Electron-acoustic supernonlinear waves and their multistability in the framework of the nonlinear Schrödinger equation

Finite-amplitude supernonlinear electron-acoustic waves (EAWs) are investigated under the nonlinear Schrödinger (NLS) equation in a plasma system that is composed of cold electron fluid, immobile ions and q-nonextensive hot electrons. Using the wave transfiguration, the NLS equation is deduced in a dynamical system. The presence of finite-amplitude nonlinear and supernonlinear EAWs is shown by phase plane analysis. The effects of the nonextensive parameter (q) and the speed of waves (v) on different traveling wave solutions of EAWs are presented. Furthermore, by introducing a small external periodic force in the dynamical system, multistability behaviors of EAWs under the NLS equation are shown for the first time in classical plasmas.     

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.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Nonlinear theory of electrostatic baryonic waves in ambiplasma

A collisionless nonmagnetized ambiplasma consisting of Maxwellian gases of protons, antiprotons, electrons, and positrons is considered. The dispersion relation for electrostatic baryonic waves is derived and analyzed and exact expressions for the linear wave phase velocities are obtained. Two types of such waves are shown to be possible in ambiplasma: acoustic and plasma ones. Analysis of the dispersion relation has allowed the ranges of parameters in which nonlinear solutions should be sought in the form of solitons to be found. A nonlinear theory of baryonic waves is developed and used to obtain and analyze the exact solution to the basic equations. The analysis is performed by the method of a fictitious potential. The ranges of phase velocities of periodic baryonic waves and soliton velocities (Mach numbers) are determined. It is shown that in the plasma under consideration, these ranges do not overlap and that the soliton velocity cannot be lower than the linear velocity of the corresponding wave. The profiles of physical quantities in a periodic wave and a soliton (wave scores) are plotted.     

Novel localized wave solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations.     

Lump and interaction solutions to the (3+1)-dimensional Burgers equation

The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.     

Acousto-Optic Solitons in Fibers

It is shown that the deceleration of light pulses down to the velocity of a sound value can be realized in a case of unidirectional parametric interaction of two electromagnetic waves with an acoustic one in the regime of forming three wave acousto-optic solitons. This nonlinear acousto-optic interaction can be realized in long distance systems like fibers. As the result of such an interaction, two types of acousto-optic envelope solitons can propagate in fibers. Modulation of the amplitude of the electromagnetic pump wave can control the soliton velocity.     

Formation of freak waves in a soliton gas described by the modified Korteweg–de Vries equation

The nonlinear dynamics of multisoliton, differently polar fields is investigated within the framework of the modified Korteweg–de Vries equation. It is shown that the occurrence of abnormally large waves (freak waves) is possible in similar fields, which is associated with the modulation instability of cnoidal waves. The statistical moments of wave fields are investigated. It is shown that an increase in the coefficient of excess due to the interaction of solitons correlates with an increase in the probability of occurrence of freak waves. It is shown that the nonlinear interaction of differently polar solitons results in variation of the distribution functions of peak characteristics: the fraction of low-amplitude waves decreases, while that of the waves with large amplitudes increases. The dependence of the intensity of the density of the characteristics of the soliton gas is shown.     

Electromagnetic envelope solitons in magnetized plasma

A multiple scales technique is employed to solve the fluid-Maxwell equations describing a weakly nonlinear circularly polarized electromagnetic pulse in magnetized plasma. A nonlinear Schrödinger-type (NLS) equation is shown to govern the amplitude of the vector potential. The conditions for modulational instability and for the existence of various types of localized envelope modes are investigated in terms of relevant parameters. Right-hand circularly polarized (RCP) waves are shown to be modulationally unstable regardless of the value of the ambient magnetic field and propagate as bright-type solitons. The same is true for left-hand circularly polarized (LCP) waves in a weakly to moderately magnetized plasma. In other parameter regions, LCP waves are stable in strongly magnetized plasmas and may propagate as dark-type solitons (electric field holes). The evolution of envelope solitons is analyzed numerically, and it is shown that solitons propagate in magnetized plasma without any essential change in amplitude and shape.     

Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons

An averaged variational principle is applied to analyze the nonlinear effect of transverse perturbations (including diffraction) on quasi-one-dimensional soliton propagation governed by various wave equations. It is shown that parameters of the spatiotemporal solitons described by the cubic Schrödinger equation and the Yajima-Oikawa model of interaction between long-and short-wavelength waves satisfy the spatial quintic nonlinear Schrödinger equation for a complex-valued function composed of the amplitude and eikonal of the soliton. Three-dimensional solutions are found for two-component “bullets” having long-and short-wavelength components. Vortex and hole-vortex structures are found for envelope solitons and for two-component solitons in the regime of resonant long/short-wave coupling. Weakly nonlinear behavior of transverse perturbations of one-dimensional soliton solutions in a self-defocusing medium is described by the Kadomtsev-Petviashvili equation. The corresponding rationally localized “lump” solutions can be considered as secondary solitons propagating along the phase fronts of the primary solitons. This conclusion holds for primary solitons described by a broad class of nonlinear wave equations.     

