Dynamics on Solitary Wave and Kink Wave Solutions for a KP-MEW Equation with Damping

This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.

In this paper, we study the exact solitary wave solutions, periodic wave solutions, and bounded rational function solution of the high-order nonlinear Schrödinger equation and the evolutional relationships between the solitary and periodic wave solutions dependent on the Hamilton energy of their amplitude. First, based on the theory and the method of planar dynamical systems, we give a detailed qualitative analysis of the planar dynamical systems corresponding to the amplitude of traveling wave solutions. Then, based on the first integral of the system, we obtain the exact solitary wave solutions, periodic wave solutions, and bounded rational function solution of the equation in various forms by the analysis method, the integral technique, and proper transformation and establish the relationship between the solutions and the Hamilton energy of their amplitude. Furthermore, we discuss the evolutional relationships between the solitary and periodic wave solutions and reveal that the solitary and periodic wave solutions of the equation are essentially determined by the energy change in the Hamilton system corresponding to their amplitude. Finally, we give some diagrams that demonstrate the evolution from periodic wave solutions to solitary wave solutions when Hamilton energy changes.

New solitary wave solutions with compact support and Jacobi elliptic function solutions for the nonlinearly dispersive Boussinesq equations

In this work, we investigated the modulated, nonlinear, ion-acoustic wave (IAW) propagating in an unmagnetized, collisionless, homogeneous plasma where a charged density source of varying velocity due to the debris particles is present. The space-time dynamics of the one dimensional nonlinear ion-acoustic wave is modelled by a forced Nonlinear Schrodinger equation (fNLSE) where the forcing function is generated because of the space debris objects. We obtained : the exact 1 solitary wave, 2 solitary wave followed by N solitary wave solutions and the 1st order rogue wave, the 2nd order rogue wave and then the Nth order rogue wave solutions of the fNLSE for some specific forms of pinned debris functions. The combined system of the nonlinear ion-acoustic wave and the debris function propagates with acceleration. These debris induced, accelerated, exact, modulated nonlinear wave solutions are new in this field of plasma physics as per our study. The condition of modulation instability of such forced system is also discussed with plot. Such exact, nonlinear wave solutions with acceleration may be useful in modelling the experimental data of real astrophysical plasma system.

Bifurcation studies on travelling wave solutions for an integrable nonlinear wave equation

By introducing a new transformation, a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system, which extends Fan's direct algebraic method to the case when r > 4. The solutions of a first-order nonlinear ordinary differential equation with a higher degree nonlinear term and Fan's direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined KdV–mKdV–GKdV equation, which is derived from a simple incompressible non-hydrostatic Boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere. As a result, by taking advantage of the new first-order nonlinear ordinary differential equation with a fifth-degree nonlinear term and an eighth-degree nonlinear term, periodic wave solutions associated with the Jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing. Most importantly, the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter, which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere, which are affected by the local excitation structures in atmosphere. In addition, as an illustrative sample, the properties of the solitary wave solution and Jacobin periodic solution are shown by some figures under the consideration of heating interaction.

