Eur. Phys. J. Plus (2024) 139: 603
https://doi.org/10.1140/epjp/s13360-024-05382-x

Regular Article

A Riemann–Hilbert approach for the focusing and defocusing mKdV equation with asymmetric boundary conditions in Few-Cycle Pulses

School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, 100875, Beijing, China

^a zhaoy23@bnu.edu.cn

Received: 30 April 2024
Accepted: 17 June 2024
Published online: 9 July 2024

Abstract

In this paper, we study the Riemann–Hilbert approach for the focusing and defocusing mKdV equation with fully asymmetric nonzero boundary conditions. The mKdV equation serves as the universal model for Few-Cycle Pulses within the long-wave approximation regime, applicable to optical pulse propagation in a medium governed by a two-level Hamiltonian without relying on the slowly varying envelope approximation. Our analysis focuses on the properties of the Jost function, allowing us to establish the associated Riemann–Hilbert problem for both the focusing and defocusing cases. Furthermore, we recover the potential from the solution of the Riemann–Hilbert problem, which plays a crucial role in calculating the long-time asymptotic behavior of the solution. In the symmetric case, the eigenvalues are given by ${\lambda }^2=k^2\pm q_0^2$. However, in the asymmetric case, the eigenvalues are represented as ${\lambda }_{\pm }^2=k^2\pm q_{\pm }^2$. As a result, we directly handle the branch cut instead of utilizing the four-sheeted Riemann surface. Naturally, this approach leads to the discontinuity of the functions across their respective branch cuts, which significantly impacts the entire development of the Riemann–Hilbert problem.