Inverse scattering transform for the fourth-order nonlinear Schrödinger equation with fully asymmetric non-zero boundary conditions

Abstract

The inverse scattering transform is utilized to address the initial-value problem for the fourth-order nonlinear Schrödinger equation characterized by fully asymmetric non-zero boundary conditions. This work considers the fully asymmetric scenario for both asymptotic amplitudes and phases. The direct problem demonstrates the establishment of the corresponding analytic properties of eigenfunctions and scattering data. The inverse scattering problem is approached using both (left and right) Marchenko integral equations and is also formulated as the Riemann–Hilbert problem on a single sheet of the scattering variable. The temporal evolution of the scattering coefficients is subsequently deduced, revealing that in contrast to solutions with uniform amplitudes, both reflection and transmission coefficients exhibit a nontrivial time dependency here. The findings of this paper are expected to be pivotal for exploring the long-time asymptotic behavior of the fourth-order nonlinear Schrödinger solutions with significant boundary conditions.