Exact solutions of a local fractional nonisospectral complex mKdV equation based on Riemann–Hilbert method with time-varying spectrum

Abstract

This article combines the Riemann–Hilbert method with fractional power-law time-varying spectrum for the first time to solve a time fractional nonisospectral complex mKdV (tfniscmKdV) equation. Firstly, the tfniscmKdV equation and its associated Lax pair are proposed. Then, based on the Lax pair, a solvable Riemann–Hilbert problem without the explicit involvement of time variable is constructed to link a formal solution of the tfniscmKdV equation. By considering the involvement of time variable, the scattering data for recovering solution of the tfniscmKdV equation is further derived. Finally, in the case of zero reflection coefficient constants, the fractional N-soliton solution of the tfniscmKdV equation is obtained, and the evolution characteristics of fractional solitons, such as variable velocity, amplitude, wave width, shorter duration, and evolutionary symmetry, are revealed by simulating some novel spatial structures of the fractional single- and double-soliton solutions. Our research indicates that the approach of extending the Riemann–Hilbert method in this article is also worth referring to for other inverse scattering integrable systems in terms of time fractional order.

Keywords

• Riemann–Hilbert method combined with fractional power-law time-varying spectrum
• Time fractional nonisospectral complex mKdV equation
• Riemann–Hilbert problem
• Scattering data
• Fractional N-soliton solution