The workshop Attraction to Solitary Waves and Related Aspects of Physics , organised by Vladimir Buslaev (St. Petersburg University), Andrew Comech (Texas A&amp;M), Alexander Komech (Universität Wien), and Boris Vainberg (UNC – Charlotte) was held February 10–16, 2008. This meeting was attended with 15 participants with broad geographic representation from Europe and America. This workshop was a blend of researchers with backgrounds in Partial Differential Equations, Harmonic Analysis, and Quantum Field Theory. The aim of the miniworkshop has been the discussion of current state of the long-time asymptotics for nonlinear Hamiltonian partial differential equations and relation to mathematical problems of Quantum Physics. The central themes were the orbital and asymptotic stability of solitary waves, quantum scattering, renormalization, and global attraction to solitary waves. Bohr's transitions as global attraction to solitary waves. According to Bohr's postulates [Boh13], an unperturbed electron runs forever along certain stationary orbit , which we denote \vert E\rangle and call quantum stationary state . Once in such a state, the electron has a fixed value of energy E , not losing the energy via emitting radiation. The electron can jump from one quantum stationary state to another, \tag{1} \vert E_{-}\rangle \longmapsto \vert E_{+}\rangle, emitting or absorbing a quantum of light with the energy equal to the difference of the energies E_{+} and E_{-} . Bohr's second postulate states that the electrons can jump from one quantum stationary state (Bohr's stationary orbit ) to another. Bohr's stationary orbits were interpreted by Schrödinger as quasistationary solitary wave solutions of the form \tag{2}\psi(x,t)=\phi(x)e^{-i\omega t}, \qquad \text{with} \quad \omega\in\mathbb R, \quad \lim_{|x|\to\infty}\phi(x)=0. We will call such solutions solitary waves . Other appropriate names are nonlinear eigenfunctions and quantum stationary states (the solution (2) is not exactly stationary, but certain observable quantities, such as the charge and current densities, are time-independent indeed). As a consequence, the electron in such a state does not emit the energy and “circles” forever around the nucleus in an atom. Bohr's quantum jumps can be interpreted dynamically as long-time asymptotics \tag{3}\Psi(t)\longrightarrow\vert E_\pm\rangle, \qquad t\to\pm\infty, for any trajectory \Psi(t) of the corresponding dynamical system, where the limiting states \vert E_\pm\rangle generally depend on the trajectory. Then the quantum stationary states should be viewed as the points of the global attractor \mathscr{A} . The attraction (3) takes the form of the long-time asymptotics \tag{4}\psi(x,t) \sim \phi_{\omega_\pm}(x)e^{-i\omega_{\pm}t}, \qquad t\to\pm\infty, that hold for each finite energy solution. Now let us describe the existing results on solitary waves in the context of dispersive Hamiltonian systems. Nonlinear wave equations. Well-posedness in the energy space. The nonlinear wave equations take their origin in Quantum Field Theory from the articles by Schiff [Sch51a, Sch51b], who considered the nonlinear Klein–Gordon equation in his research on the classical nonlinear meson theory of nuclear forces. The mathematical analysis of this equation is started by Jörgens [Jör61]and Segal [Seg63a, Seg63b], who studied its global well-posedness in the energy space. Since then, this equation (alongside with the nonlinear Schrödinger equation) has been the main playground for developing tools to handle more general nonlinear Hamiltonian systems. Local attraction to zero. The asymptotics of type (4)were discovered first with \psi_\pm=0 in the scattering theory. Segal [Seg66] and then Morawetz and Strauss[Str68, MS72] studied the (nonlinear) scattering for solutions of nonlinear Klein–Gordon equation in \mathbb R^3 . We may interpret these results as local (referring to small initial data) attraction to zero: \tag{5}\psi(x,t)\sim\psi_\pm=0,\qquad t\to\pm\infty. The asymptotics (5) hold on an arbitrary compact set and represent the well-known local energy decay. These results were further extended in [GS79, Kla82, GV85, Hör91]. Solitary waves. Apparently, there could be no global attraction to zero ( global referring to arbitrary initial data) if there are solitary wave solutions of the form \phi_\omega(x)e^{-i\omega t} . The existence of solitary wave solutions \psi_\omega(x,t)=\phi_\omega(x)e^{-i\omega t}, \qquad \omega\in\R, \quad\phi_\omega\in H^ 1(\R^n), with H^{1}(\mathbb R^n) being the Sobolev space, to the nonlinear Klein–Gordon equation (and nonlinear Schrödinger equation) in \mathbb R^n , in a rather generic situation, was established in [Str77] (a more general result was obtained in[BL83a, BL83b]). Typically, such solutions exist for <jats:tex-mat

