Controllable behaviors of nonautonomous solitons on background of continuous wave and cnoidal wave in $$\mathcal {}$$-symmetric dimer with inhomogeneous effect

We obtain exact \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric nonautonomous soliton solutions on background waves. These solutions indicate that dispersion and nonlinear coefficients influence form factors of nonautonomous solitons such as amplitude, width and center; however, linear coupling coefficient and gain/loss parameter only influence phase of solitons. Based on these solutions, the controllable behaviors such as postpone, sustainment and restraint on continuous wave background in an exponential decreasing dispersion system are discussed. Moreover, the propagation behaviors of solitons on the cnoidal wave background in different dispersion systems are also studied.     

Spatiotemporal solitons on cnoidal wave backgrounds in three media with different distributed transverse diffraction and dispersion

The (3+1)-dimensional nonlinear Schrödinger equation with different distributed transverse diffraction and dispersion is studied based on the similarity transformation, and exact bright soliton solution on cnoidal wave backgrounds is derived. Moreover, three kinds of dynamical behaviors of these soliton solutions in three different dispersion/diffraction decreasing media with the Gaussian, hyperbolic, and Logarithmic profiles are discussed. Solitons interact with cnws and/or the change of characteristics of solitons by an addition of cnws are studied. Result of comparison with three media indicates that for the same parameters, the bright soliton in the Gaussian profile is compressed to the utmost degree. These results are potentially useful for future experiments in the optical communications, long-haul telecommunication networks, and Bose–Einstein condensations.     

On the integrability and Riemann theta functions periodic wave solutions of the Benjamin Ono equation

In this paper, the complete integrability of the Benjamin Ono equation is systematically studied. Its bilinear equation, soliton solutions, bilinear Bäcklund transformation and Lax pair are successfully obtained, by virtue of generalized Bell’s polynomials scheme. Moreover, by using multidimensional Riemann theta functions, the periodic wave solutions of the Benjamin Ono equation are constructed. Further, the asymptotic behaviors of the periodic wave solutions are presented with a limiting procedure, which shows the relations between the periodic wave solutions and soliton solutions.     

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.     

Interaction between a stokes wave packet and a solitary wave

This paper is a theoretical and experimental study of the propagation of a short gravity wave packet (modulated Stokes wave) over a solitary wave. The theoretical approach used here relies on a nonlinear WKB-type perturbation method. This method yields a theory of gravity waves that can describe both short and long waves simultaneously. We obtain explicit analytical solutions describing the interaction between the soliton and the short wave packet: phase shifts, variations of wavelengths and of frequencies (Doppler effects). In the experimental part of this work the phase shift experienced by the Stokes wave is measured. The theoretical conclusions are confirmed.     

On Boussinesq's paradigm in nonlinear wave propagation

Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of ‘Boussinesq's paradigm’. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the ‘mechanics’ of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered. To cite this article: C.I. Christov et al., C. R. Mecanique 335 (2007).     

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…     

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.

     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

General high-order rational solutions and their dynamics in the (3+1)-dimensional Jimbo–Miwa equation

General high-order rational solutions are derived for the (3+1)-dimensional Jimbo–Miwa equation based on the Hirota bilinear form. The solutions are presented in terms of Gram determinants; the elements of determinants are connected to Schur polynomials and have simple algebraic expressions. Their dynamic behaviors are researched using three-dimensional imagery and contour plots. It is revealed that different kinds of solutions appear in (x, y) plane and (y, z) plane. When one of these internal parameters in the rational solutions is sufficiently large, in (x, y) plane Lump solutions appear with obvious geometric structures, which are deconstructed by a first-order Lump such as triangle, pentagon, and nonagon, among others; in (y, z) plane rational line soliton solutions with maximum background amplitude changing over time appear. These findings might help us comprehend the nonlinear wave propagation processes in the many nonlinear physical systems.

     

Hybrid solutions to Mel’nikov system

Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.

     

Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium

A (3+1)-dimensional coupled nonlinear Schrödinger equation with different inhomogeneous diffractions and dispersion is investigated, and rogue wave and combined breather solutions are constructed. Different diffractions and dispersion of medium lead to the repeatedly excited behaviors of rogue wave and combined breather in the dispersion/diffraction decreasing system. These repeated behaviors including complete excitation, rear excitation, peak excitation and initial excitation are discussed.     

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear Schrodinger (NLS) equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.     

Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model

Nonlinear Dynamics - In this article, the authors simulate and study dark and bright soliton solutions of 1D and 2D regularized long wave (RLW) models. The RLW model occurred in various fields such...     

A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.

     

Gravity waves under nonuniform pressure over a free surface. Exact solutions

Plane periodic oscillations of an infinitely deep fluid are studied in the case of a nonuniform pressure distribution over its free surface. The fluid flow is governed by an exact solution of the Euler equations in the Lagrangian variables. The dynamics of an oscillating standing soliton are described, together with the scenario of the soliton evolution and the birth of a wave of an anomalously large amplitude against the background of the homogeneous Gerstner undulation (freak wave model). All the flows are nonuniformly vortical.     

Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation

In this paper, we consider an extended nonlinear Schrödinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.     

Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation

The (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation of B-type (BKP) is hereby investigated. New soliton solutions and soliton-like similarity solutions are constructed for the (2+1)-dimensional BKP equation. The similarity solutions are not travelling wave solutions when the arbitrary functions involved are chosen appropriately. Painlevé test shows that there are two solution branches, one of which has the resonance ?2. And four similarity reductions for the BKP equation are given out through nontrivial variable transformations. Moreover, abundant soliton behaviour modes of the solutions, such as soliton fusion and soliton reflection, are discussed in detail.     

Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.