Fifth order evolution equation for long wave dissipative solitons

Third and fifth order nonlinear wave equations which arise in the theory of water waves possess solitary and periodic traveling waves. Solitary waves also arise in systems with dissipation and instability where a balance between these effects allows the existence of dissipative solitons. Here we search for a model equation to describe long wave dissipative solitons including fifth order dispersion. The equation found includes quadratic and cubic nonlinearities. For periodic solutions in a small box we characterize the rate of growth, and show that they do not blow up in finite time. Analytic solutions are constructed for special parameter values.     

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the combination of the third- and fifth-order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are close to solitons of the modified Korteweg-de Vries equation. However, the height of large-amplitude solutions has a limit approaching which solitary waves widen and acquire a table like shape similar to soluitons of the Gardner equation. Numerical calculations confirm that the collision of solitons of the derived equation is inelastic. Inelasticity is the most pronounced in the interaction of unipolar pulses. The direction of the shift of the phase of the higher-amplitude soliton owing to the interaction of solitons of different polarities depends on the amplitudes of the pulses.     

负折射介质中孤波间相互作用

以描述负折射介质中超短脉冲传输的归一化非线性薛定谔方程为模型,采用对称分步傅里叶算法研究了负折射介质中亮、暗孤波间的相互作用.数值模拟发现:当孤波的初始频移为零时,亮孤波间的相互作用与常规介质中类似;当孤波的初始频移不为零时,其传输速度和相互作用明显受三阶色散和自陡峭效应的影响,主要表现为相互排斥.而负折射介质中暗孤波间的相互作用与常规介质中的相互作用类似,无论暗孤波是否存在初始频移,暗孤波间的相互作用在三阶色散和自陡峭的影响下都表现为相互排斥.结果表明,通过调节三阶色散和自陡峭系数可以在一定程度上抑制负折射介质中亮、暗孤波间的相互作用.该研究结果为负折射介质在未来高速通信中的应用提供了理论依据.     

负折射介质中孤波间相互作用

以描述负折射介质中超短脉冲传输的归一化非线性薛定谔方程为模型,采用对称分步傅里叶算法研究了负折射介质中亮、暗孤波间的相互作用.数值模拟发现:当孤波的初始频移为零时,亮孤波间的相互作用与常规介质中类似;当孤波的初始频移不为零时,其传输速度和相互作用明显受三阶色散和自陡峭效应的影响,主要表现为相互排斥.而负折射介质中暗孤波间的相互作用与常规介质中的相互作用类似,无论暗孤波是否存在初始频移,暗孤波间的相互作用在三阶色散和自陡峭的影响下都表现为相互排斥.结果表明,通过调节三阶色散和自陡峭系数可以在一定程度上抑制负折射介质中亮、暗孤波间的相互作用.该研究结果为负折射介质在未来高速通信中的应用提供了理论依据.     

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and(2+1)-dimensional breaking soliton equations via a generalized exponential rational function(GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons,double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons,elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.     

Nonlinear Longitudinal Strain Waves in a Solid Exposed to Pulsed Laser Radiation

The propagation of longitudinal strain waves in a solid with quadratic nonlinearity of elastic continuum was studied in the context of a model that takes into account the joint dynamics of elastic displacements in the medium and the concentration of the laser-induced point defects. The input equations of the problem are reformulated in terms of only the total displacements of the medium points. In this case, the presence of structural defects manifests itself in the emergence of a delayed response of the system to the propagation of the strain-related perturbations, which is characteristic of media with relaxation or memory. The model equations describing the nonlinear displacement wave were derived with allowance made for the values of the relaxation parameter. The influence of the generation, relaxation, and the strain-induced drift of defects and the flexoelectricity on the propagation of this wave was analyzed. It is shown that, for short relaxation times of defects, the strain can propagate in the form of both shock fronts and solitary waves (solitons). Exact solutions depending on the type of relation between the coefficients in the equation and describing both the shock-wave structures and the evolution of solitary waves are presented. In the case of longer relaxation times, shock waves do not form and the strain wave propagates only in the form of solitary waves or a train of solitons. The contributions of the finiteness of the defect-recombination rate and the flexoelectricity to linear elastic moduli and spatial dispersion are determined.     

