﻿ 基于首次积分法求解两个非线性薛定谔方程的精确解

# 基于首次积分法求解两个非线性薛定谔方程的精确解The First Integral Method for Solving Exact Solutions of Two Nonlinear Schrodinger Equations

Abstract: The first integral method proposed by Feng is very reliable integral method for solving nonlinear partial differential equations, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the generalized nonlinear Schrodinger equation and the high order dispersion nonlinear Schrodinger equation are studied by using the first integral me-thod. By introducing the travelling wave transformations, two nonlinear Schrodinger equations have been transformed into ordinary differential equations. Then according to the division theorem of polynomial, exact travelling wave solutions of two nonlinear Schrodinger equations are obtained.

1. 引言

$i{u}_{t}-{r}_{2}{u}_{xx}+{c}_{3}{|u|}^{2}u=i{\left[\left({s}_{0}+{s}_{2}{|u|}^{2}\right)u\right]}_{x}-{c}_{5}{|u|}^{4}u,$ (1)

$i{q}_{t}-a{q}_{xx}-b{q}_{xxxx}+c\left({|q|}^{2}+d{|q|}^{4}\right)q=0.$ (2)

2. 首次积分法的基本介绍

$P\left(u,{u}_{t},{u}_{x},{u}_{xx},{u}_{tt},{u}_{xt},{u}_{xxx},\cdots \right)=0.$ (3)

$u\left(x,t\right)=u\left(\xi \right),\text{\hspace{0.17em}}\xi =x-ct.$ (4)

$\frac{\partial }{\partial t}\left(.\right)=-c\frac{\partial }{\partial \xi }\left(.\right),\text{\hspace{0.17em}}\frac{\partial }{\partial x}\left(.\right)=\frac{\partial }{\partial \xi }\left(.\right),\text{\hspace{0.17em}}\frac{{\partial }^{2}}{\partial {t}^{2}}\left(.\right)={c}^{2}\frac{{\partial }^{2}}{\partial {\xi }^{2}}\left(.\right),\cdots$ (5)

$Q\left(u,{u}_{\xi },{u}_{\xi \xi },\cdots \right)=0.$ (6)

$u\left(x,t\right)=u\left(\xi \right).$ (7)

$X\left(\xi \right)=u\left(\xi \right),\text{\hspace{0.17em}}Y\left(\xi \right)={u}_{\xi }\left(\xi \right).$ (8)

$\left\{\begin{array}{l}{X}_{\xi }\left(\xi \right)=Y\left(\xi \right),\hfill \\ {Y}_{\xi }\left(\xi \right)=F\left(X\left(\xi \right),Y\left(\xi \right)\right).\hfill \end{array}$ (9)

3. 应用

$u\left(x,t\right)=\phi \left(\xi \right){\text{e}}^{i\left(x-\omega t\right)}.$ (10)

$\left(2{r}_{2}-\lambda -{s}_{0}\right){\phi }^{\prime }-3{s}_{2}{\phi }^{2}{\phi }^{\prime }=0,$ (11)

${r}_{2}{\phi }^{″}+\left(\omega +{s}_{0}-{r}_{2}\right)\phi -\left({c}_{3}+{s}_{2}\right){\phi }^{3}+{c}_{5}{\phi }^{5}=0.$ (12)

${\phi }^{″}+\frac{\omega +{r}_{2}-\lambda }{{r}_{2}}\phi +\frac{{c}_{3}}{{r}_{2}}{\phi }^{3}+\frac{{c}_{5}}{{r}_{2}}{\phi }^{5}=0.$ (13)

$\left\{\begin{array}{l}{X}^{\prime }\left(\xi \right)=Y\left(\xi \right),\hfill \\ {Y}^{\prime }\left(\xi \right)={k}_{1}X\left(\xi \right)+{k}_{2}X{\left(\xi \right)}^{3}+{k}_{3}X{\left(\xi \right)}^{5}.\hfill \end{array}$ (14)

$q\left(x,t\right)=\varphi \left(\xi \right){\text{e}}^{i\left(kx-\omega t\right)}.$ (15)

