A variety of (3 + 1)‐dimensional Burgers equations derived by using the Burgers recursion operator

A new variety of (3 + 1)‐dimensional Burgers equations is presented. The recursion operator of the Burgers equation is employed to establish these higher‐dimensional integrable models. A generalized dispersion relation and a generalized form for the one kink solutions is developed. The new equations generate distinct solitons structures and distinct dispersion relations as well. Copyright © 2014 John Wiley & Sons, Ltd.     

Propagation of the nonlinear damped Korteweg‐de Vries equation in an unmagnetized collisional dusty plasma via analytical mathematical methods

In this article, we consider the problem formulation of dust plasmas with positively charge, cold dust fluid with negatively charge, thermal electrons, ionized electrons, and immovable background neutral particles. We obtain the dust‐ion‐acoustic solitary waves (DIASWs) under nonmagnetized collision dusty plasma. By using the reductive perturbation technique, the nonlinear damped Korteweg‐de Vries (D‐KdV) equation is formulated. We found the solutions for nonlinear D‐KdV equation. The constructed solutions represent as bright solitons, dark solitons, kink wave and antikinks wave solitons, and periodic traveling waves. The physical interpretation of constructed solutions is represented by two‐ and three‐dimensional graphically models to understand the physical aspects of various behavior for DIASWs. These investigation prove that proposed techniques are more helpful, fruitful, powerful, and efficient to study analytically the other nonlinear nonlinear partial differential equations (PDEs) that arise in engineering, plasma physics, mathematical physics, and many other branches of applied sciences.     

Dressing the boundary: On soliton solutions of the nonlinear Schrödinger equation on the half‐line

Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schrödinger equation on the half‐line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux‐dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self‐modulated solitons on a constant background. Half‐line solitons in both cases are explicitly computed. In particular, the boundary‐bound solitons, which are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.     

Moving gap solitons in periodic potentials

We address the existence of moving gap solitons (traveling localized solutions) in the Gross–Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit solutions of the coupled‐mode system. We show, however, that exponentially decaying traveling solutions of the Gross–Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled‐mode formalism and are estimated by using techniques of spatial dynamics and local center‐stable manifold reductions. Existence of bounded traveling solutions of the Gross–Pitaevskii equation with a single bump surrounded by oscillatory tails on a large interval of the spatial scale is proven by using these techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability of peakons for the Degasperis‐Procesi equation

The Degasperis‐Procesi equation can be derived as a member of a one‐parameter family of asymptotic shallow‐water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa‐Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis‐Procesi equation on the line. By constructing a Lyapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations. © 2007 Wiley Periodicals, Inc.     

The variational approximation for 2‐dimension solitons at cubic‐quintic nonlinearity in quasiperiodic potentials

We apply the variational approximation to study the dynamics of solitary waves of the nonlinear Schrödinger equation with compensative cubic‐quintic nonlinearity for asymmetric 2‐dimension setup. Such an approach allows to study the behavior of the solitons trapped in quasisymmetric potentials without an axial symmetry. Our analytical consideration allows finding the soliton profiles that are stable in a quasisymmetric geometry. We show that small perturbations of such states lead to generation of the oscillatory‐bounded solutions having 2 independent eigenfrequencies relating to the quintic nonlinear parameter. The behavior of solutions with large amplitudes is studied numerically. The resonant case when the frequency of the time variations (time managed) potential is near of the eigenfrequencies is studied too. In a resonant situation, the solitons acquire a weak time decay.     

Three‐Dimensional Surface Water Waves Governed by the Forced Benney–Luke Equation

This paper studies the propagation of three‐dimensional surface waves in water with an ambient current over a varying bathymetry. When the ambient flow is near the critical speed, under the shallow water assumptions, a forced Benney–Luke (fBL) equation is derived from the Euler equations. An asymptotic approximation of the water's reaction force over the varying bathymetry is derived in terms of topographic stress. Numerical simulations of the fBL equation over a trough are compared to those using a forced Kadomtsev–Petviashvilli equation. For larger variations in the bathymetry that upstream‐radiating three‐dimensional solitons are observed, which are different from the upstream‐radiating solitons simulated by the forced Kadomtsev–Petviashvilli equation. In this case, we show the fBL equation is a singular perturbation of the forced Kadomtsev–Petviashvilli equation which explains the significant differences between the two flows.     

Two-dimensional solitons in irregular lattice systems

We compute and study localized nonlinear modes (solitons) in the semi-infinite gap of the focusing two-dimensional nonlinear Schrödinger (NLS) equation with various irregular lattice-type potentials. The potentials are characterized by large variations from periodicity, such as vacancy defects, edge dislocations, and a quasicrystal structure. We use a spectral fixed-point computational scheme to obtain the solitons. The eigenvalue dependence of the soliton power indicates parameter regions of self-focusing instability; we compare these results with direct numerical simulations of the NLS equation. We show that in the general case, solitons on local lattice maximums collapse. Furthermore, we show that the Nth-order quasicrystal solitons approach Bessel solitons in the large-N limit.     

General N‐Dark–Dark Solitons in the Coupled Nonlinear Schrödinger Equations

N‐dark–dark solitons in the integrable coupled NLS equations are derived by the KP‐hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright–bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed‐nonlinearity case, two dark–dark solitons can form a stationary bound state.     

