N-fold Darboux transformation and explicit solutions in terms of the determinant for the three-field Blaszak–Marciniak lattice

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

“…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].…”

Modulation instability and rogue waves for the sixth-order nonlinear Schro¨dinger equation with variable coefficients on a periodic background

“…In the current soliton theory, the problem of rogue wave on a periodic background is a subject with considerable attention. By means of the nonlinearization of spectral problem [14] and Darboux transformation approach [15][16][17][18][19], periodic standing waves of various equations have been investigated, such as mKdV equation [20], the NLS equation [21][22][23][24][25][26], the Hirota equation [27], the sine-Gordon equation [28] and the fifth-order Ito equation [29]. However, all the above mentioned are equations with constant coefficients, and in this paper we focus on a variable-coefficient equation.…”