Karachawa方程

Karachawa方程是一个模拟有表面张力的水波运动的非线性偏微分方程：

$$u_{t}+\mu*u_{xxx}+2*\beta*u_{xxxxx}=0$$

## 解析解

$$u(x, t) = (1/676)*(-338*\sqrt(-13*\mu)*_C3-69*\mu^3)/\mu+(105/338)*\mu^2*tanh(_C1-(1/26)*\sqrt(-13*\mu)*x+_C3*t)^2-(105/676)*\mu^2*tanh(_C1-(1/26)*\sqrt(-13*\mu)*x+_C3*t)^4$$

$$u(x, t) = (1/676)*(338*\sqrt(-13*\mu)*_C3-69*\mu^3)/\mu+(105/338)*\mu^2*coth(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^2-(105/676)*\mu^2*coth(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^4$$

$$u(x, t) = (1/676)*(338*\sqrt(-13*\mu)*_C3-69*\mu^3)/\mu+(105/338)*\mu^2*tanh(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^2-(105/676)*\mu^2*tanh(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^4$$

$$u(x, t) = -(1/676)*(-338*\sqrt(13)*\sqrt(\mu)*_C3+69*\mu^3)/\mu-(105/338)*\mu^2*cot(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^2-(105/676)*\mu^2*cot(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^4$$

$$u(x, t) = -(1/676)*(-338*\sqrt(13)*\sqrt(\mu)*_C3+69*\mu^3)/\mu-(105/338)*\mu^2*tan(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^2-(105/676)*\mu^2*tan(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^4$$

$$u(x, t) = -(1/676)*(338*\sqrt(13)*\sqrt(\mu)*_C3+69*\mu^3)/\mu-(105/338)*\mu^2*tan(_C1+(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^2-(105/676)*\mu^2*tan(_C1+(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^4$$

$$u(x, t) = (1/209560)*(-(13/5)*(2015*\mu-(195*I)*\mu*\sqrt(31))^(3/2)*_C3-(4991/1300*(2015*\mu-(195*I)*\mu*\sqrt(31)))*\mu^3+10478*\sqrt(2015*\mu-(195*I)*\mu*\sqrt(31))*_C3*\mu+961*\mu^4)/\mu^2+(7/676)*\mu*((651/20)*\mu-(123/20*I)*\mu*\sqrt(31))*sech(_C1-(1/260)*\sqrt(2015*\mu-(195*I)*\mu*\sqrt(31))*x+_C3*t)^2-(651/1352)*\mu*((11/20)*\mu-(3/20*I)*\mu*\sqrt(31))*sech(_C1-(1/260)*\sqrt(2015*\mu-(195*I)*\mu*\sqrt(31))*x+_C3*t)^4$$

$$u(x, t) = -(1/523900)*(-(13/2)*(2015*\mu+(195*I)*\mu*\sqrt(31))^(3/2)*_C3+(23529/650*(2015*\mu+(195*I)*\mu*\sqrt(31)))*\mu^3-26195*\sqrt(2015*\mu+(195*I)*\mu*\sqrt(31))*_C3*\mu-175863*\mu^4)/(\mu*((21/10)*\mu+(3/10*I)*\mu*\sqrt(31)))+(-(217/338)*\mu^2+(7/16900)*\mu*(2015*\mu+(195*I)*\mu*\sqrt(31)))*coth(_C1-(1/260)*\sqrt(2015*\mu+(195*I)*\mu*\sqrt(31))*x+_C3*t)^2-(651/1352)*\mu*((11/20)*\mu+(3/20*I)*\mu*\sqrt(31))*coth(_C1-(1/260)*\sqrt(2015*\mu+(195*I)*\mu*\sqrt(31))*x+_C3*t)^4$$

$$p[46] := 8.0741721983973002048*10^5+11257.587038449976187*I+(1813.0402209066405653-19.865040422291120617*I)*JacobiNS(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^1.5-1199.5620*JacobiNS(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^4$$

$$p[47] := 8.0741721983973002048*10^5+11257.587038449976187*I+(15.667532401561428902-15.998034611025966429*I)*JacobiSN(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^1.5+(0.48758338627653809364e-2+0.84722715715396426655e-2*I)*JacobiSN(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^4$$

## 行波图

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## 参考文献

- *谷超豪 《孤立子理论中的达布变换及其几何应用》上海科学技术出版社
- *阎振亚著 《复杂非线性波的构造性理论及其应用》科学出版社 2007年
- 李志斌编著 《非线性数学物理方程的行波解》科学出版社
- 王东明著 《消去法及其应用》科学出版社 2002
- *何青 王丽芬编著 《Maple 教程》科学出版社 2010 ISBN 9787030177445
- Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
- Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
- Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
- Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
- Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
- Dongming Wang, Elimination Practice,Imperial College Press 2004
- David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
- George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759