Coupled derivative nonlinear Schrödinger III equation: Darboux transformation and higher-order rogue waves in a two-mode nonlinear fiber

Localized wave solutions to the vector nonlinear Schrödinger equation with nonzero backgrounds

“…The authors gave the explicit expression of eigenfunction and analyzed the dynamics of the soliton–RW solutions by choosing different parameters. We can see that the eigenfunctions have been the key ingredient for constructing new solutions to nonlinear integrable systems such as Sasa–Satsuma equation [47], coupled GI equation [48–50], and three‐wave resonance equation [51]. In this work, starting with the nonzero seed solution in the exponential form of variable $$$ t $$$, we will give the eigenfunction of the $$$ n $$$‐component NLS equation on the basis of the Cayley–Hamilton theorem.…”

Darboux transformation, infinite conservation laws, and exact solutions for the nonlocal Hirota equation with variable coefficients

“…[4][5][6][7][8] Through the concerted efforts of many researchers, many effective methods have been developed, such as the Hirota bilinear method, [9][10][11] the Lie symmetry analysis method, [12] the Riemann-Hilbert method, [13,14] neural network method, [15] and the Darboux transformation. [16][17][18] The Darboux transformation is a canonical transformation of the spectral problem for NLEEs. The idea is to construct exact solutions of the integrable equations by means of solutions of the eigenfunctions of the NLEEs associated with the Lax pair.…”

The multi elliptic‐localized solutions and their asymptotic behaviors for the mKdV equation

“…Then, a large number of articles have appeared on equations with solutions expressed as elliptic functions, such as the Hirota equation, 31,32 the sine-Gordon equation, 33 and the derivative NLS equation. 34 In the past 40 years, a large number of scholars focused on utilizing the theta functions to represent the solutions under the background of elliptic functions. In the 1980s, Its, Rybin, and Sail 35 used the algebro-geometric approach to construct the finite-gap solutions of the NLS equation in terms of the theta functions.…”

“…In the past 2 years, an increasing number of scholars have focused on solutions under the background of periodic functions. Then, a large number of articles have appeared on equations with solutions expressed as elliptic functions, such as the Hirota equation, 31,32 the sine‐Gordon equation, 33 and the derivative NLS equation 34 …”