Novel Particular Solutions, Breathers, and Rogue Waves for an Integrable Nonlocal Derivative Nonlinear Schrödinger Equation

A determinant representation of the n -fold Darboux transformation for the integrable nonlocal derivative nonlinear Schödinger (DNLS) equation is presented. Using the proposed Darboux transformation, we construct some particular solutions from zero seed, which have not been reported so far for locally integrable systems. We also obtain explicit breathers from a nonzero seed with constant amplitude, deduce the corresponding extended Taylor expansion, and obtain several first-order rogue wave solutions. Our results reveal several interesting phenomena which differ from those emerging from the classical DNLS equation.

The Darboux Transformation and N-Soliton Solutions of Coupled Cubic-Quintic Nonlinear Schrödinger Equation on a Time-Space Scale

The coupled cubic-quintic nonlinear Schrödinger (CQNLS) equation is a universal mathematical model describing many physical situations, such as nonlinear optics and Bose–Einstein condensate. In this paper, in order to simplify the process of similar analysis with different forms of the coupled CQNLS equation, this dynamic system is extended to a time-space scale based on the Lax pair and zero curvature equation. Furthermore, Darboux transformation of the coupled CQNLS dynamic system on a time-space scale is constructed, and the N-soliton solution is obtained. These results effectively combine the theory of differential equations with difference equations and become a bridge connecting continuous and discrete analysis.

Discrete N-fold Darboux transformation and infinite number of conservation laws of a four-component Toda lattice

In this paper, we investigate a four-component Toda lattice (TL), which may be used to model the wave propagation in lattices just like the famous TL. By means of the Lax pair and gauge transformation, we construct the [Formula: see text]-fold Darboux transformation (DT), which enables us to obtain multi-soliton or multi-solitary wave solution without complex iterative process. Through the obtained DT, [Formula: see text]-fold explicit exact solutions of the system and their figures with proper parameters are presented from which we find the [Formula: see text]-fold solution shows two-solitary wave structure, the amplitude and shape of the wave change with time. Finally, we derive an infinite number of conservation laws formulaically to illustrate the integrability of the system.

Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation

A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.

A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.

Multi-Elliptic Rogue Wave Clusters of the Nonlinear Schrodinger Equation on Different Backgrounds

In this work we analyze the multi-elliptic rogue wave clusters as new solutions of the nonlinear Schr\"odinger equation (NLSE). Such structures are obtained on uniform backgrounds by using the Darboux transformation scheme of order $n$ with the first $m$ evolution shifts that are equal, nonzero, and eigenvalue-dependent, while the imaginary parts of all eigenvalues tend to one. We show that an Akhmediev breather of $n-2m$ order appears at the origin of the $(x,t)$ plane and can be considered as the central rogue wave of the cluster. We show that the high-intensity narrow peak, with characteristic intensity distribution in its vicinity, is enclosed by $m$ ellipses consisting of the first-order Akhmediev breathers. The number of maxima on each ellipse is determined by its index and the solution order. Since rogue waves in nature usually appear on a periodic background, we utilize the modified Darboux transformation scheme to build these solutions on a Jacobi elliptic dnoidal background. We analyze the minor semi-axis of all ellipses in a cluster as a function of an absolute evolution shift. We show that the cluster radial symmetry in the $(x,t)$ plane is violated when the shift values are increased above a threshold. We apply the same analysis on Hirota equation, to examine the influence of a free real parameter and Hirota operator on the cluster appearance. The same analysis can be extended to the infinite hierarchy of extended NLSEs.