Darboux–Bäcklund transformation, breather and rogue wave solutions for Ablowitz–Ladik equation

A novel analytical approach to the Benjamin–Ono equation

“…Recent advancements have yielded numerous approaches to exploring exact solutions for nonlinear evolution equations. These methods include the quantum inverse scattering method [1], inverse scattering transform method [2,3], multiple exp-function method [4,5], Hirota's direct method [6,7], modified extended tanh-function method [8,9], binary Darboux transformation [10], and others [11][12][13][14][15][16][17].…”

Dynamical Behaviors of Lumpoff and Rogue Wave Solutions for Nonlocal Gardner Equation

“…e role of nonlinearity in waves is quite signi cant, mostly throughout non-linear sciences; research development towards exact solutions of partial di erential nonlinear equations has always been a major endeavor for the past couple of years. All the while, several powerful, e cient, and reliable methods for attempting to seek exact analytical solutions to traveling waves have been established: for example, the N-soliton solution [7], a fractional-order multiple-model type-3 fuzzy control [8], a calculation methodology for geometrical characteristics [9], a combination of group method of data handling and computational uid dynamics [10], the truss optimization with metaheuristic algorithms [11], the extended generalized Darboux method [12], the Lax pair technique [13], intermolecular interactions to the modern acid-base theory [14], the deep learning for Feynman's path integral [15], the Darboux-Bäcklund technique [16], the Hirota bilinear technique [17], the multiple exp-function method [18], optimal structure design [19], the experimental study on circular steel [20,21], an influence of seismic orientation on the statistical distribution [22], effects of actual loading waveforms on the fatigue behaviors [23], and so on [24][25][26][27]. Hirota bilinear method is an efficient instrument to construct exact solutions of NLEEs, and there exist a lot of completely integrable equations that are investigated in this procedure, for example, the generalized bilinear equations [28], the Kadomtsev-Petviashvili (KP)-Benjamin-Bona-Mahony equation [29], the optimal Galerkin-homotopy asymptotic method [30], the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation [31], the KP equation [32], the B-type KP equation [33], the optical interactions within magnetic layered structures [34], the (2 + 1)dimensional breaking soliton equation [35], and the bidirectional Sawada-Kotera equation …”

Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

“…Searching for explicit exact solutions, especially soliton solutions, is used for depicting and explaining such nonlinear phenomena described by the discrete nonlinear lattice models. Methods of constructing the soliton solutions of the discrete integrable models have been proposed and developed such as the discrete inverse scattering method [13], the discrete Hirota bilinear formalism method [14,15], and the discrete Darboux transformation (DT) method [16][17][18][19][20][21][22][23][24][25]. Among them, the discrete DT based on corresponding Lax representation is a useful technique to solve the discrete nonlinear models and its main idea is to keep the corresponding Lax pair of these discrete equations unchanged.…”

“…Compared with the common discrete DT which can only give the usual soliton (US) solutions, the discrete generalized ðm, N − mÞ-fold DT [24,25] is a generalization of the common discrete DT and the main advantage of this method is that it can give not only the US solution but also the rational solutions (e.g., rouge wave solutions and rational soliton (RS) solutions) and mixed interaction solutions of US and rational solution [24,25]. For the US solutions of discrete nonlinear models, there have been a lot of literature studies [16][17][18][19][20][21][22][23] but the study of rational solutions and mixed interaction solutions of US and rational solutions is still not sufficient and systematic. As far as we know, the asymptotic behaviors of RS solutions and mixed interaction soliton solutions of US and RS by using asymptotic analysis have not been studied yet.…”