Darboux transformation and soliton solutions of a nonlocal Hirota equation

Abstract

Starting from local coupled Hirota equations, we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem. The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair. By Lax pair, an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found. The solutions with specific properties are distinct from those of the local Hirota equation. In order to further describe the properties and the dynamic features of the solutions explicitly, several kinds of graphs are depicted.

Superposed periodic kink and pulse solutions of coupled nonlinear equations

“…nonlocal mKdV [25] and nonlocal Hirota [26,27] equations as well as the coupled nonlocal Ablowitz-Musslimani variant of NLS and the coupled nonlocal mKdV [28].…”

Generalized perturbation (n, N − n) fold Darboux transformation for a nonlocal Hirota equation with variable coefficients

“…Zuo et al obtained the higher order solutions of nonlocal Hirota equation via the Hirota bilinear method [34]. Xia et al obtained the higher order soliton solutions of nonlocal Hirota equation by applying the classical Darboux transformation method [35]. Yang et al constructed the infinitely-many conservation laws based on the Lax pair and applying Darboux transformation, they derived the three-soliton solutions, the higher-order breather solutions and breather-to-soliton transition condition of the Hirota equation with variable coefficients [36].…”

Trajectory equation of a lump before and after collision with other waves for generalized Hirota–Satsuma–Ito equation

“…[1][2][3] The exact solution of NLPDEs can describe various physical properties exhibited by the equation, therefore, the search for effective methods of solving NLPDEs has received a lot of attention. Until now, many useful methods have been proposed, such as Darboux transformation, [4,5] Bäcklund transformation, [7] inverse scattering transformation, [8] variable separation method, [9,10] Hirota bilinear method, [11][12][13] etc. Among them, Hirota bilinear method is most studied by many scholars owing to its simplicity and directness.…”