Compound solitons in fiber Bragg grating

Single soliton and compound solitons are described by coupled-mode equation. It is noted that three parameters, which are dimensionless group velocity, normalized frequency, and grating strength, influence formed solitons by emulation. The novel designs of parallel and serial multi-grating are advanced, and the compound solitons formed from parallel multi-grating are linear superposition; the compound solitons formed from serial multi-grating are nonlinear superposition, and finally two general formulae are obtained. Furthermore, it is theoretically shown that the compound grating solitons are prominent and flexible signals in optical communication.     

Soliton perturbation theory for a higher order Hirota equation

Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified.     

Efficient parallel implementation of the nonparaxial beam propagation method

An efficient parallel implementation of a nonparaxial beam propagation method for the numerical study of the nonlinear Helmholtz equation is presented. Our solution focuses on minimizing communication and computational demands of the method which are dependent on a nonparaxiality parameter. Performance tests carried out on different types of parallel systems behave according theoretical predictions and show that our proposal exhibits a better behavior than those solutions based on the use of conventional parallel fast Fourier transform implementations. The application of our design is illustrated in a particularly demanding scenario: the study of dark solitons at interfaces separating two defocusing Kerr media, where it is shown to play a key role.     

Domain wall arrays, fronts, and bright and dark solitons in a generalized derivative nonlinear Schrödinger equation

We obtain some exact solutions of a generalized derivative nonlinear Schrödinger equation, including domain wall arrays (periodic solutions in terms of elliptic functions), fronts, and bright and dark solitons. In certain parameter domains, fundamental bright and dark solitons are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. Moreover, we also find the chirping reversal phenomena of fronts, and bright and dark solitons, and discuss two different ways to produce the chirping reversal.     

Existence,stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation

We revisit the topic of the existence and azimuthal modulational stability of solitary vortices (alias vortex solitons) in the two-dimensional (2D) cubic–quintic nonlinear Schrödinger equation. We develop a semi-analytical approach, assuming that the vortex soliton is relatively narrow, which allows one to effectively split the full 2D equation into radial and azimuthal 1D equations. A variational approach is used to predict the radial shape of the vortex soliton, using the radial equation, yielding results very close to those obtained from numerical solutions. Previously known existence bounds for the solitary vortices are recovered by means of this approach. The 1D azimuthal equation of motion is used to analyze the modulational instability of the vortex solitons. The semi-analytical predictions – in particular, the critical intrinsic frequency of the vortex soliton at the instability border – are compared to systematic 2D simulations. We also compare our findings to those reported in earlier works, which featured some discrepancies. We then perform a detailed computational study of collisions between stable vortices with different topological charges. Borders between elastic and destructive collisions are identified.     

New schemes for the coupled nonlinear Schrödinger equation

In this paper, we present three new schemes for the coupled nonlinear Schrödinger equation. The three new schemes are multi-symplectic schemes that preserve the intrinsic geometry property of the equation. The three new schemes are also semi-explicit in the sense that they need not solve linear algebraic equations every time-step, which is usually the most expensive in numerical simulation of partial differential equations. Many numerical experiments on collisions of solitons are presented to show the efficiency of the new multi-symplectic schemes.     

Painlevé analysis,soliton solutions and lump-type solutions of the (3+1)-dimensional generalized KP equation

The Painlevé analysis is applied and the multi-soliton criterion is presented to test the integrability of the (3+1)-dimensional generalized KP equation derived from a Hirota bilinear equation. It is shown that the considered equation does not pass the well known Painlevé test and it is only integrable in a conditional sense. Solitary wave solutions are shown to interact each other like solitons in multiple wave collisions unless some additional conditions are imposed. Moreover, we analyze a class of analytical rational lump-type solutions in detail, which are generated from positive quadratic polynomial function and rationally localized in many directions in the space, based upon the Hirota bilinear form.     

Method of lines solution of the Korteweg-de vries equation

The Korteweg-de Vries equation (KdVE) is a classical nonlinear partial differential equation (PDE) originally formulated to model shallow water flow. In addition to the applications in hydrodynamics, the KdVE has been studied to elucidate interesting mathematical properties. In particular, the KdVE balances front sharpening and dispersion to produce solitons, i.e., traveling waves that do not change shape or speed. In this paper, we compute a solution of the KdVE by the method of lines (MOL) and compare this numerical solution with the analytical solution of the kdVE. In a second numerical solution, we demonstrate how solitons of the KdVE traveling at different velocities can merge and emerge. The numerical procedure described in the paper demonstrates the ease with which the MOL can be applied to the solution of PDEs using established numerical approximations implemented in library routines.     