$-4bk{\beta }^{3}{\varphi }^{″}+\left(-\lambda +2a\beta k+4b\beta {k}^{3}\right){\varphi }^{\prime }=0.$ (16)

$-b{\beta }^{4}{\varphi }^{\left(4\right)}+\left(a{\beta }^{2}+b{\beta }^{2}{k}^{2}\right){\varphi }^{″}+\left(\omega -a{k}^{2}-b{k}^{4}\right)\varphi +c{\varphi }^{3}+dc{\varphi }^{5}=0.$ (17)

${\varphi }^{″}-\frac{4bk{p}_{3}{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}}\varphi -\frac{4bck{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}}{\varphi }^{3}-\frac{4bcdk{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}}{\varphi }^{5}=0.$ (18)

$\left\{\begin{array}{l}{X}^{\prime }\left(\xi \right)=Y\left(\xi \right),\hfill \\ {Y}^{\prime }\left(\xi \right)=\frac{4bk{p}_{3}{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}}X\left(\xi \right)+\frac{4bck{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}}X{\left(\xi \right)}^{3}+\frac{4bcdk{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}}X{\left(\xi \right)}^{5}.\hfill \end{array}$ (19)

${k}_{1}=\frac{4bk{p}_{3}{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}},{k}_{2}=\frac{4bck{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}},{k}_{3}=\frac{4bcdk{\beta }^{3}}{{p}_{1}b{\beta }^{4}-4{p}_{2}bk{\beta }^{3}}$ 方程(19)和(14)有相同的首次积分，要找到它们的精确解，我们仅需讨论其中一个方程即可，下面我们将给出对方程(14)的求解过程。

$P\left(X,Y\right)=\underset{i=0}{\overset{m}{\sum }}{a}_{i}\left(X\right){Y}^{i}.$ (20)

$P\left(X\left(\xi \right),Y\left(\xi \right)\right)=\underset{i=0}{\overset{m}{\sum }}{a}_{i}\left(X\left(\xi \right)\right)Y{\left(\xi \right)}^{i}=0.$ (21)

${\frac{\text{d}P}{\text{d}\xi }|}_{\left(20\right)}={\left(\frac{\text{d}P}{\text{d}X}\frac{\text{d}X}{\text{d}\xi }+\frac{\text{d}P}{\text{d}Y}\frac{\text{d}Y}{\text{d}\xi }\right)|}_{\left(20\right)}=\left(h\left(X\right)+g\left(X\right)Y\right)\left(\underset{i=0}{\overset{m}{\sum }}{a}_{i}\left(X\right){Y}^{i}\right).$ (22)

3.1. 情形一

$\underset{i=0}{\overset{1}{\sum }}{{a}^{\prime }}_{i}\left(X\right){Y}^{i+1}+\underset{i=0}{\overset{1}{\sum }}i{a}_{i}\left(X\right){Y}^{i-1}\left({Y}^{\prime }\left(\xi \right)\right)=\left(h\left(X\right)+g\left(X\right)Y\right)\left(\underset{i=0}{\overset{1}{\sum }}{a}_{i}\left(X\right){Y}^{i}\right).$ (23)

${{a}^{\prime }}_{1}\left(X\right)=g\left(X\right){a}_{1}\left(X\right),$ (24)

${{a}^{\prime }}_{0}\left(X\right)=h\left(X\right){a}_{1}\left(X\right)+g\left(X\right){a}_{0}\left(X\right),$ (25)

${a}_{1}\left(X\right)\left({k}_{1}X+{k}_{2}{X}^{3}+{k}_{5}{X}^{5}\right)=h\left(X\right){a}_{0}\left(X\right).$ (26)

${a}_{0}\left(X\right)=\frac{1}{3}A{X}^{3}+\frac{1}{2}B{X}^{2}+CX+D.$ (27)

$\left\{\begin{array}{l}\begin{array}{l}\frac{1}{3}{A}^{2}={k}_{3}\hfill \\ \frac{5}{6}AB=0\hfill \\ \frac{4}{3}AC+\frac{1}{2}{B}^{2}={k}_{2}\hfill \end{array}\hfill \\ AD+\frac{3}{2}BC=0\hfill \\ BD+{C}^{2}={k}_{1}\hfill \\ CD=0\hfill \end{array}$ (28)

