Solitons, breathers and periodic rogue waves for the variable-coefficient seventh-order nonlinear Schrödinger equation

Abstract

Through Darboux transformation (DT) method, Several nonlinear wave solutions of seventh-order variable-coefficient nonlinear Schrödinger (vcNLS) equation are obtained, including solitons, breathers and rogue periodic waves. When the coefficients are linear, parabolic and periodic functions, the parabolic, cubic and quasi-periodic solitons and breathers can be constructed. Then we investigate
their effects on the solutions, the variation of the coefficients affects the shapes of the solutions. On this basis, the interactions between two solitons are studied and the interactions between two types of breathers are constructed. Next, through the approaches of the nonlinearization of spectral problem and DT method, rogue waves on the background of the Jacobi elliptic functions dn and cn for a seventh-order vcNLS equation are constructed. When the coefficients are selected as linear, exponential and periodic functions, the nonlinear dynamics of two kinds of rogue periodic waves are analysed.

Rogue waves for the (2+1)-dimensional Myrzakulov–Lakshmanan-IV equation on a periodic background

“…They combined the method of the nonlinearization of the Lax pair with the Darboux transformation to obtain the rogue periodic wave of the focused nonlinear Schrödinger (NLS) equation [5]. Then, rogue wave on a periodic background of the modified Korteweg-de Vries (mKdV) equation [6,7], Ito equation [8], fourth-, fifth-, sixth-, seven-order NLS equation [9][10][11][12], the sine-Gordon equation [13], and the Hirota equation [14,15] has been studied similarly. In recent years, the same method has been used to study the (2+1) dimensional nonlinear evolution equation [16,17].…”