Inverse scattering transforms and soliton solutions of nonlocal reverse‐space nonlinear Schrödinger hierarchies

Abstract

The aim of the paper is to construct nonlocal reversespace nonlinear Schrödinger (NLS) hierarchies through nonlocal group reductions of eigenvalue problems and generate their inverse scattering transforms and soliton solutions. The inverse scattering problems are formulated by Riemann-Hilbert problems which determine generalized matrix Jost eigenfunctions. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A solution formulation to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless transforms, is presented and applied to-soliton solutions of the nonlocal NLS hierarchies.

The multi-component nonisospectral KdV hierarchies associated with a novel kind of N-dimensional Lie algebra

“…In fact, the idea and the method used here are universal for other isospectral and nonisospectral problems. The Riemann-Hilbert method for multi-component systems based on higher-order matrix spectral problems has been discussed (see [48][49][50]). It is quite intriguing for us to consider the application of the Riemann-Hilbert approach to the multi-component KdV systems.…”

“…which is equivalent to the coupled nonisospectral KdV hierarchy (36). Based on ( 39)-( 40), we select α 1 = 1, α 2 = 0, and then (49) becomes…”

Integrable nonlocal nonlinear Schrödinger equations associated with 𝑠𝑜(3,ℝ)

“…The resulting model equations can relate function values at point x in space to its function values at a mirror-reflection space point -x [4]. This has opened new avenues for studying NLS type integrable equations [20]. One popular example of nonlocal dynamics is pantograph modeling, which has long history in pantograph mechanics and pantograph transport [5].…”

Nonlocal PT-Symmetric Integrable Equations of Fourth-Order Associated with so(3, ℝ)

“…Associated with simple Lie algebras, the wellknown integrable equations include the KdV equation, the nonlinear Schrödinger equation and the derivative nonlinear Schrödinger equation [6][7][8]. Nonlocal integrable equations have also been recently explored in soliton theory, including scalar equations [9,10] and vector generalizations (see, e.g., [11,12]).…”