Nonlocal modified KdV equations and their soliton solutions by Hirota Method

• Published in: Communications in nonlinear science & numerical simulation 2019-02, Vol.67, p.427-448
• Main Authors: Gürses, Metin, Pekcan, Aslı
• Format: Article
• Language: English
• Subjects: Ablowitz–Musslimani reduction; Graph theory; Hirota bilinear form; Korteweg-Devries equation; Mathematical analysis; Nonlinear equations; Nonlocal mKdV equations; Simulation; Soliton solutions
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2018.07.013

•Soliton solutions of the coupled mKdV system are given by using the Hirota direct method.•Search for possible integrable nonlocal reductions of the mKdV system.•Introducing a general method for finding soliton solutions of nonlocal integrable equations.•By using the Ablowitz–Musslimani reduction formulas soliton solutions of nonlocal mKdV and complex mKdV equations are presented.•Particular examples of solutions of nonlocal mKdV equations are given. We study the nonlocal modified Korteweg–de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz–Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz–Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of nonlocal mKdV and nonlocal complex mKdV equations. The soliton solutions of these equations are of two types. We give one-soliton solutions of both types and present only first type of two- and three-soliton solutions. We illustrate our solutions by plotting their graphs for particular values of the parameters.