二维非线性复Ginzburg-Landau方程的一种解法
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摘要
为了得到一类二维非线性复Ginzburg-Landau方程的周期行波解,采用变化后的F-展开法,即根据齐次平衡原则,利用F-展开法的思想求出其行波解。由于在平面中考虑问题,首先引入了两个波速和一个频率,将原来的奇阶偏导和偶阶偏导共存的偏微分方程化为奇阶和偶阶导数共存的非线性常微分方程;其次根据非线性项和最高阶偏导数齐次平衡可确定复值函数中的最高次项,将常微分方程表示为一类Riccati方程的解的多项式形式的方程;再令多项式的各次幂系数为零,利用Maple数学软件解出用Riccati方程中的待定常数表示的波速、频率与各系数之间的关系,再把结果代入多项式的幂级数中去;最后应用Riccati方程已知的三角函数和双曲函数表示的解,得到方程的多个包络波形式的精确解。
In this paper,we investigate the problem of how to get periodic traveling wave solutions to a two-dimensional nonlinear complex Ginzburg-Landau equation.We extend the F-expansion methods,and obtain the wave solutions according to the homogeneous balance principle.Since we only concern the plane case,our approach could be divided into following steps:first involve twowave velocity and a frequency,transfer the original odd and even order partial derivatives of the coexistence of partial differential equations into odd and even order derivatives,coexistence of nonlinear ordinary differential equations.Then the highest item of the value function could be determined according to the nonlinear term and highest order partial derivative homogeneous balance.The ordinary differential equation could be expressed as a kind of solutions of the Riccati equation in the form of polynomial equation.We set all the coefficients of the polynomial to zero,using any mathematical software(such as Mathematica or Maple)to work out the undetermined constants in the Riccati equation of wave velocity,frequency and the relationship between coefficients.We take the results to the powers of polynomial.At the end,we apply the solutions of Riccati equation which already known from trigonometric functions and hyperbolic functions and we get the exact solutions of the equation in the form of multiple envelope wave.
In this paper,we investigate the problem of how to get periodic traveling wave solutions to a two-dimensional nonlinear complex Ginzburg-Landau equation.We extend the F-expansion methods,and obtain the wave solutions according to the homogeneous balance principle.Since we only concern the plane case,our approach could be divided into following steps:first involve twowave velocity and a frequency,transfer the original odd and even order partial derivatives of the coexistence of partial differential equations into odd and even order derivatives,coexistence of nonlinear ordinary differential equations.Then the highest item of the value function could be determined according to the nonlinear term and highest order partial derivative homogeneous balance.The ordinary differential equation could be expressed as a kind of solutions of the Riccati equation in the form of polynomial equation.We set all the coefficients of the polynomial to zero,using any mathematical software(such as Mathematica or Maple)to work out the undetermined constants in the Riccati equation of wave velocity,frequency and the relationship between coefficients.We take the results to the powers of polynomial.At the end,we apply the solutions of Riccati equation which already known from trigonometric functions and hyperbolic functions and we get the exact solutions of the equation in the form of multiple envelope wave.
引文
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[6]谢春娥,高平.Ginzburg-Landau方程的周期解[J].广州大学学报:自然科学版,2008,7(2):32-35.Xie C E,Gao P.The time-periodic solution for a GinzburgLandau equation[J].Journal of Guangzhou University:Natural Science,2008,7(2):32-35.
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[8]罗森月,杨荣晖,钟澎洪.三维复Ginzburg-Landau方程的一些精确解[J].肇庆学院学报:自然科学版,2010,31(2):6-8.Luo S Y,Yang R H,Zhong P H.Some exact solutions to a kind of 3Dcomplex Ginzburg-Landau equation[J].Journal of Zhaoqing University:Natural Science,2010,31(2):6-8.
