五阶变系数非线性薛定谔方程的暗孤子解研究  

The Study on the Dark-Soliton Solutions to the Fifth-Order Variable-Coefficient Nonlinear Schrodinger Equation

作　　者：吴素琴 程燕 许道军 李国望 WU Su-qin;CHENG Yan;XU Dao-jun;LI Guo-wang(Mathematic Teaching,Army Academy of Artillery and Air Denfense of PLA,Hefei 230031,China)

机构地区：[1]陆军炮兵防空兵学院基础部数学教研室

出　　处：《数学的实践与认识》2019年第19期250-259,共10页Mathematics in Practice and Theory

摘　　要：高阶非线性薛定谔方程的孤子解研究是孤子理论最前沿的研究课题之一,在光纤通信中具有重要应用.研究了一个五阶变系数非线性薛定谔方程,方程可以用来描述阿托秒脉冲在光纤中的传播.通过Hirota双线性方法和辅助函数,计算得到方程的双线性形式及其暗孤子解,讨论了暗孤子的传播及碰撞的性质,并得到如下结论:第一,暗孤子的传播速度是由方程的二阶、三阶、四阶和五阶项的系数决定的,暗孤子的振幅则是由这些系数和波数共同决定;第二,当遇上系数为常数、线性函数、二次函数或三角函数时,方程的暗孤子则相应的具有线性、抛物线性、三次函数形式和周期性的性质;第三,孤子在碰撞过程中,其振幅、速度都保持不变,仅仅在相位上发生了相移,因此其碰撞为弹性碰撞.Soliton analysis of the high-order nonlinear Schrodinger equations has been an advancing research in soliton area,which has a great application in optical fiber communications.In this paper,a fifth-order variable-coefficient nonlinear Schrodinger equation is investigated,which describes the propagation of the attosecond pulses in an optical fiber.Via the Hirota method and auxiliary functions,bilinear forms and dark one-,two,threesoliton solutions are obtained.Propagation and interaction of the solitons are discussed and the following conclusions are derived:Firstly,solitonic velocities are only related to the coefficients of the second-,third-,fourth-and fifth-order terms of the equation,while the solitonic amplitudes are related to these coefficients as well as the wave number.Secondly,when the above coefficients are the constants,or the linear,quadratic and trigonometric functions of the space,we obtain the linear,parabolic,cubic and periodic dark solitons,respectively.Thirdly,interactions of the solitons can be regarded to be elastic because the solitonic amplitudes remain unchanged except for some phase shifts after each interaction.

关 键 词：光纤 五阶变系数非线性薛定谔方程 HIROTA双线性方法 双线性形式 暗孤子解 

分 类 号：O17[理学—数学]