solutions, which correspond to two pure imaginary simple zeros at $k=\mathrm{i} k_j$, $j=1,2$, of $a_1(k)$, see \ref{['zrlca']}. These graphs are plotted from the exact formula \ref{['usI']} with $A=1$, $B=0.243$, and $(\gamma_1,\gamma_2)=(1,1)$ (top left), $(\gamma_1,\gamma_2)=(1,-1)$ (top right), $(\gamma_1,\gamma_2)=(-1,1)$ (bottom left), and $(\gamma_1,\gamma_2)=(-1,-1)$ (bottom right).
  • Figure 2: The graphs of the two-soliton solutions, which correspond to two simple zeros at $k=p_1$ and $k=-\bar{p}_1$ of $a_1(k)$, see \ref{['zrlcb']}. These graphs are plotted from the exact formula \ref{['usII']} with $A=1$, $B=0.26$, and $\eta_1=1$ (left), and $\eta_1=-1$ (right).
  • Figure 3: The graphs of the two-soliton solutions, which correspond to one double zero at $k=\mathrm{i}\ell_1$ of $a_1(k)$, see \ref{['zrlcc']}. These graphs are plotted from the exact formula \ref{['usIII']} with $A=1$, $B=\frac{1}{4}$, and $\nu_1=1$ (left), and $\nu_1=-1$ (right).

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.1: Conservation law
  • Proposition 2.3
  • proof
  • Proposition 2.4: $a_j(k)$ in terms of $b(k)$, Cases I--III
  • proof
  • Remark 2.2
  • ...and 19 more