[Paper Review] Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform

Jia-Liang Ji, Zuo-Nong Zhu|arXiv (Cornell University)|Mar 13, 2016

Nonlinear Waves and Solitons2 references3 citations

TL;DR

This paper presents exact soliton and breather solutions for an integrable nonlocal modified Korteweg-de Vries (mKdV) equation using the inverse scattering transform (IST). It derives one-soliton, two-soliton, and breather solutions with novel properties—such as exponential amplitude growth or decay and phase shifts—distinguishing them from classical mKdV solutions, particularly under nonlocal PT-symmetric reductions.

ABSTRACT

It is well known that the nonlinear Schrödinger (NLS) equation is a very important integrable equation. Ablowitz and Musslimani introduced and investigated an integrable nonlocal NLS equation through inverse scattering transform. Very recently, we proposed an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can also be found in a paper of Ablowitz and Musslimani. We have constructed the Darboux transformation and soliton solutions for the nonlocal mKdV equation. In this paper, we will investigate further the nonlocal mKdV equation. We will give its exact solutions including soliton and breather through inverse scattering transformation. These solutions have some new properties, which are different from the ones of the mKdV equation.

Motivation & Objective

• To investigate the integrable nonlocal mKdV equation through the inverse scattering transform (IST), extending prior work on Darboux transformations.
• To derive exact solutions, including solitons and breathers, for the nonlocal mKdV equation with nonlocal PT-symmetry.
• To analyze the long-time behavior and singularity structure of solutions, highlighting differences from the classical mKdV equation.
• To demonstrate that the nonlocal mKdV equation supports soliton solutions with exponential amplitude growth or decay, unlike the classical case.
• To provide a systematic construction of solutions via IST, including the derivation of Wronskian identities and spectral data relations.

Proposed method

• Formulates the Lax pair for the nonlocal mKdV equation using a 2×2 matrix system with spectral parameter $k$, involving $\sigma_3$, $\mathbf{Q}$, and auxiliary matrices $\mathbf{V}_1$, $\mathbf{V}_2$.
• Applies the inverse scattering transform by defining Jost solutions $\phi$, $\bar{\phi}$, $\psi$, $\bar{\psi}$ with specific asymptotic behaviors at $x \to \pm\infty$.
• Derives the scattering data via Wronskian relations and spectral symmetry, including $a(k)$, $b(k)$, and their complex conjugates.
• Constructs the Gelfand–Levitan–Marchenko integral equation to reconstruct the potential $q(x,t)$ from scattering data.
• Imposes the nonlocal reduction $r(x,t) = -q(-x,-t)$ to recover the nonlocal mKdV equation from the Lax pair system.
• Solves the inverse problem to obtain one-soliton, two-soliton, and breather solutions in terms of exponential and hyperbolic/trigonometric functions of $x$ and $t$.

Experimental results

Research questions

• RQ1 How do soliton solutions of the nonlocal mKdV equation differ in behavior from those of the classical mKdV equation?
• RQ2 What is the long-time asymptotic behavior of soliton solutions under the nonlocal PT-symmetric reduction?
• RQ3 Under what conditions do singularities appear in the solutions of the nonlocal mKdV equation?
• RQ4 Can breather solutions be constructed for the nonlocal mKdV equation, and what are their structural properties?
• RQ5 How does the inverse scattering transform apply to nonlocal integrable systems with nonlocal reductions?

Key findings

• One-soliton solutions exhibit exponential amplitude decay or growth depending on the sign of $\alpha - \beta$, with solutions decaying when $\alpha < \beta$ and growing when $\alpha > \beta$.
• Two-soliton solutions of the bright-bright type show asymmetric amplitude evolution: one soliton may grow exponentially while the other decays, depending on the parameters $\alpha_j$, $\beta_j$.
• In the bright-dark soliton case, both solitons can exhibit exponential growth or decay, with phase shifts occurring during interaction but no change in speed.
• Breather solutions exist only under specific symmetry conditions: $k_1 = -k_2^*$ and $\bar{k}_1 = -\bar{k}_2^*$, with a special case yielding a periodic, localized breather when $\eta_1 = \eta_2 = \zeta_1 = \zeta_2 = \mu$.
• The breather solution is given explicitly by $q(x,t) = 4\mu \frac{\sinh(\xi_+ among \sin(\xi_-) - \cosh(\xi_+)\cos(\xi_-)}{\cosh^2(\xi_+) + \sin^2(\xi_-)}$ with $\xi_\pm = -2\mu(x \pm 8\mu^2 t)$, showing periodic oscillations in space and time.
• Solutions become singular if $\eta_1 \neq \eta_2$ or $\zeta_1 \neq \zeta_2$ in the breather case, but remain regular under symmetric parameter choices.