[9]陈兆蕙,李泽华.二维非线性耦合复Ginzburg-Landau方程组的周期波解[J].宝鸡文理学院学报:自然科学版,2011,31(2):6-10.Chen Z H,Li Z H.Periodic wave solutions for 2D-nonlinear coupled Ginzburg-Landau equations[J].Journal of Baoji University of Arts and Science:Natural Science,2011,31(2):6-10.
[10]陈兆蕙,李泽华.3维非线性复Ginzburg-Landau方程的同宿波解[J].徐州师范大学学报:自然科学版,2012,30(1):14-17.Chen Z H,Li Z H.Homoclinic wave solutions for 3Dnonlinear Ginzburg-Landau equation[J].Journal of Xuzhou Normal University:Natural Science,2012,30(1):14-17.
[11]李向正,张金良,王明亮.Ginzburg-Landau方程的一种解法[J].河南科技大学学报:自然科学版,2004,25(6):78-81.Li X Z,Zhang J L,Wang M L.Method to solve GinzburgLandau equation[J].Journal of Henan University of Science and Technology:Natural Science,2004,25(6):78-81.
[2]Doering C,Gibbon J D,Holm D,et al.Low-dimensional behavior in the complex Ginzburg-Landau equation[J].Nonlinearity,1988(1):279-309.
[3]Abdul-Majid W.Explicit and implicit solutions for the onedimensional cubic and quintic complex Ginzburg-Landau equations[J].Appl Math Lett,2006,19(10):1007-1012.
[4]Dai Z D,Li Z T,Liu Z J,et al.Exact homoclinic wave and solition solutions for the 2DGinzburg-Landau equation[J].Phys Lett A,2008,372(17):3010-3014.
[5]李自田.Ginzburg-Landau方程的周期波解与孤子解[J].曲靖师范学院学报:自然科学版,2008,27(6):30-33.Li Z T.Periodic wave and soliton solutions of the GinzburgLandau equation[J].Journal of Qujing Normal University:Natural Science,2008,27(6):30-33.
[6]谢春娥,高平.Ginzburg-Landau方程的周期解[J].广州大学学报:自然科学版,2008,7(2):32-35.Xie C E,Gao P.The time-periodic solution for a GinzburgLandau equation[J].Journal of Guangzhou University:Natural Science,2008,7(2):32-35.
[7]Zhong P H,Yang R H,Yang G S.Exact periodic and blow up solutions for 2D Ginzburg-Landau equation[J].Phys Lett A,2008,373(1):19-22.
[8]罗森月,杨荣晖,钟澎洪.三维复Ginzburg-Landau方程的一些精确解[J].肇庆学院学报:自然科学版,2010,31(2):6-8.Luo S Y,Yang R H,Zhong P H.Some exact solutions to a kind of 3Dcomplex Ginzburg-Landau equation[J].Journal of Zhaoqing University:Natural Science,2010,31(2):6-8.
[9]陈兆蕙,李泽华.二维非线性耦合复Ginzburg-Landau方程组的周期波解[J].宝鸡文理学院学报:自然科学版,2011,31(2):6-10.Chen Z H,Li Z H.Periodic wave solutions for 2D-nonlinear coupled Ginzburg-Landau equations[J].Journal of Baoji University of Arts and Science:Natural Science,2011,31(2):6-10.
[10]陈兆蕙,李泽华.3维非线性复Ginzburg-Landau方程的同宿波解[J].徐州师范大学学报:自然科学版,2012,30(1):14-17.Chen Z H,Li Z H.Homoclinic wave solutions for 3Dnonlinear Ginzburg-Landau equation[J].Journal of Xuzhou Normal University:Natural Science,2012,30(1):14-17.
[11]李向正,张金良,王明亮.Ginzburg-Landau方程的一种解法[J].河南科技大学学报:自然科学版,2004,25(6):78-81.Li X Z,Zhang J L,Wang M L.Method to solve GinzburgLandau equation[J].Journal of Henan University of Science and Technology:Natural Science,2004,25(6):78-81.