$D=0,B=0,A=±\sqrt{3{k}_{3}},C=±\sqrt{{k}_{1}},$ (29)

${k}_{1}{k}_{3}=\frac{3}{16}{k}_{2}^{2}.$ (30)

$Y=-\frac{1}{3}A{X}^{3}-CX.$ (31)

${\phi }^{\prime }\left(\xi \right)=-\frac{1}{3}A\phi {\left(\xi \right)}^{3}-C\phi \left(\xi \right).$ (32)

${Z}^{\prime }=aZ+b{Z}^{p},$ (33)

$Z\left(\xi \right)={\left[\frac{-a/b}{{\xi }_{{}_{0}}{\text{e}}^{a\left(1-p\right)}+1}\right]}^{\frac{1}{p-1}},$ (34)

$=\left\{\begin{array}{l}{\left\{\frac{-a}{2b}\left[1+\mathrm{tanh}\left(\frac{a\left(p-1\right)}{2}\xi -\frac{\mathrm{ln}{\xi }_{0}}{2}\right)\right]\right\}}^{\frac{1}{p-1}}\text{if}\text{\hspace{0.17em}}{\xi }_{0}>0,\hfill \\ {\left\{\frac{-a}{2b}\left[1+\mathrm{coth}\left(\frac{a\left(p-1\right)}{2}\xi -\frac{\mathrm{ln}\left(-{\xi }_{0}\right)}{2}\right)\right]\right\}}^{\frac{1}{p-1}}\text{if}\text{\hspace{0.17em}}{\xi }_{0}<0,\hfill \\ {\left\{\frac{-a}{b}\right\}}^{\frac{1}{p-1}}\text{}\text{if}\text{\hspace{0.17em}}{\xi }_{0}=0.\hfill \end{array}$ (35)

$u\left(\xi \right)=\left\{\begin{array}{l}{\left\{±\frac{1}{2}\sqrt{\frac{3{k}_{1}}{{k}_{3}}}\left[1+\mathrm{tanh}\left(\sqrt{{k}_{1}}\xi -\frac{\mathrm{ln}{\xi }_{0}}{2}\right)\right]\right\}}^{\frac{1}{2}}\text{if}\text{\hspace{0.17em}}{\xi }_{0}>0,\hfill \\ {\left\{±\frac{1}{2}\sqrt{\frac{3{k}_{1}}{{k}_{3}}}\left[1+\mathrm{coth}\left(\sqrt{{k}_{1}}\xi -\frac{\mathrm{ln}\left(-{\xi }_{0}\right)}{2}\right)\right]\right\}}^{\frac{1}{2}}\text{if}\text{\hspace{0.17em}}{\xi }_{0}<0,\hfill \\ {\left\{±\sqrt{\frac{3{k}_{1}}{{k}_{3}}}\right\}}^{\frac{1}{2}}\text{if}\text{\hspace{0.17em}}{\xi }_{0}=0.\hfill \end{array}$ (36)

$u\left(x,t\right)=\left\{\begin{array}{l}{\left\{±\frac{1}{2}\sqrt{\frac{3{k}_{1}}{{k}_{3}}}\left[1+\mathrm{tanh}\left(\sqrt{{k}_{1}}\left(x-\lambda t\right)-\frac{\mathrm{ln}{\xi }_{0}}{2}\right)\right]\right\}}^{\frac{1}{2}}{\text{e}}^{\left(x-\omega t\right)}\text{if}\text{\hspace{0.17em}}{\xi }_{0}>0,\hfill \\ {\left\{±\frac{1}{2}\sqrt{\frac{3{k}_{1}}{{k}_{3}}}\left[1+\mathrm{coth}\left(\sqrt{{k}_{1}}\left(x-\lambda t\right)-\frac{\mathrm{ln}\left(-{\xi }_{0}\right)}{2}\right)\right]\right\}}^{\frac{1}{2}}{\text{e}}^{\left(x-\omega t\right)}\text{if}\text{\hspace{0.17em}}{\xi }_{0}<0,\hfill \\ {\left\{±\sqrt{\frac{3{k}_{1}}{{k}_{3}}}\right\}}^{\frac{1}{2}}{\text{e}}^{\left(x-\omega t\right)}\text{if}\text{\hspace{0.17em}}{\xi }_{0}=0.\hfill \end{array}$ (37)