Varieties of solitons and solitary waves

A summary is presented of the principal types of completely integrable partial differential equations having soliton solutions. Each type is derived from an appropriate physical model of an electromagnetic wave problem, with the intention to show how known mathematical results apply to a coherent class of physical problems in electromagnetic waves. The non-linear Schrödinger (NS) equation appears when the induced non-linear dielectric polarization is expanded in a series of powers of the electric field, only the linear and third-order polarizations are retained, and the temporal spectrum of the wave is a narrow band far removed from any resonance of the medium. The sine-Gordon equation appears from a similar optical model of propagation in a dielectric consisting of identical 2-level atomic systems, but resonance occurs between the carrier frequency of the wave and the transition frequency of the atoms. The Boussinesq and Korteweg– de Vries equations appear at different levels of approximation to a potential wave on a transmission line having a non-linear capacitance such that the charge stored is a non-linear function of the line potential. In all cases the evolution variable is the propagation distance; the transverse variable is time, but in the case of the NS equation it may alternatively be a spatial coordinate, giving rise to the possibility of spatial solitons as well as temporal solitons for NS-type problems. Two examples are derived of non-integrable Hamiltonian systems having spatial solitary waves, namely the second-order cascade interaction and vector spatial solitary waves of the third-order interaction, and a brief survey of the analytical solutions for the plane waves and solitary waves of these two types is presented. Finally, the addition of a second spatial dimension to the non-linear transmission line problem leads to the Kadomtsev–Petviashvili equations, and a further approximation for weakly modulated travelling waves leads to the Davey–Stewartson equations. Both of these completely integrable systems support combined spatial–temporal solitons.     

Lump waves,solitary waves and interaction phenomena to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we investigated the $(2+1)$-dimensional Konopelchenko–Dubrovsky equation. The lump waves, solitary waves as well as interaction between lump waves and solitary waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the rational solutions are obtained by taking a long wave limit, and we also discussed the lump solutions and rogue wave solutions. Moreover, all these solutions are presented via 3-dimensional plots and density plots with choosing some special parameters to show the dynamic graphs.     

Nonlinear three-wave equation for a polydisperse gas-liquid mixture

A new nonlinear three-wave equation of the sixth order for pressure in a polydisperse gas-liquid mixture with bubbles of two sizes is obtained, and its stationary solutions are analyzed. The equation includes two nonlinear quadratic terms of different derivative orders. These terms are of fundamental importance for obtaining good correspondence between the solutions to this equation and the experimentally observed solitary waves in the form of two coupled solitons.     

Electrostatic Solitary Waves in Pair-ion Plasmas with Trapped Electrons

Electrostatic solitons in an unmagnetized pair-ion plasma comprising adiabatic fluid positive and negative ions and non-isothermal electrons are investigated using both arbitrary and small amplitude techniques. An energy integral equation involving the Sagdeev potential is derived, and the basic properties of large amplitude solitary structures are investigated. Various features of solitons differ in different existence domains. The effects of ion adiabaticity, particle concentration, and resonant electrons on the profiles of Sagdeev potential and corresponding solitary waves are investigated. The generalized Korteweg-de Vries equation with mixed-nonlinearity is derived by expanding the Sagdeev potential. Asymptotic solutions for different orders of nonlinearity are discussed for solitary waves. The present work is applicable to understanding the wave phenomena and associated nonlinear electrostatic perturbations in pair/bi-ion plasmas which may occur in space and laboratory plasmas.     

Novel solitary and shock wave solutions for the generalized variable-coefficients (2+1)-dimensional KP-Burger equation arising in dusty plasma

The 2-D generalized variable-coefficient Kadomtsev-Petviashvili-Burgers equation representing many types of acoustic waves in cosmic and/or laboratory dusty plasmas is reduced by the modified classical direct similarity reduction method to nonlinear ordinary differential equation of fourth-order. Using the extended Riccati equation mapping method for solving the reduced equation, many new shock wave, solitary wave and periodic wave solutions are obtained with some constraints between the variable coefficients. Finally, some physical interpretations for the obtained solutions as, bright and dark solitons, periodic solitary wave, and shock wave in dust plasma and quantum plasma are achieved.     

New Exact Jacobi Elliptic Function Solutions of Three—Dimensional Nonlinear Helmholtz Equation in a Nonlinear Kerr—Type Medium

In this letter the three-dimensional nonlinear Helmholtz equation is investigated.which describes electromagnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic function solutions are obtained,by using our extended Jacobian elliptic function expansion method.When the modulus m-→1 or 0,the corresponding solitary waves including bright solitons,dark solitons and new line solitons and singly periodic solutions can be also found.     

Soliton dynamics in the complex modified KdV equation with a periodic potential

We study the existence and stability of stationary and moving solitary waves in a periodically modulated system governed by an extended cmKdV (complex modified Korteweg-de Vries) equation. The proposed equation describes, in particular, the co-propagation of two electromagnetic waves with different amplitudes and orthogonal linear polarizations in a liquid crystal waveguide, the stronger (nonlinear) wave actually carrying the soliton, while the other (a nearly linear one) creates an effective periodic potential. A variational analysis predicts solitons pinned at minima and maxima of the periodic potential, and the Vakhitov-Kolokolov criterion predicts that some of them may be stable. Numerical simulations confirm the existence of stable stationary solitary waves trapped at the minima of the potential, and show that persistently moving solitons exist too. The dynamics of pairs of interacting solitons is also studied. In the absence of the potential, the interaction is drastically different from the behavior known in the NLS (nonlinear Schrödinger) equation, as the force of the interaction between the cmKdV solitons is proportional to the sine, rather than cosine, of the phase difference between the solitons. In the presence of the potential, two solitons placed in one potential well form a persistently oscillating bound state.     