Ion acoustic solitary waves and periodic waves in an unmagnetized plasma with superthermal (kappa-distributed) electrons and positrons are investigated through a non-perturbative approach. Model equations are transformed to a planar dynamical system. Then by using the bifurcations of phase portraits of this planar dynamical system, we have established that our model has solitary wave and periodic wave solutions. We have obtained two analytical solutions for these solitary and periodic waves depending on the parameters. From these solitary wave and periodic wave solutions, we have shown the combined effects of temperature ratio (σ) of electrons and positrons, spectral index (κ), speed of the traveling wave (v), and density ratio (p) of positrons and electrons on the characteristics of ion acoustic solitary and periodic waves. The spectral index, density ratio, speed of the traveling wave, and temperature ratio significantly affect the characteristics of ion acoustic solitary and periodic structures. The present study might be helpful to understand the salient features of nonlinear ion acoustic solitary and periodic structures in the interstellar medium.

The solitary structures of the ion-acoustic waves have been considered in a plasma consisting of warm adiabatic ions and non-thermal electrons (due to the presence of fast energetic electrons) having a vortex-like velocity distribution function (due to the presence of trapped electrons), immersed in a uniform (space-independent) and static (time-independent) magnetic field. The nonlinear dynamics of ion-acoustic waves in such a plasma is governed by the Schamel's modified Korteweg–de Vries–Zakharov–Kuznetsov (S-ZK) equation. This equation admits solitary wave solutions having a profile sech4. When the coefficient of the nonlinear term of this equation vanishes, the vortex-like velocity distribution function of electrons simply becomes the non-thermal velocity distribution function of electrons and the nonlinear behaviour of the same ion-acoustic wave is described by a Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation. This equation admits solitary wave solutions having a profile sech2. A combined S–KdV–ZK equation more efficiently describes the nonlinear behaviour of an ion-acoustic wave when the vortex-like velocity distribution function of electrons approaches the non-thermal velocity distribution function of electrons, i.e. when the contribution of trapped electrons tends to zero. This combined S-KdV-ZK equation admits an alternative solitary wave solution having a profile different from either sech4or sech2. The condition for the existence of this alternative solitary wave solution has been derived. It is found that this alternative solitary wave solution approaches the solitary wave solution (the sech2profile) of the KdV-ZK equation when the contribution of trapped electrons tends to zero. The three-dimensional stability of these solitary waves propagating obliquely to the external uniform and static magnetic field has been investigated by the multiple-scale perturbation expansion method of Allen and Rowlands. The instability condition and the growth rate of the instability have been derived at the lowest order. It is also found that the instability condition and growth rate of instability of the alternative solitary waves are exactly the same as those of the solitary waves as determined from the KdV-ZK equation (the sech2profile) when the contribution of trapped electrons tends to zero.

