Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Bo Xu,Sheng Zhang

doi:10.3390/sym13091593

Abstract

Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.

Highlights

The two-scale fractal theory developed by He et al [17,18,19] is effectively established for dealing with many discontinuous problems; the local fractional derivative (LFD) [5,20,21] can be adopted to solve some non-differentiable problems
When the time variable t is limited to the Cantor set, we show in Figures 1–5 the one-fractal-soliton solution (67) by selecting the parameters ξ = 0.5, η = 1, k0 = 1.5, and δ0 = 1
Infinitely many conservation laws of Equation (3), determined by Equation (68), one end of which is the partial derivative of the local fractional order in time, and the other end is the partial derivative of integral order in space, are obtained

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In 2017, Yang et al [28] presented a family of special functions defined on the fractal sets and obtained non-differentiable exact solutions of some nonlinear local fractional ordinary differential equations. Its generalized forms [42,43,44,45] were solved by the RH approach [33], to our knowledge, the local time-fractional NLS-type Equation (3) has not been studied by such a method. In this paper, referring to the idea of the RH approach [33], we Symmetry 2021, 13, 1593 shall transform the solution of Equation (3) into the solution of a related RH problem, and derive the long-time asymptotic solution and N-fractal-soliton solution of Equation (3).

Lax Pair and the Related RH Problem

Solutions of the Related RH Problem and Time-Dependence of Scattering Data

Long-Time Asymptotic Solution and N-Fractal-Soliton Solution

Dynamical

Infinitely

Conclusions