(2+1)-Dimensional Spatial Localized Modes in Cubic-Quintic Nonlinear Media with the PT-Symmetric Potentials

We obtain exact spatial localized mode solutions of a (2+1)-dimensional nonlinear Schrödinger equation with constant diffraction and cubic-quintic nonlinearity in PT-symmetric potential, and study the linear stability of these solutions. Based on these results, we further derive exact spatial localized mode solutions in a cubic-quintic medium with harmonic and PT-symmetric potentials. Moreover, the dynamical behaviors of spatial localized modes in the exponential diffraction decreasing waveguide and the periodic distributed amplification system are investigated.     

Nonautonomous Solitons in the(3+1)-Dimensional Inhomogeneous Cubic-Quintic Nonlinear Medium

We have constructed explicit nonautonomous soliton solutions of the generalized nonlinear Schrdinger equation in the(3+1)-dimensional inhomogeneous cubic-quintic nonlinear medium.The gain parameter has no effects on the motion of the soliton's phase or their velocities,and it affects just the evolution of their peaks.As two examples,we discuss the propagation of nonautonomous solitons in the periodic distributed amplification system and the exponential dispersion decreasing system.Results show that the presence of the chirp not only makes the intensity of solitons weaken more promptly,but also broadens their width.     

(2+1)-Dimensional Analytical Solutions of the Combining Cubic-Quintic Nonlinear Schrdinger Equation

We investigate analytical solutions of the(2+1)-dimensional combining cubic-quintic nonlinear Schrdinger(CQNLS) equation by the classical Lie group symmetry method.We not only obtain the Lie-point symmetries and some(1+1)-dimensional partial differential systems,but also derive bright solitons,dark solitons,kink or anti-kink solutions and the localized instanton solution.     

Localized Excitations in a Sixth-Order （1＋1）-Dimensional Nonlinear Evolution Equation

In this letter, by means of the Lax pair, Darboux transformation, and variable separation approach, a new exact solution of a sixth-order （1＋1）-dimensional nonlinear evolution equation, which includes some arbitrary functions, is obtained. Abundant new localized excitations can be found by selecting appropriate functions and they are illustrated both analytically and graphically.     

Localized Excitations in a Sixth-Order (1+1)-Dimensional Nonlinear Evolution Equation

In this letter, by means of the Lax pair, Darboux transformation, and variable separation approach, a new exact solution of a sixth-order (1 1)-dimensional nonlinear evolution equation, which includes some arbitrary functions,is obtained. Abundant new localized excitations can be found by selecting appropriate functions and they are illustrated both analytically and graphically.     

(1+1)维强非局域非线性介质中的高阶模呼吸子

基于强非局域非线性介质中的Snyder-Mitchell模型,利用分离变量法得到了(1 1)维光束传输的厄米-高斯型解析解.比较厄米-高斯型解析解与非局域非线性薛定谔方程的数值解,证实了,在强非局域条件下,该厄米-高斯型解与数值解完全吻合.对厄米-高斯光束的传输特性进行研究,结果表明,光束束宽会出现周期性的压缩或者展宽现象.并且得到了实现厄米-高斯光束稳定传输的临界功率、厄米-高斯孤子解及传输常量,临界功率与厄米-高斯光束的阶数无关,但传输常量随阶数的增加而增加.高斯呼吸子和高斯孤子就是基模厄米-高斯呼吸子和基模厄米-高斯孤子.     

Folded Localized Excitations in a Generalized (2+1)-Dimensional Perturbed Nonlinear Schr(o)dinger System

Starting from a special Backlund transform and a variable separation approach, a quite general variableseparation solution of the generalized (2+ 1)-dimensional perturbed nonlinear Schrodinger system is obtained. In additionto the single-valued localized coherent soliton excitations like dromions, breathers, instantons, peakons, and previouslyrevealed chaotic localized solution, a new type of multi-valued (folded) localized excitation is derived by introducingsome appropriate lower-dimensional multiple valued functions.     

Folded Localized Excitations in a Generalized (2+1)-Dimensional Perturbed Nonlinear SchrÖdinger System

Starting from a special Bäcklund transform and a variable separation approach, a quite general variable separation solution of the generalized (2+1)-dimensional perturbed nonlinear Schrödinger system is obtained. In addition to the single-valued localized coherent soliton excitations like dromions, breathers, instantons, peakons, and previously revealed chaotic localized solution, a new type of multi-valued (folded) localized excitation is derived by introducing some appropriate lower-dimensional multiple valued functions.     

Folded Localized Excitations of the （2＋1）-Dimensional Broer-Kaup Equation

Considering that the multi-valued (folded) localized excitations may appear in many (2 1)-dimensional soliton equations because some arbitrary functions can be included in the exact solutions, we use some special types of muliti-valued functions to construct folded solitrary waves and foldons in the (2 1)-dimensional Broer-Kaup equation.These folded excitations are invesigated both analytically and graphically in an alternative way.     