This thesis examines two aspects of the non-equilibrium dynamics of Bose-Einstein condensates (BECs). Our particular interests are the relaxation of near-integrable systems, and the inter-particle correlations present in BECs, both in equilibrium and following an abrupt change in the Hamiltonian. In the first part of this thesis we investigate the stability of solitary wave solutions to a system of two coupled one-dimensional non-linear Schrodinger equations (CNLSE). This system has an integrable point known as the Manakov[1] model which has analytic dark-bright soliton solutions[2]. We break the integrability of this model by varying the inter-species interaction strength, and employ variational and numerical methods in order to find dark-bright solitary wave solutions, away from the integrable point. The time evolution of these states is calculated via numerical integration of the CNLSE, which allows us to assess the stability of these solutions in isolation and to quantify their robustness against collisions with a dark soliton. We find that there is a broad region of the parameter space in which solutions for black-bright (stationary) solitary waves can be found. In this domain, the integrability breaking is revealed only during collisions. Prior to a collision, the numerical solutions are indistinguishable from true solitons, however, the collisional stability of these solitary waves is significantly affected by the extent to which integrability is broken. We find that there is a smooth transition between the non-dispersive particle-like nature of soliton interactions for integrable systems, and the destructive collisions which take place when the system is perturbed far from the integrable point. In the case of moving (grey-bright) solitary waves, we find a more restricted region of the parameter space which admits solitary wave solutions. In this domain we are able to find approximately stable solitary wave solutions using a variational ansatz. The collisional stability of these grey-bright solitary waves is also found to depend upon the extent to which integrability is broken. We conclude that the observation of stable, long-lived dark-bright solitary waves is not sufficient to indicate near-integrability of the two-component system. However, near-integrability may inferred if dark-bright solitary waves are observed to survive multiple collisions. These results are not only of theoretical interest, but may also inform future experiments which explore the behaviour of solitary waves in two-component Bose-Einstein condensates. Following this we investigate the properties of magnetic solitary wave solutions to the CNLSE. We find that the analytic ‘magnetic soliton’ solutions derived in reference [3] are dynamically unstable and relax to non-stationary magnetic solitary wave states over a short timescale. We investigate the collisional dynamics of these magnetic solitary waves and find that these excitations are remarkably robust. In the second part of this thesis we calculate the fluctuations in the number difference between two sub-regions of a Bose-Einstein condensate. This study is motivated by recent experiments performed by the Truscott group at the Australian National University (ANU), which indicate sub- Poissonian fluctuations even for condensates with large thermal depletion. This result cannot easily be reconciled with existing literature, which predicts strongly super-Poissonian fluctuations below the critical temperature (Tc). We develop one-dimensional, and three-dimensional, models to describe these experiments and employ the Bogoliubov formalism to explore the quantum fluctuations of the condensate, which have not previously been studied in detail. We calculate the relative number fluctuations between two halves of the condensate, and also between multiple pairs of spatial bins. For harmonically confined condensates at zero temperature, quantum correlations result in a suppression of the relative number fluctuations between the two halves of the condensate below the shot-noise level. However, the presence of even a small thermal component results in strongly super-Poissonian fluctuations below Tc; this is in agreement with the existing theoretical and experimental results. In addition, we find that the fluctuations of the harmonically confined condensate are non-uniform, with the relative number fluctuations being peaked towards the edges of the condensate. The recent studies of the Truscott group measure the fluctuations after free expansion. We therefore calculate the propagation of correlations following a sudden removal of the harmonic confinement. We find that free expansion results in a strong decrease in the relative number fluctuations. Condensates which exhibit super-Poissonian fluctuations when harmonically confined may therefore display number- squeezing after expansion. A second set of experiments by the Truscott group measured the density fluctuations present in small ‘atom laser’ pulses of atoms out-coupled from their BEC. We also model this procedure and find that immediately after the out-coupling process, the number fluctuations of the out-coupled atomic pulse are approximately Poissonian, irrespective of the correlations present in the atomic reservoir. However, the competing effects of interactions between the atomic pulse and the condensate reservoir, and free expansion of the pulse after out-coupling, can result in the measured fluctuations being either number squeezed or super-Poissonian. Our results will guide future experiments planned by the Truscott group.

In the present research, we explored the various kinds of optical solitons and many other solitary wave solutions for the nonlinear Akbota equation by utilizing the symbolic computational simulation on the basis of the improved F-expansion approach. The nonlinear Akbota equation has applications in physics and engineering. The examined solitary wave and soliton solutions have interesting physical structures, including anti-kink wave solitons, bright solitons, kink wave solitons, dark solitons, periodic wave solitons, peakon bright solitons, peakon dark solitons, mixed bright–dark periodic solitons, mixed solitons in bright–dark form, and solitary wave structures. The newly extracted soliton solutions in this study shed light on the fact that the utilized approach is more efficient, concise, powerful, effective, straightforward, and simple, and we can also utilize it for other higher order nonlinear complex models. The extracted solutions will be helpful to understand the nonlinear phenomena in various areas of nonlinear sciences and engineering, including quantum physics, laser optics, nonlinear optics, optical fibers, ocean engineering, and electronic engineering. The physical interpretation of the extracted solutions is visualized in two-dimensional, three-dimensional, and contour graphics based on numerical simulation by using the computer software Mathematica. The presented research will be helpful for further investigation of analytical solitary wave and soliton solutions to the complex, higher order nonlinear evolution equations.

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.

