Novel soliton motion with time-varying wave-width and amplitude as well as velocity through a reverse-space-time nonlocal variable-coefficient mKdV equation

Abstract

Nonlocal integrable systems have received widespread attention because of their special physical effects. In this article, we present a novel reverse-space-time nonlocal variable-coefficient mKdV (RSTVmKdV) equation and reveal its N-soliton solution by Riemann–Hilbert (RH) method. We notice that the revealed soliton solutions are tilted and exhibit the following behavioral characteristics: their wave width, amplitude, and velocity vary over time. This is novel for the isospectral soliton equation. In terms of specific content, firstly, we give the Lax pair of a system of coupled mKdV equations with time-varying coefficients and construct the related solvable RH problem. Then, we analyze the symmetry of eigenvalues $\eta \in C_{+}$ and $\overline{\eta }\in C_{-}$. Finally, we derive the general formula for the N-soliton solution and the specific expression for single soliton solution of the RSTVmKdV equation. Through mathematical analysis and simulation, we find that when the related eigenvalues ${\eta }_1$ and ${\overline{\eta }}_1$ are not conjugate, the single-soliton solution depicted is not symmetric, and when $|{\eta }_1|<|{\overline{\eta }}_1|$, the soliton tilts towards the negative x-axis, while when $|{\eta }_1|>|{\overline{\eta }}_1|$, the soliton tilts towards the positive x-axis. In addition, $\alpha \left(t\right)$ and $\beta \left(t\right)$ affect the amplitude of solitons in the case of $|{\eta }_1|\ne |{\overline{\eta }}_1|$, while $\alpha \left(t\right)$, $\beta \left(t\right)$, and $\gamma \left(t\right)$ affect the wave width, speed, and propagation direction of solitons. The results indicate that the method proposed in this paper can be used to derive and solve other reverse-space-time nonlocal integrable systems with time-varying coefficients and explore their rich soliton behavior.