Interactions of Localized Wave Structures on Periodic Backgrounds for the Coupled Lakshmanan–Porsezian–Daniel Equations in Birefringent Optical Fibers

Nonlinear waves on periodic backgrounds play an important role in physical systems. In this study, nonlinear waves that include solitons, breathers, rogue waves, and semi-rational solutions on periodic backgrounds for the coupled Lakshmanan-Porsezian-Daniel equations are investigated. Moreover, the interactions between different types of nonlinear waves are examined and their dynamic behaviors are studied. In particular, it is observed that bright-dark rogue waves interact with bright-dark breathers or solitons on periodic backgrounds, four-petaled breathers interact with two eye-shaped breathers on periodic backgrounds, and a four-petal rogue wave interplays with a rogue wave on periodic backgrounds. Furthermore, it is found that the value of the parameter γ₃ affects the weak and strong interactions of these nonlinear waves. These results may be useful in the study of nonlinear wave dynamics in coupled nonlinear wave models.     

Breathers and solitons on two different backgrounds in a generalized coupled Hirota system with four wave mixing

We study breathers and solitons on different backgrounds in optical fiber system, which is governed by generalized coupled Hirota equations with four wave mixing effect. On plane wave background, a transformation between different types of solitons is discovered. Then, on periodic wave background, we find breather-like nonlinear localized waves of which formation mechanism are related to the energy conversion between two components. The energy conversion results from four wave mixing. Furthermore, we prove that this energy conversion is controlled by amplitude and period of backgrounds. Finally, solitons on periodic wave background are also exhibited. These results would enrich our knowledge of nonlinear localized waves' excitation in coupled system with four wave mixing effect.     

Nonlinear theory of ionic sound waves in a hot quantum-degenerate electron-positron-ion plasma

A collisionless nonmagnetized e-p-i plasma consisting of quantum-degenerate gases of ions, electrons, and positrons at nonzero temperatures is considered. The dispersion equation for isothermal ionic sound waves is derived and analyzed, and an exact expression is obtained for the linear velocity of ionic sound. Analysis of the dispersion equation has made it possible to determine the ranges of parameters in which nonlinear solutions in the form of solitons should be sought. A nonlinear theory of isothermal ionic sound waves is developed and used for obtaining and analyzing the exact solution to the system of initial equations. Analysis has been carried out by the method of the Bernoulli pseudopotential. The ranges of phase velocities of periodic ionic sound waves and soliton velocities are determined. It is shown that in the plasma under investigation, these ranges do not overlap and that the soliton velocity cannot be lower than the linear velocity of ionic sound. The profiles of physical quantities in a periodic wave and in a soliton are constructed, as well as the dependences of the velocity of sound and the critical velocity on the ionic concentration in the plasma. It is shown that these velocities increase with the ion concentration.     

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.     

On diffraction and dispersion effect on three wave interaction

The effect of diffraction and dispersion on three wave coupling is investigated. The instability leading to space modulation of packets is obtained. This instability and its nonlinear stage is similar to modulational instability of quasimonochromatic waves. The localized structures (solitons and waveguide) can be a result of the instability. We demonstrate that one-dimensional and two-dimensional such structures are unstable. The existence of stable three-dimensional solitons is shown.     

On the extension of solutions of the real to complex KdV equation and a mechanism for the construction of rogue waves

Rogue waves are more precisely defined as waves whose height is more than twice the significant wave height. This remarkable height was measured (by Draupner in 1995). Thus, the need for constructing a mechanism for the rogue waves is of great utility. This motivated us to suggest a mechanism, in this work, that rogue waves may be constructed via nonlinear interactions of solitons and periodic waves. This suggestion is consolidated here, in an example, by studying the behavior of solutions of the complex (KdV). This is done here by the extending the solutions of its real version.     

铁磁薄膜中非线性静磁表面波的传播

本文用微扰理论导出了横向磁化条件下铁磁薄膜中非线性静磁表面波满足的运动方程和它的解析解。获得非线性色散关系,揭示了传播功率致使静磁表面波频带压缩。研究了群色散和非线性频移随频率和薄膜厚度的变化规律。证明了横向磁化时非线性MSSW不能以静磁孤子的形式存在。 关键词：     

Multiple-mode wave solutions in Vakhnenko equation

A new type of wave solutions, called as multiple-mode waves, which can be expressed in the superposition forms of more than two types of single-mode waves of Vakhnenko equation have been investigated in this Letter. A new general method for obtaining the multiple-mode waves is proposed, based on which four cases of the possible forms of wave solutions with two-mode have been derived. The explicit expressions of the two-mode waves as well as the existence conditions have been presented, which may be the nonlinear combinations between periodic waves, solitons, compactons, etc., with different wave speeds, respectively. It is pointed out that more complicated multiple-mode waves with more than three single-mode waves can be derived accordingly, which can be used to reveal the evolution of interactions between different types of waves, especially between various solitons.