有偏压光折变晶体中的高斯孤子

利用变分方法求解了小光强光折变非线性薛定谔方程，得到了高斯光束在外加正偏压光折变晶体中的演化特性，以及高斯光束宽度压缩与展宽的动态振荡规律，给出了高斯光束的宽度、振幅和波前曲率的表达式，研究了势函数与光束宽度的关系.结果表明，小光强高斯光束在正偏压光折变晶体中传播时，其宽度和振幅将经历周期地变化或者展宽发散，在一定条件下高斯光束在晶体中可以形状不变地传播，形成“高斯孤子”，这些特性在工程实践中具有很高的实用价值. 关键词： 高斯孤子 高斯光束 变分方法 光折变非线性方程     

线性与非线性光晶格中偶极孤立子的稳定性

利用变分法研究了线性和非线性交叉光晶格中偶极玻色-爱因斯坦凝聚(BEC)体系中物质波孤立子的稳定性.选用柱对称高斯型试探波函数,得出参数的Euler-Lagrange方程和体系的有效作用势能,根据有效势能是否具有局域最小值判断体系是否具有稳定孤立子解.结果表明,由于存在接触相互作用的空间调制,在排斥和吸引偶极相互作用下,均能形成稳定的孤立子解.给出了参数空间中存在稳定解的区域和物质波波包宽度随时间的变化曲线.     

非均匀交换各向异性铁磁介质的非线性表面自旋波

利用Landau-Lifshitz 方程，研究了具有非均匀交换各向异性的半无限大铁磁体的非线性表 面自旋波理论。导出了部分钉扎纯交换铁磁介质的磁化强度所满足的边界条件和非线性表面 自旋波的色散关系，并获得了自旋波振幅沿z方向驻波的一维非线性Schrdinger方程和包 络振幅沿平面传播的二维非线性Schrdinger方程，结果表明铁磁体磁化强度的包络振幅随时空变化的性质是由二维非线性Schrdinger方程决定的。因此预言铁磁介质的表面非线性激发应是二维孤波的形式。对于弱非线性表面自旋波，对非线性Schrdinger方程存在孤子形式解的可能性作了讨论. 关键词： 表面自旋波 Landau-Lifshitz方程 非线性Schrdinger方程 孤子     

Bäcklund transformations,nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the $n$th Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type $N$-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.     

Vortex solitons in the semi-infinite gap of optically induced periodic lattices

Vortex solitons with a ring vortex core residing in a single lattice site in the semi-infinite gap of square optical lattices are reported. These solitons are no longer bound states of the Bloch-wave unit (Bloch-wave distribution in one lattice site) at the band edge of the periodic lattice, and consequently they do not bifurcate from the corresponding band edge. For saturable nonlinearity, one family of such solitons is found, and its existing curve forms a closed loop, which is very surprising. For Kerr nonlinearity, two families of such vortex solitons are found.     

非均匀系统中非自治孤子的色散和非线性管理

基于包含自发拉曼散射和外电光调制效应的非自治非线性薛定谔方程,采用简单的变换方法,解析研究了三种非均匀系统中非自治孤子的管理和传输。结果发现,在非均匀的非线性渐增或色散渐减光纤系统中都存在精确的啁啾非自治孤子解,都可以实现孤子的放大和压缩,但具有不同的速度、频移和啁啾特性;而在非均匀的色散和非线性均渐减光纤系统中,可以支持无啁啾的非自治孤子,该孤子具有不变的脉宽和振幅以及振荡衰减的速度,孤子的频移仅由自发拉曼散射决定。同时,数值模拟结果进一步证实在三种非均匀管理系统中都可支持非自治孤子的传输。该研究结果为实际非均匀孤子管理系统中实现孤子的压缩和传输提供了理论依据。     

INTEGRABILITY OF CERTAIN DEFORMED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

A systematic investigation of certain higher order or deformed soliton equations with (1 + 1) dimensions, from the point of complete integrability, is presented. Following the procedure of Ablowitz, Kaup, Newell and Segur (AKNS) we find that the deformed version of Nonlinear Schrodinger equation, Hirota equation and AKNS equation admit Lax pairs. We report that each of the identified deformed equations possesses the Painlevé property for partial differential equations and admits trilinear representation obtained by truncating the associated Painlevé expansions. Hence the above mentioned deformed equations are completely integrable.     

EXPLICIT SOLITON ASYMPTOTICS FOR THE NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

There exists a particular class of boundary value problems for integrable nonlinear evolution equations formulated on the half-line, called linearizable. For this class of boundary value problems, the Fokas method yields a formalism for the solution of the associated initial-boundary value problem, which is as efficient as the analogous formalism for the Cauchy problem. Here, we employ this formalism for the analysis of several concrete initial-boundary value problems for the nonlinear Schrödinger equation. This includes problems involving initial conditions of a hump type coupled with boundary conditions of Robin type.