Darboux transformation of the variable coefficient nonlocal equation

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

“…The explict solutions of nonlinear PDE play an important role in nonlinear science and engineering, and there are many kinds of effective methods that have been established to construct explict solutions for nonlinear equations, such as symmetry reductions [30,31], bilinear method [41][42][43], Darboux transformation [44], and Painlevé analysis [45], but most of them are more difficult to find interaction solutions which are an important and meaningful research topic [46][47][48]. Fortunately, the above consistent tanh expansion (CTE) method can be easily applied to investigate interaction solutions between a soliton and any other types of nonlinear excitations.…”

N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

“…The NLS equation is widely used in physics [8][9][10][11][12][13][14][15], nonlinear optics [16,17], and soft condensed matter physics [18] and there has been a vast amount of literature involving the NLS equation over the years. The applications of the DT in higher multicomponent NLS equations spatial dimensions and nonlocal equations have attracted much attention over the years [19][20][21][22].…”

A variety of exact solutions for fractional (2+1)-dimensional Heisenberg ferromagnetic spin chain in the semi classical limit

“…A large variety of these equations are utilized to describe important phenomena in different scientifi field like, plasma physics [1,2], condensed matter physics [3], convective fluid [5], optical fiber [6,7], solid state physics [8,9], hydrodynamic [10], water waves [11] and many other branches of engineering [12][13][14]. In past years, to fin the exact solutions of NLSEs many powerful technique have been developed such as, the inverse scattering transformation [15], the homotopy perturbation method [16,17], the Darboux transformation method [18,19], the Sine-Gordon expansion method [20], Bernoulli sub-equation method [21], the modifie auxiliary equation mapping method [22,23], the Riccati equation mapping method [4], the extended sinh-Gordon equation expansion method [24],the modify extended direct algebraic method [25].…”