Elliptic‐ and hyperbolic‐function solutions of the nonlocal reverse‐time and reverse‐space‐time nonlinear Schrödinger equations

• Published in: Mathematical methods in the applied sciences 2022-11, Vol.45 (17), p.10877-10890
• Main Authors: Li, Bo‐Wen, Xu, Tao, Zhang, Tian‐Li, An, Li‐Cong, Chen, Yang
• Format: Article
• Language: English
• Subjects: Hyperbolic functions; hyperbolic‐function solutions; Infinity; Jacobian elliptic‐function solutions; Mathematical analysis; nonlocal nonlinear Schrödinger equation; Schrodinger equation; Solitary waves
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8422

In this paper, we obtain the stationary elliptic‐ and hyperbolic‐function solutions of the nonlocal reverse‐time and reverse‐space‐time nonlinear Schrödinger (NLS) equations based on their connection with the standard Weierstrass elliptic equation. The reverse‐time NLS equation possesses the bounded dn$$ \mathrm{dn} $$‐, cn$$ \mathrm{cn} $$‐, sn$$ \mathrm{sn} $$‐, sech$$ \operatorname{sech} $$‐, and tanh$$ \tanh $$‐function solutions. Of special interest, the tanh$$ \tanh $$‐function solution can display both the dark‐ and antidark‐soliton profiles. The reverse‐space‐time NLS equation admits the general Jacobian elliptic‐function solutions (which are exponentially growing at one infinity or display the periodical oscillation in x$$ x $$), the bounded dn$$ \mathrm{dn} $$‐ and cn$$ \mathrm{cn} $$‐function solutions, as well as the K$$ K $$‐shifted dn$$ \mathrm{dn} $$‐ and sn$$ \mathrm{sn} $$ function solutions. In addition, the hyperbolic‐function solutions may exhibit an exponential growth behavior at one infinity, or show the gray/bright‐soliton profiles.