Various aspects of an unusual evolutionary equation are analysed. This pseudodifferential equation arises as an approximate model for nonlinear dispersive waves in a two-fluid system where the interface is subject to capillarity and the depth of the upper fluid is much smaller than the depth of the lower, more dense fluid. The character of solitary and periodic waves of permanent form is shown to depend primarily on a parameter γ = 1 2 α / ( β C ) 1 / 2 e ( 0 , 1 ) , where α and β are constants representing two different types of dispersion and C > 0 represents wave velocity relative to the velocity of infinitesimal long waves. The asymptotic properties of solitary waves are examined first, being demonstrated to differ markedly whenever γ > 0 from those when γ = 0. When γ is close to 1, moreover, solitary waves have protracted oscillations at their outskirts. In all cases γ € (0,1), solitary-wave solutions of the original equation are not single signed; but their Fourier transforms are positive functions, and this property is made the basis of existence theories using positive-operator methods. Periodic solutions are proved to exist by consideration of the nonlinear equation for their Fourier series, which is posed in a cone of positive sequences from the space Then solitary-wave solutions are treated by a comparable strategy, which also relies on Leray-Schauder degree theory applied to a positive-operator equation but must circumvent the difficulty that the operator in question is not compact. Finally, the existence and stability of solitary waves is shown to be inferable by comparatively simple means, with use of the implicit-function theorem, in the case that 7 is sufficiently small.

Piezoelectric elements have been commonly used because of their wide applications in sensors, transducers, and some micro intelligent structures. However, in the fields of aviation, aerospace, and automation, some relevant equipment works in a harsh environment and is susceptible to the temperature change, thereby leading its performances to be greatly affected. Therefore, the problem of nonlinear wave relating to piezoelectric circular rods in different temperature fields is studied by modeling and numerical analysis. Firstly, based on the theory of finite deformation, we take infinite piezoelectric circular rod as a research object and consider the effects of transverse inertia and equivalent Poisson's ratio under the thermoelectric coupling action. Using the Hamilton principle and introducing the Euler equation, the longitudinal wave equation of piezoelectric circular rod is obtained. Secondly, Jacobi elliptic cosine function and Jacobi elliptic sine function expansion method are used to solve the wave equation of the piezoelectric circular rod, and the solitary wave solution and the exact periodic solution of the wave equation are obtained. It is found that the periodic solution can be reduced into a solitary wave solution under certain conditions, and it is proved theoretically that there may be solitary wave stably propagating in a piezoelectric circular rod. Finally, the dispersion curves of different wave velocity ratios and the curves about influences of temperature field on the waveform, amplitude and wave number of the piezoelectric rod are obtained by Matlab. The numerical results show that the wave velocity decreases with the increase of temperature when the wave velocity ratio is constant. Given the temperature is constant, it can be found that with the increase of the ratio, the amplitude of solitary wave gradually increases while the wavelength gradually decreases. In addition, the images obtained show that although temperature change can cause the characteristics of solitary waves to change, the solitary waves are always symmetrical bell shaped waves in the propagation process, reflecting the stability characteristics under the combined action of nonlinear and dispersion effects. Therefore, the variation of temperature field can influence and control some propagation characteristics of solitary waves. Moreover, the wave theory has been widely used in the nondestructive testing of structures and the improving of information transmission quality due to its special stability.

Nonlinear two-dimensional Kadomtsev–Petviashvili (KP) equation governs the behaviour of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions. By using the reductive perturbation method, the two-dimensional dust-acoustic solitary waves (DASWs) in unmagnetized cold plasma consisting of dust fluid, ions and electrons lead to a KP equation. We derived the solitary travelling wave solutions of the two-dimensional nonlinear KP equation by implementing sech–tanh, sinh–cosh, extended direct algebraic and fraction direct algebraic methods. We found the electrostatic field potential and electric field in the form travelling wave solutions for two-dimensional nonlinear KP equation. The solutions for the KP equation obtained by using these methods can be demonstrated precisely and efficiency. As an illustration, we used the readymade package of Mathematica program 10.1 to solve the original problem. These solutions are in good agreement with the analytical one.