Inverse scattering transform for a defocusing local–nonlocal Lakshmanan–Porsezian–Daniel equation under nonzero boundary conditions: N-soliton solutions and asymptotic analysis

Abstract

In this paper, we present the inverse scattering transform for a defocusing local–nonlocal Lakshmanan–Porsezian–Daniel equation under nonzero boundary conditions. By characterizing the analytic and symmetric properties of the Jost eigenfunctions and scattering data, an adjoint spectral problem with auxiliary eigenfunctions is introduced, thereby constructing a matrix Riemann–Hilbert formula. In particular, the reverse-space nonlocal reduction introduces an additional spectral symmetry, resulting in coupling relations among the eigenfunctions, scattering data and norming constants. The residue conditions, trace formula and theta condition are then obtained for scattering data with an arbitrary number of simple zeros. When the reflection coefficients vanish, the Riemann–Hilbert problem is reduced to a linear algebraic system, from which determinant form of the N-soliton solutions is derived. Then, several types of localized waves are derived, including dark solitons generated by on-circle eigenvalues, beating solitons associated with off-circle quartets, and dark-beating interactions under suitable spectral constraints. In addition, the asymptotic behaviors are investigated for two typical interaction patterns, namely the interaction between two beating solitons and the interaction among one non-propagating dark soliton and two moving dark solitons. The asymptotic analysis shows that corresponding solutions are characterized by different asymptotic solutions in distinct regions. Moreover, off-circle eigenvalue quartets and on-circle eigenvalues generate different asymptotic wave structures, yielding abundant interaction patterns. These results enrich our understanding of the nonlinear wave dynamics arising from the nonzero background, reverse-space nonlocality and higher-order effects.

Inverse scattering transform for the defocusing local–nonlocal Lakshmanan–Porsezian–Daniel equation with nonzero boundary conditions. A Construction of the Riemann–Hilbert problem and reflection less soliton solutions. B Symmetric discrete spectrum for the mixed interaction shown in (C). C Interaction of dark solitons with a localized beating structure. D Classification of soliton structures according to the location of discrete eigenvalues relative to the spectral circle. E Comparison of exact and asymptotic profiles before and after interaction