3.2. 情形二

${{a}^{\prime }}_{2}\left(X\right)=g\left(X\right){a}_{2}\left(X\right),$ (38)

${{a}^{\prime }}_{1}\left(X\right)=h\left(X\right){a}_{2}\left(X\right)+g\left(X\right){a}_{1}\left(X\right),$ (39)

${{a}^{\prime }}_{0}\left(X\right)+2{a}_{2}\left(X\right)\left({k}_{1}X+{k}_{2}{X}^{3}+{k}_{5}{X}^{5}\right)=h\left(X\right){a}_{1}\left(X\right)+g\left(X\right){a}_{0}\left(X\right),$ (40)

${a}_{1}\left(X\right)\left({k}_{1}X+{k}_{2}{X}^{3}+{k}_{5}{X}^{5}\right)=h\left(X\right){a}_{0}\left(X\right).$ (41)

${a}_{1}\left(X\right)=\frac{1}{3}A{X}^{3}+\frac{1}{2}B{X}^{2}+CX+D.$ (42)

$\begin{array}{c}{a}_{0}\left(X\right)=\left(\frac{1}{18}{A}^{2}-\frac{1}{3}{k}_{3}\right){X}^{6}+\left(\frac{1}{6}AB\right){X}^{5}+\left(\frac{1}{3}AC+\frac{1}{8}{B}^{2}+\frac{1}{2}{k}_{2}\right){X}^{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(\frac{1}{3}AD+\frac{1}{2}BC\right){X}^{3}+\left(\frac{1}{2}BD+\frac{1}{2}{C}^{2}-{k}_{1}\right){X}^{2}+DCX+E.\end{array}$ (43)

$\left\{\begin{array}{l}\begin{array}{l}\begin{array}{l}\frac{1}{18}{A}^{3}=\frac{2}{3}A{k}_{3}\hfill \\ \frac{2}{9}{A}^{2}B=\frac{5}{6}B{k}_{3}\hfill \\ \frac{7}{18}{A}^{2}C+\frac{7}{24}A{B}^{2}=\frac{5}{6}A{k}_{2}+\frac{4}{3}C{k}_{3}\hfill \\ \frac{1}{3}{A}^{2}D+ABC+\frac{1}{8}{B}^{2}=B{k}_{2}+D{k}_{3}\hfill \end{array}\hfill \\ \frac{7}{6}ABD+\frac{5}{6}A{C}^{2}+\frac{1}{8}{B}^{2}C=\frac{4}{3}A{k}_{1}+\frac{3}{2}C{k}_{2}\hfill \\ \frac{4}{3}ACD+\frac{1}{2}D{B}^{2}+B{C}^{2}=\frac{3}{2}B{k}_{1}+2D{k}_{2}\hfill \end{array}\hfill \\ \frac{3}{2}BCD+\frac{1}{2}{C}^{2}+AE=2C{k}_{1}\hfill \\ D{C}^{2}+BE=D{k}_{1}\hfill \\ EC=0,\hfill \end{array}$ (44)

$D=0,B=0,A=±2\sqrt{3{k}_{3}},C=±2\sqrt{{k}_{1}},$ (45)

${k}_{1}{k}_{3}=\frac{3}{16}{k}_{2}^{2}.$ (46)

$Y=-\frac{1}{6}A{X}^{3}-\frac{1}{2}CX.$ (47)

${\phi }^{\prime }\left(\xi \right)=-\frac{1}{6}A\phi {\left(\xi \right)}^{3}-\frac{1}{2}C\phi \left(\xi \right).$ (48)

