扎哈羅夫-庫茲涅佐夫方程

扎哈羅夫-庫茲涅佐夫方程（Zhakharov-Kutznezov equation)是一個非線性偏微分方程。

${\frac {\partial u}{\partial t}}+a*u(x,y,t)*{\frac {\partial u}{\partial x}}+b*u^{2}(x,y,t)*{\frac {\partial u}{\partial x}}+c*{\frac {\partial ^{3}u}{\partial x^{3}}}+d*{\frac {\partial ^{3}u}{\partial x\partial y^{2}}}=0$

解析解

扎哈羅夫-庫茲涅佐夫方程有許多解析解，包括

$u[1]:=i/((l*\xi +m*\eta )*t^{(}1/3))-1/(2*b)$

$u[4]:={\sqrt {(}}-a2/a[4])*sec({\sqrt {(}}-a2)*(l*\xi +m*\eta ))$

$u[5]:=e*{\sqrt {(}}a2/a4)*tan({\sqrt {(}}(1/2)*a2)*(l*\xi +m*\eta ))$

$u[7]:=-a1/(2*a2)+e*a1*sinh(2*{\sqrt {(}}a2)*(l*\xi +m*\eta ))/(2*a2)$

$u[8]:=e*{\sqrt {(}}-a2/a4)*tanh({\sqrt {(}}-(1/2)*a2)*(l*\xi +m*\eta ))$

$u[9]:=-a2*sech({\sqrt {(}}(1/2)*a2)*(l*\xi +m*\eta ))^{2}/a3$

$u[11]:=1/(a3*(l*\xi +m*\eta )^{2})$

其中

$\xi :=x/t^{(}1/3)+a^{2}*t^{(}2/3)/(4*b)+3*c[4]/(c[1]*t^{(}1/3))$

$\eta :=y/t^{(}1/3)+3*c[3]/(c[1]*t^{(}1/3))$

$b:=-6*(c*l^{2}+d*m^{2})$

參數：params := a1 = 1, a2 = 1, a3 = 2.3, a4 = 3.4, d = 1.2, e = 1.3, m = 5.35, c[1] = 5.22, c[2] = 3.23, c[3] = 1.3, c3 = 3.33, c[4] = 1.35, c4 = 1.44, l = 2.1, a[1] = 3.1, a[2] = 3.23, a0 = 5.34, a = .7, c = 1.1；

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  Zakharov-Kutznezov equation plot1
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  Zakharov-Kutznezov equation plot4
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  Zakharov-Kutznezov equation plot5
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  Zakharov-Kutznezov equation plot7
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  Zakharov-Kutznezov equation plot8
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  Zakharov-Kutznezov equation plot9
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  Zakharov-Kutznezov equation plot11

參考文獻

1. *谷超豪 《孤立子理論中的達布變換及其幾何應用》 上海科學技術出版社
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3. 李志斌編著 《非線性數學物理方程的行波解》 科學出版社
4. 王東明著 《消去法及其應用》 科學出版社 2002
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6. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
7. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
8. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
9. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
11. Dongming Wang, Elimination Practice,Imperial College Press 2004
12. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
13. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759