Dissipative structures of the Kuramoto–Sivashinsky equation

The features of dissipative structure formation, which is described by the periodic boundary value problem for the Kuramoto–Sivashinsky equation, are investigated. A numerical algorithm based on the pseudospectral method is presented. The efficiency and accuracy of the proposed numerical method are proved using the exact solution of the equation under study. Using the proposed method, the process of dissipative structure formation, which is described by the Kuramoto–Sivashinsky equation, is studied. The quantitative and qualitative characteristics of this process are described. It is shown that there is a value of the control parameter for which the dissipative structure formation processes occur. Via cyclic convolution, the average value of the control parameter is found. In addition, the dependence of the amplitude of the formed structures on the value of the control parameter is analyzed.     

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.     

The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation

In this paper, the $N$-soliton solution is constructed for the ($2+1$)-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons and their interactions are obtained by choosing special parameters. Especially, by selecting appropriate parameters on the multi-soliton solutions, the two soliton can reduce to a periodic soliton or a lump soliton, the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton, while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons. Detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters. Finally, an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method, and the relevant interaction and propagation characteristics are discussed graphically. The results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two-dimensional nonlinear wave equations.     

Nonlinear variants of the improved Boussinesq equation with compact and noncompact structures

Variants of the improved Boussinesq equation with positive and negative exponents are investigated. It is formally shown that these nonlinear variants give rise to compact and non-compact physical structures, where compactons, solitons, solitary patterns and periodic solutions are developed. The presented sine/cosine ansatz is reliable.     

A group-theory method to find stationary states in nonlinear discrete symmetry systems

In the field of nonlinear optics, the self-consistency method has been applied to searching optical solitons in different media. In this paper, we generalize this method to other systems, adapting it to discrete symmetry systems by using group theory arguments. The result is a new technique that incorporates symmetry concepts into the iterative procedure of the self-consistency method, that helps the search of symmetric stationary solutions. An efficient implementation of this technique is also presented, which restricts the computational work to a reduced section of the entire domain and is able to find different types of solutions by specifying their symmetry properties. As a practical application, we develop an efficient algorithm for solving the nonlinear Schrödinger equation with a discrete symmetry potential.     

Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation

In this paper, we propose a multi-symplectic splitting method to solve the coupled nonlinear Schrödinger (CNLS) equation by using the idea of splitting the multi-symplectic partial differential equation (PDE). Numerical experiments show that the proposed method can simulate the propagation and collision of solitons well. The corresponding errors in global energy and momentum are also presented to show the good preservation property of the proposed method during long-time numerical calculation.     

Collocation method using quadratic B-spline for the RLW equation

A Collocation method is presented here for the Regularized Long Wave (RLW) equation by using Quadratic B-splines at mid points as element shape functions. A linear stability analysis shows the scheme to be unconditionally stable. Test problems, including the migration and interaction of solitary waves, are used to validate the method which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The temporal evaluation of a Maxwellian initial pulse is then studied, and then we prove that the number of solitons which are generated from Maxwellian initial conditions are determined and we compare our results with earlier studies.     

Master equations with a unique stationary state

An effective method which allows one to verify whether a master equation on the Gibbs space has a unique stationary state is proposed. For a unique stationary state the formula establishing its dependence on matrix elements of the generator of the master equation is obtained. Examples of one-step processes with a unique stationary state are studied.     

Numerical solution of GRLW equation using Sinc-collocation method

In this paper, we present a meshfree technique for the numerical solution of the generalized regularized long wave (GRLW) equation. This approach is based on a global collocation method using Sinc basis functions. The propagation of single solitons and the interaction of two solitary waves are used to validate the method which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the method.     

含外力项时变系数KdV方程与时变系数耦合KdV方程组的孤子解

应用孤子拟解法研究了含外力项时变系数KdV方程与一类时变系数耦合KdV方程组.首先将方程经过变量代换转换为齐次方程,然后将孤子解假设为双曲正割函数的形式带入方程或方程组,最后借助Maple软件完成复杂的计算来确定假设的孤子解的待定系数,从而得到孤子解存在的条件及其孤子解.结果显示:孤子拟解法计算简便且能得到方程的亮孤子解.     

Interaction of solitons in a Boussinesq equation with dissipation

We investigate numerically an equation of Boussinesq type with square and cubic nonlinearity. In the model equation, dissipation is added and we investigate the physical properties of the modified problem. The technique applied here is the Christov spectral method in L ²(?∞, ∞). In previous works of the author, it was found that this technique was effective, accurate and computationally efficient for problems of this kind. Localized solutions are obtained numerically for the case of the moving frame which are used as initial conditions for the time-dependent problem. We investigate the propagation, head-on and overcome interaction of solitons. The issue of the phase shift is introduced and is been evaluated numerically.     

Exact wave solutions for a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation

In this work, we investigate a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation. Based on the simplified Hirota bilinear method, we first construct its soliton solutions. Meanwhile, we correct the formula of N-soliton solution for this equation. On the basis of these solitons we further calculate its lump solutions, periodic waves. Meanwhile, rogue waves as well as interaction solutions of this equation are also obtained by a direct algebraic method. Some figures are given to display the behavior of these solutions.