On the formation of gap solitons in nonlinear electromagnetic metamaterials

A nonlinear (Kerr‐type) electromagnetic metamaterial, characterized by a double‐Lorentz model of its frequency‐dependent linear effective dielectric permittivity and magnetic permeability, is considered. The formation of gap solitons in the low‐ and high‐frequency band gaps of this metamaterial is investigated analytically. Evolution equations governing the gap solitons, of the form of a nonlinear Klein‐Gordon and a nonlinear Schrödinger equation, are obtained, and the structure of their solutions is discussed.     

Optical soliton solutions for the generalized Kudryashov equation of propagation pulse in optical fiber with power nonlinearities by three integration algorithms

In this paper, we employ three integration algorithms, namely, the well‐known Kudryashov method, the new Kudryashov method, and the unified Riccati equation expansion method to extract optical soliton solutions for the generalized Kudryashov equation with power nonlinearities. Straddled soliton, bright solitons, dark solitons, and singular solitons have been found.     

Numerical solution of a coupled Korteweg–de Vries equations by collocation method

A numerical method for solving the coupled Korteweg‐de Vries (CKdV) equation based on the collocation method with quintic B‐spline finite elements is set up to simulate the solution of CKdV equation. Invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm. Simulation of single soliton, interaction of two solitons, and birth of solitons are presented. A linear stability analysis shows the scheme to be unconditionally stable. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Stability of peakons for the Novikov equation

In this paper we study the orbital stability of the peaked solitons to the Novikov equation, which is an integrable Camassa–Holm type equation with cubic nonlinearity. We show that the shapes of these peaked solitons are stable under small perturbations in the energy space.     

Some properties of the spinor soliton

In this paper, the solitons of nonlinear Dirac equation are discussed in detail, and several functions which reflect their characteristics are computed. The numerical results show that, the nonlinear Dirac equation has only finite meaningful solitons, and these solitons have 1/2-spin and positive mass; the spinor soliton has two kinds of parity states, and each parity state has two kinds of energy states; the larger the self-coupling coefficientw, the more the excitation states, and ifw is less than a critical value, then the meaningful soliton does not exist. These properties may have relations with some fundamental particles.     

Exponential Asymptotics for Line Solitons in Two‐Dimensional Periodic Potentials

As a first step toward a fully two‐dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two‐dimensional periodic potentials. For this two‐dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one‐dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence‐relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch‐band edge; and for each rational slope, two line‐soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline‐soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.     

Dynamics of Strongly Nonlinear Kinks and Solitons in a Two‐Layer Fluid

Perturbation theory is developed for interaction of strongly nonlinear solitary waves close to the limiting, tabletop solitons (Π‐solitons). The method is based on representing each soliton as a compound of two kinks so that the interaction of N solitons is treated as the interaction of 2N kinks. As an example the Miyata–Choi–Camassa equations for a two‐layer fluid is considered. Equations for kink coordinates are obtained and analyzed. Some nontrivial features of two‐soliton interaction characteristic of the strongly nonlinear case are established.     

Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. II. moving soliton

We analyze a mechanism and features of a numerical instability (NI) that can be observed in simulations of moving solitons of the nonlinear Schrödinger equation (NLS). This NI is completely different than the one for the standing soliton. We explain how this seeming violation of the Galilean invariance of the NLS is caused by the finite‐difference approximation of the spatial derivative. Our theory extends beyond the von Neumann analysis of numerical methods; in fact, it critically relies on the coefficients in the equation for the numerical error being spatially localized. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1024–1040, 2016     

Soliton Solutions for a Generalized Inhomogeneous Variable-Coefficient Hirota Equation with Symbolic Computation

Under investigation in this paper is a generalized inhomogeneous variable- coefficient Hirota equation. Through the Hirota bilinear method and symbolic computation, the bilinear form and analytic one-, two- and N-soliton solutions for such an equation are obtained, respectively. Properties of those solitons in the inhomogeneous media are discussed analytically. We get the soliton with the property that the larger the amplitude is, the narrower and slower the pulse is. Dynamics of that soliton can be regarded as a repulsion of the soliton by the external potential barrier. During the interaction of two solitons, we observe that the larger the value of the coefficient β in the equation is, the larger the distance of the two solitons is.     

On the optimal focusing of solitons and breathers in long‐wave models

Conditions of optimal (synchronized) collisions of any number of solitons and breathers are studied within the framework of the Gardner equation (GE) with positive cubic nonlinearity, which in the limits of small and large amplitudes tends to other long‐wave models, the classic and the modified Korteweg–de Vries equations. The local solution for an isolated soliton or breather within the GE is obtained. The wave amplitude in the focal point is calculated exactly. It exhibits a linear superposition of partial amplitudes of the solitons and breathers. The crucial role of the choice of proper soliton polarities and breather phases on the cumulative wave amplitude in the focal point is demonstrated. Solitons are most synchronized when they have alternating polarities. The straightforward link to the problem of synchronization of envelope solitons and breathers in the focusing nonlinear Schrödinger equation is discussed (then breathers correspond to envelope solitons propagating above a condensate).     

Solutions of Kadomtsev–Petviashvili equation with power law nonlinearity in 1+3 dimensions

This paper studies the solution of the Kadomtsev–Petviasvili equation with power law nonlinearity in 1+3 dimensions. The Lie symmetry approach as well as the extended tanh‐function and G′/G methods are used to carry out the analysis. Subsequently, the soliton solution is obtained for this equation with power law nonlinearity. Both topological as well as non‐topological solitons are obtained for this equation. Copyright © 2010 John Wiley & Sons, Ltd.