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

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.

Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

“…For concrete details, see [30] where N-soliton solutions of the derived nonlocal integrable equations have also been systematically studied by solving Riemann-Hilbert problems. With the developments in fractional calculus [31], the question of how to extend the existing methods for fractional differential equations is a natural research trend [32][33][34][35][36][37][38].…”

Invariant Solutions and Conservation Laws of the Time-Fractional Telegraph Equation

“…Fractional derivatives supply a perfect and elegant tool to describe diverse materials and processes with memory and hereditary characteristic of a variety of different phenomena. By Xu et al [13], Zhang et al [14], Xu and Zhang [15], and Zhang et al [16], the analytical solutions of the fractional differential equations have been obtained. The telegraph equations describe the current and voltage of wave propagation of electric signals in a cable transmission line to find distance and time, and has many applications such as neutron transport [17], random walk of suspension flows [18], signal analysis for transmission, propagation of electrical signals [19] etc.…”

Application of Jordan canonical form and symplectic matrix in fractional differential models

“…Fractional calculus has played an irreplaceable role in many fields and attracted more and more attention [1][2][3][4][5][6][7][8][9][10][11][12][13]. In this paper, we mainly have two contributions.…”