4. 总结

[1] Feng, Z.S. (2002) The First Integral Method to Study the Burgers Korteweg-de Vries Equation. Physics Letters A, 35, 343-349.
https://doi.org/10.1088/0305-4470/35/2/312

[2] Feng, Z.S. (2002) On Explicit Exact Solutions to the Compound Burgers Korteweg-de Vries Equation. Physics Letters A, 293, 57-66.
https://doi.org/10.1016/S0375-9601(01)00825-8

[3] Feng, Z.S. (2008) Travelling Wave Behavior for a generalized Fisher Equation. Chaos, Solitons & Fractals, 38, 481-488.
https://doi.org/10.1016/j.chaos.2006.11.031

[4] Feng, Z.S. and Knobel, R. (2007) Travelling Waves to a Burgers Korteweg-de Vries Equation with Higher Order Nonlinearities. Journal of Mathematical Analysis and Applications, 328, 1435-1450.
https://doi.org/10.1016/j.jmaa.2006.05.085

[5] Abdoon, M.A. (2015) Programming First Integral Method General Formula for the Solving Linear and Nonlinear Equations. Applied Mathematics, 6, 568-575.
https://doi.org/10.4236/am.2015.63051

[6] Seadawy, A. and Sayed, A. (2017) Soliton Solutions of Cubic-Quintic Nonlinear Schrodinger and Variant Boussinesq Equations by the First Integral Method. Filomat, 214, 4199-4208.
https://doi.org/10.2298/FIL1713199S

[7] Ibrahim, S. and El-Ganaini, A. (2007) The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions. Abstract and Applied Analysis, 18, 1187-1197.

[8] Ma, W.X. (1993) Travelling Wave Solutions to a Seventh Order Generalized KdV Equation. Physics Letters A, 180, 221-224.
https://doi.org/10.1016/0375-9601(93)90699-Z

[9] Liu, S., Fu, Z., Liu, S.D. and Zhao, Q. (2001) Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289, 69-74.
https://doi.org/10.1016/S0375-9601(01)00580-1

[10] Zhang, S. (2006) The Periodic Wave Solutions for the (2 + 1) Dimensional Konopelchenko Dubrovsky Equations. Chaos, Solitons & Fractals, 30, 1213-1220.
https://doi.org/10.1016/j.chaos.2005.08.201

[11] He, J.H. and Zhang, L.N. (2008) Generalized Solitary Solution and Compac-ton-Like Solution of the Jaulent-Miodek Equations Using the Exp-Function Method. Physics Letters A, 372, 1044-1047.
https://doi.org/10.1016/j.physleta.2007.08.059

[12] Wang, D.S. (2009) A systematic Method to Construct Hirotas Transforma-tions of Continuous Soliton Equations and Its Applications. Computers and Mathematics with Applications, 58, 146-153.
https://doi.org/10.1016/j.camwa.2009.03.077

[13] Li, Y., Shan, W.R., Shuai, T.P. and Rao, K. (2015) Bifurcation Analysis and Solutions of a Higher Order Nonlinear Schrodinger Equation. Mathematical Problems in Engineering, 3, 1-10.
https://doi.org/10.1155/2015/408586

[14] Geng, Y.X. and Li, J.B. (2010) Exact Explicit Traveling Wave Solutions for Two Nonlinear Schrodinger Type Equations. Applied Mathematics and Computation, 217, 1509-1521.
https://doi.org/10.1016/j.amc.2009.06.031

[15] Xu, G.Q. (2011) New Types of Exact Solutions for the Fourth-Order Disperesive Cubic-Quintic Nonlinear Schrodinger Equation. Applied Mathematics and Computation, 217, 5967-5971.
https://doi.org/10.1016/j.amc.2010.12.008

[16] Ding, T.R. and Li, C.Z. (1996) Ordinary Differential Equations. Peking Univer-sity Press, Peking.

[17] Ma, W.X. and Fuchsstciner, B. (1996) A Explicit and Exact Solutions to Kolmogorov-Petrovskli-Piskunov Equation. International Journal of Non-Linear Mechanics, 31, 329-338.
https://doi.org/10.1016/0020-7462(95)00064-X

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