Folded Localized Excitations of the (2 + 1)-Dimensional Broer-Kaup Equation

Considering that the multi-valued (folded) localized excitations may appear in many (2 1)-dimensional soliton equations because some arbitrary functions can be included in the exact solutions, we use some special types of muliti-valued functions to construct folded solitrary waves and foldons in the (2 1)-dimensional Broer-Kaup equation.These folded excitations are invesigated both analytically and graphically in an alternative way.     

Oscillating Solitons for (2+1)-Dimensional Nonlinear Models

Using extended homogeneous balance method and variable separation hypothesis,we found new variableseparation solutions with three arbitrary functions of the (2 1)-dimensional dispersive long-wave equations.Based on derived solutions,we revealed abundant oscillating solitons such as dromion,multi-dromion,solitoff,solitary waves,and so on,by selecting appropriate functions.     

Rich Localized Coherent Structures of the (2+1)-Dimensional Broer-Kaup-Kupershmidt Equation

Using a Backlund transformation and the variable separation approach, we find there exist abundant localized coherent structures for the (2 + 1)-dimensional Broer-Kaup-Kupershmidt (BKK) system. The abundance of the localized structures for the model is introduced by the entrance of an arbitrary function of the seed solution. For some specialselections of the arbitrary function, it is shown that the localized structures of the BKK equation may be dromions, lumps, ring solitons, peakons, or fractal solitons etc.     

Localized Spatial Soliton Excitations in (2+1)-Dimensional NonlinearSchrödinger Equation with Variable Nonlinearity and an ExternalPotential

We report on the localized spatial soliton excitations in the multidimensional nonlinear Schrödinger equation with radially variable nonlinearity coefficient and an external potential. By using Hirota's binary differential operators, we determine a variety of external potentials and nonlinearity coefficients that can support nonlinear localized solutions of different but desired forms. For some specific external potentials and nonlinearity coefficients, we discuss features of the corresponding (2+1)-dimensional multisolitonic solutions, including ring solitons, lump solitons, and soliton clusters.     

（2＋1）-Dimensional Envelope Solitons in Nonlinear Magnetic Metamaterials

The dynamics of nonlinear magnetoinductive waves in a two-dimensional monoatomic lattice of Split ring resonators （SRRs） with Kerr nonlinear interaction between nearest neighbors is studied analytically. The soliton excitation genuine of the discreteness and nonlinearity in such a system based on an extended quasidiscreteness approach are obtained.     

Nonpropagating Solitary Waves in (2+1)-Dimensional Nonlinear Systems

By means of extended homogeneous balance method and variable separation approach, quite a general variable separation solution of the (2+1)-dimensional Broer-Kaup-Kupershmidt equation is derived. From the variable separation solution and by selecting appropriate functions, a new class of (2+1)-dimensional nonpropagating solitary waves are found. The novel features exhibited by these new structures are first revealed.     

Exotic Localized Coherent Structures of the （2＋1）—Dimensional Dispersive Long—Wave Equation

This article is concerned with the extended homogeneous balance method for studying the abundant localized solution structures in the (2 1)-dimensional dispersive long-wave equations uty ηxx (u^2)xy/2=0,ηt (uη u uxy)x=0.Starting from the homogeneous balance method,we find that the richness of the localized coberent structures of the model is caused by the entrance of two variable-separated arbitrary functions.for some special selections of the arbitrary functions,it is shown that the localized structures of the model may be dromions,lumps,breathers,instantons and ring solitons.     

Non-Lie Symmetry Groups of (2+1)-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2 1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik-Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.     

Non-Lie Symmetry Groups of （2＋1）-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.     

Quasi-periodic and Non-periodic Waves in (2+1)-Dimensional Nonlinear Systems

New exact quasi-periodic and non-periodic solutions for the (2 1)-dimensional nonlinear systems are studied by means of the multi-linear variable separation approach (MLVSA) and the Jacobi elliptic functions with the space-time-dependent modulus. Though the result is valid for all the MLVSA solvable models, it is explicitly shown for the long-wave and short-wave interaction model.     

Exotic Localized Coherent Structures of New (2+1)-Dimensional Soliton Equation

The variable separation approach is used to obtain localized coherent structures of the new (2 1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryfunctions of the seed solutions, the abundance of the localized structures of this model are derived. Some special types ofsolutions solitoff, dromions, dromion lattice, breathers and instantons are discussed by selecting the arbitrary functionsappropriately. The breathers may breath in their amplititudes, shapes, distances among the peaks and even the numberof the peaks.