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一类分数阶修正的不稳定Schrödinger方程的新精确解

刘静静 孙峪怀

刘静静,孙峪怀. 一类分数阶修正的不稳定Schrödinger方程的新精确解 [J]. 应用数学和力学,2022,43(10):1185-1194 doi: 10.21656/1000-0887.420228
引用本文: 刘静静,孙峪怀. 一类分数阶修正的不稳定Schrödinger方程的新精确解 [J]. 应用数学和力学,2022,43(10):1185-1194 doi: 10.21656/1000-0887.420228
LIU Jingjing, SUN Yuhuai. New Exact Solutions to a Class of Fractional-Order Modified Unstable Schrödinger Equations[J]. Applied Mathematics and Mechanics, 2022, 43(10): 1185-1194. doi: 10.21656/1000-0887.420228
Citation: LIU Jingjing, SUN Yuhuai. New Exact Solutions to a Class of Fractional-Order Modified Unstable Schrödinger Equations[J]. Applied Mathematics and Mechanics, 2022, 43(10): 1185-1194. doi: 10.21656/1000-0887.420228

一类分数阶修正的不稳定Schrödinger方程的新精确解

doi: 10.21656/1000-0887.420228
基金项目: 四川省教育厅自然科学基金(重点项目)(2012ZA135)
详细信息
    作者简介:

    刘静静(1996—),女,硕士生(E-mail: 1462181092@qq.com

    孙峪怀(1963—),男,教授,博士,硕士生导师(通讯作者. E-mail: sunyuhuai63@163.com

  • 中图分类号: O175

New Exact Solutions to a Class of Fractional-Order Modified Unstable Schrödinger Equations

  • 摘要:

    研究了分数阶修正的不稳定Schrödinger方程(FMUSE),该方程描述了光脉冲在非均匀光纤系统中传播的色散、非线性、增益或吸收变化的普适问题。首先适当地利用广义分数波变换将FMUSE转化为常微分方程,分离实部和虚部并分别令为零,得到了色散关系。再利用修改的(G'/G)-展开法,求得了一系列带参数的新精确解析解,其中包括三角函数解、双曲函数解和有理函数解,并给出了保证解存在的约束条件。最后当参数取特殊值时得到暗孤波和周期波解。

  • 图  1  $ \lambda = 3,{\kern 1pt} \mu = 2,{\kern 1pt} K = 1,{\kern 1pt} {\kern 1pt} v = 1,{\kern 1pt} {\kern 1pt} \beta = - 1,{\kern 1pt} $ $c = 1,{\kern 1pt} {\kern 1pt} s = - 15,{\kern 1pt} {\kern 1pt} \alpha = {1}/{2}$时,${q}_{2.1.1}$的图像

    Figure  1.  The graphic corresponding to ${q}_{2.1.1}$ ($ \lambda = 3,{\kern 1pt} \mu = 2,{\kern 1pt} K = 1,{\kern 1pt} {\kern 1pt} v = 1,{\kern 1pt} {\kern 1pt} \beta = - 1,{\kern 1pt} $ $c = 1,{\kern 1pt} {\kern 1pt} s = - 15,{\kern 1pt} {\kern 1pt} \alpha = {1}/{2}$)

    图  2  $\lambda = 2,\;\mu = 2,\;K = 1,\;v = 1,\;\beta = - 1,\; c = 1,\;s =-5,\;\alpha = {1}/{2}$时,$ {q_{2.2.1}} $的图像

    Figure  2.  The graphic corresponding to $ {q_{2.2.1}} $ ($\lambda = 2,{\kern 1pt} \mu = 2,{\kern 1pt} K = 1, v = 1, $$ {\kern 1pt} \beta = - 1,{\kern 1pt}$ $c = 1,{\kern 1pt} {\kern 1pt} s = - 5,{\kern 1pt} {\kern 1pt} \alpha = {1}/{2}$)

    图  3  $\lambda = 2,\; \mu = 2, \;K = 1, \; v = 1,\; \beta = - 1,\;c = 1, \; s = - 5, \; \alpha = {1}/{2}$时,$ {q_{2.2.2}} $的图像

    Figure  3.  The graphic corresponding to $ {q_{2.2.2}} $ ($\lambda = 2, \mu = 2, K = 1, v = 1, $$ \beta = - 1,$$c = 1, s = - 5, \alpha = {1}/{2}$)

    图  4  $ \lambda = 2, \mu = 2, K = 1, v = 1, \beta = - 1, $ $c = 1, \alpha = {1}/{2}$时,$ {q_{4.2.1}} $的图像

    Figure  4.  The graphic corresponding to $ {q_{4.2.1}} $($\lambda = 2, \mu = 2, K = 1, v = 1, \beta = - 1, $$c = 1, \alpha = {1}/{2}$)

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出版历程
  • 收稿日期:  2021-08-03
  • 修回日期:  2021-09-06
  • 网络出版日期:  2022-10-09
  • 刊出日期:  2022-10-31

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