Rogue wave solutions for the generalized fifth-order nonlinear Schrödinger equation on the periodic background

Rogue waves of the sixth-order nonlinear Schrödinger equation on a periodic background

“…In this paper, the rogue waves of the sixth-order NLS equation on the periodic background of Jacobian elliptic functions dn and cn are constructed by means of nonlinearization of a spectral problem and Darboux transformation approach. If we compare the results in this paper to the results of some other well-known NLS equations such as [5,[14][15][16], we find that all the above papers have similar expressions for solutions like (56), (58) and similar plots for solutions, which confirms the correctness of our results. However, our current study is just dependent upon the spectral problem in the AKNS system.…”

“…Such rogue waves are a kind of wave formed on the periodic background of the Jacobian elliptic functions dn and cn. By means of the nonlinearization of a spectral problem [6] and the Darboux transformation approach [7][8][9][10][11][12], periodic standing waves of various equations have been investigated, such as the mKdV equation [13], the NLS equation [5,[14][15][16], the fifth-order Ito equation [17], the sine-Gordon equation [18] and the Hirota equation [19].…”

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].…”

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

“…So far, several methods have been derived to obtain rogue waves on periodic background, such as DT method, algebraic geometry method, and PINN deep learning method [29][30][31][32]. The periodic background of elliptic function and traveling wave have been widely studied, and periodic rogue wave solutions of many nonlinear evolution equations have been constructed, such as sine-Gordon equation [33], Gerdjikov-Ivanov equation [34], Hirota equation [35] and higher-order NLS equations [30,[36][37][38][39].…”