Exact solutions and localized excitations of (3+1)-dimensional Gross—Pitaevskii system

Periodic wave solutions and solitary wave solutions to a generalized (3+1)-dimensional Gross--Pitaevskii equation with time-modulated dispersion, nonlinearity, and potential are derived in terms of an improved homogeneous balance principle and a mapping approach. These exact solutions exist under certain conditions via imposing suitable constraints on the functions describing dispersion, nonlinearity, and potential. The dynamics of the derived solutions can be manipulated by prescribing specific time-modulated dispersions, nonlinearities, and potentials. The results show that the periodic waves and solitary waves with novel behaviors are similar to similaritons reported in other nonlinear systems.     

Decay of solitons in an isotropic collisionless quasineutral plasma with isothermal pressure

Soliton-type solutions of the complete unreduced system of transport equations describing the plane-parallel motions of an isotropic collisionless quasineutral plasma in a magnetic field with constant ion and electron temperatures are studied. The regions of the physical parameters for fast and slow magnetosonic branches, where solitons and generalized solitary waves—nonlocal soliton structures in the form of a soliton “core” with asymptotic behavior at infinity in the form of a periodic low-amplitude wave—exist, are determined. In the range of parameters where solitons are replaced by generalized solitary waves, soliton-like disturbances are subjected to decay whose mechanisms are qualitatively different for slow and fast magnetosonic waves. A specific feature of the decay of such disturbances for fast magnetosonic waves is that the energy of the disturbance decreases primarily as a result of the quasistationary emission of a resonant periodic wave of the same nature. Similar disturbances in the form of a soliton core of a slow magnetosonic generalized solitary wave essentially do not emit resonant modes on the Alfvén branch but they lose energy quite rapidly because of continuous emission of a slow magnetosonic wave. Possible types of shocks which are formed by two types of existing soliton solutions (solitons and generalized solitary waves) are examined in the context of such solutions.     

Interactions between neighboring combined solitary waves

In this paper, the interactions of three types of adjacent combined solitary waves, which are conveniently called Types I, II, and III combined solitary wave, respectively, are numerically investigated. The results show that their interactions exhibit quite different properties. For Type I combined solitary waves, the interaction is quite weaker than that of dark solitons for the standard nonlinear Schrödinger (NLS) equation. Interestingly, the interaction can be well suppressed when they are reduced to the pure dark ones. But for Type II combined solitary waves, the interaction is much stronger than those of Types I and III combined solitary waves and is very difficult to be suppressed. Surprisingly, two adjacent Type III combined solitary waves, both brightlike and darklike ones, hardly interplay each other. These results imply that Type I pure dark solitary waves and Type III combined solitary waves may be regarded as appropriate candidates for information carriers. In addition, the propagation of pulse trains composed of combined solitary waves is investigated.     

Exact solutions and localized excitations of a （3＋ 1）-dimensional Gross-Pitaevskii system

Periodic wave solutions and solitary wave solutions to a generalized (3+1)-dimensional Gross-Pitaevskii equation with time-modulated dispersion, nonlinearity, and potential are derived in terms of an improved homogeneous balance principle and a mapping approach. These exact solutions exist under certain conditions via imposing suitable constraints on the functions describing dispersion, nonlinearity, and potential. The dynamics of the derived solutions can be manipulated by prescribing specific time-modulated dispersions, nonlinearities, and potentials. The results show that the periodic waves and solitary waves with novel behaviors are similar to the similaritons reported in other nonlinear systems.     

Bifurcation and solitary waves of the nonlinear wave equation with quartic polynomial potential

For the nonlinear wave equation with quartic polynomial potential, bifurcation and solitary waves are investigated. Based on the bifurcation and the energy integral of the two-dimensional dynamical system satisfied by the travelling waves, it is very interesting to find different sufficient and necessary conditions in terms of the bifurcation parameter for the existence and coexistence of bright, dark solitary waves and shock waves. The method of direct integration is developed to give all types of solitary wave solutions. Our method is simpler than other newly developed ones. Some results are similar to those obtained recently for the combined KdV－mKdV equation.     

Consistent Riccati expansion solvability,symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves

We study a forced variable-coefficient extended Korteweg-de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.     

Solitary Waves of a Perturbed sine—Gordon Equation

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves,different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained.All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.