Darboux transformation of generalized coupled KdV soliton equation and its odd-soliton solutions

Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge transformation of the spectral problem. By using Darboux transformation, the generalized odd-soliton solutions of the generalized coupled KdV soliton equation are given and presented in determinant form. As an application, the first two cases are given.     

The concave or convex peaked and smooth soliton solutions of Camassa-Holm equation

ＩｎｔｒｏｄｕｃｔｉｏｎＣａｍａｓｓａ ,Ｈｏｌｍ[1]ｏｂｔａｉｎｅｄａｃｌａｓｓｏｆｎｅｗｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｓｈａｌｌｏｗｗａｔｅｒｅｑｕａｔｉｏｎ ,ｉ.ｅ.,Ｃａｍａｓｓａ＿Ｈｏｌｍｅｑｕａｔｉｏｎ2ｕｔ+ 2ｋｕｘ-12 ｕｘｘｔ+ 6ｕｕｘ =ｕｘｕｘｘｘ+ 12 ｕｕｘｘｘ. ( 1 )Ｆｏｒｅｖｅｒｙｋ,ｔｈｅＥｑ .( 1 )ｉｓａｃｌａｓｓｏｆｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍ .Ｔｈｉｓｃｌａｓｓｏｆｅｑｕａｔｉｏｎｉｓａｃｌａｓｓｏｆｎｏｔｏｎｌｙｓｔｒａｎｇｅｂｕｔａｌｓｏ…     

Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation

Nonlinear Dynamics - Based on Darboux transformation, we generate breather solutions sitting on zero background via module resonance for complex modified KdV equation. Then, some novel soliton...     

Propagation properties of optical soliton in an erbium-doped tapered parabolic index nonlinear fiber: soliton control

In this paper, with the aid of symbolic computation, we investigate the generalized nonlinear Schrödinger Maxwell–Bloch equation, which describes the propagation of the optical soliton through an inhomogeneous two-level dielectric tapered fiber medium. By virtue of the Darboux transformation method, two-soliton solutions are generated based on the constructed Lax pair and figures are plotted to illustrate the properties of the obtained solutions. Moreover, through manipulating the dispersion and nonlinearity profiles, various soliton control systems are investigated which is promising for potential applications in the design of soliton compressor, soliton amplification and high-speed optical devices in ultralarge capacity transmission systems. This means that we are able to control the soliton types with suitably selected values of the parameters. Additionally more soliton control techniques are proposed and investigated. We expect that the above analysis could be observed in future experiments.     

Attactors of dissipative soliton equation

ＡＴＴＡＣＴＯＲＳＯＦＤＩＳＳＩＰＡＴＩＶＥＳＯＬＩＴＯＮＥＱＵＡＴＩＯＮＴｉａｎＬｉ－ｘｉｎ（田立新）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＰｈｙｓｉｃｓ,ＪｉａｎｇｓｕＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇ...     

Stability of moving soliton lattices

A quasistationary dynamic state of such systems as a long Josephson junction, a superconducting film of periodically modulated thickness and some others is known to be a sequence of regularly spaced solitons moving with a constant velocity. Corresponding self-similar solutions of the nonconservative sine-Gordon equation are investigated. Their stability and a small oscillation spectrum are analyzed. Current-voltage characteristics are obtained for the case of coherent oscillations.     

Perturbation theories for sine-Gordon soliton dynamics

Three recent perturbation assumptions for soliton dynamics on Josephson junctions are compared with direct numerical integrations of the perturbed sine-Gordon equation. The McLaughlin-Scott theory yields the better prediction of soliton transmission or pinning at a microshort in presence of loss and bias without or with surface resistance loss.     

Optical chirped soliton in metamaterials

We present the iterative classical point symmetry analysis of a shallow water wave equation in \(2+1\) dimensions and that of its corresponding nonisospectral, two-component Lax pair. A few reductions arise and are identified with celebrate equations in the Physics and Mathematics literature of nonlinear waves. We pay particular attention to the isospectral or nonisospectral nature of the reduced spectral problems.     

Lump,soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [1] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation’s dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas.

     

The expression of soliton solution for Sine-Gordon equation

In this paper, we study the inverse scattering solution for Sine-Grodon equation A particularly concise expression of the soliton solution is obtained, and the single soliton and double soliton solutions are discussed.     

Bounded states for breathers–soliton and breathers of sine–Gordon equation

Nonlinear Dynamics - The Wronskian solutions to the sine–Gordon (sG) equation that can provide interaction of different kinds of solutions are revisited. And a novel expression of N-soliton...     

On the stability of plane wave and soliton solutions to the Zakharov equations

Plane wave and soliton solutions of the two types of Zakharov equation (two dimensional and simplified one directional) are considered. Stability properties in one dimensional space are seen to be similar. This is interesting, as the first type of equation is not solvable whereas the second is. The soliton solutions of both are one dimensionally stable but those of the full Zakharov equations are unstable with respect to perpendicular perturbations. Regions of stability of nonlinear wave and shock wave solutions in parameter space as well as growth rates of instabilities are given.     

Darboux transformation for an integrable generalization of the nonlinear Schr?dinger equation

A Darboux transformation for an integrable generalization of the coupled nonlinear Schr?dinger equation is derived with the help of the gauge transformation between the Lax pair. As a reduction, a Darboux transformation for an integrable generalization of the nonlinear Schr?dinger equation is obtained, from which some new solutions for the integrable generalization of the nonlinear Schr?dinger equation are given.     

The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM

Interference wave is an important research target in the field of navigation, electromagnetic and earth science. In this work, the nonlinear property of neural network is used to study the interference wave and the bright and dark soliton solutions. The generalized broken soliton-like equation is derived through the generalized bilinear method. Three neural network models are presented to fit explicit solutions of generalized broken soliton-like equations and Boiti–Leon–Manna–Pempinelli-like equation with 100% accuracy. Interference wave solutions of the generalized broken soliton-like equation and the bright and dark soliton solutions of the Boiti–Leon–Manna–Pempinelli-like equation are obtained with the help of the bilinear neural network method. Interference waves and the bright and dark soliton solutions are shown via three-dimensional plots and density plots.

     

Antidark solitons and soliton molecules in a (3 + 1)-dimensional nonlinear evolution equation

Nonlinear Dynamics - We investigate a (3 + 1)-dimensional nonlinear evolution equation which is a higher-dimensional generalization of the Korteweg–de Vries equation. On the...     

Bilinear forms and soliton interactions for two generalized KdV equations for nonlinear waves

Korteweg–de Vries (KdV)-type equations can describe the nonlinear waves in fluids, plasmas, etc. In this paper, two generalized KdV equations are under investigation. Bilinear forms of which are constructed with the Bell polynomials and an auxiliary variable. \(N\) -soliton solutions are given through the Hirota direct method. Via the asymptotic analysis, the soliton interactions of the first generalized KdV equation are analyzed, which turn out to be elastic. Singular breather solutions have been derived from the two-soliton solutions. The collision between soliton and singular breather appears to be elastic, and the bound states of soliton and singular breather are exhibited. Unlike the first one, the other generalized KdV equation can only support the bound states of solitons, for the regular and singular solitons alike.     

General soliton solutions to a nonlocal long-wave–short-wave resonance interaction equation with nonzero boundary condition

Nonlinear Dynamics - Under investigation in this work is a newly proposed nonlocal long-wave–short-wave resonance interaction (LSRI) equation with the self-induced parity-time (PT) symmetric...