扎哈罗夫-库兹涅佐夫方程

扎哈罗夫-库兹涅佐夫方程（Zhakharov-Kutznezov equation)是一个非线性偏微分方程^[1]。

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

解析解

扎哈罗夫-库兹涅佐夫方程有许多解析解，包括^[2]

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

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

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

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

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

参数：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；

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

参考文献

^ Touqeer Nawaza, Ahmet Yıldırımb, Syed Tauseef Mohyud-Din：Analytical solutions Zakharov–Kuznetsov equations,Advanced Powder Technology,Volume 24, Issue 1, January 2013, Pages 252–256
^ YAN Zhi-Lian and LIU Xi-Qiang Symmetry Reductions and Explicit Solutions for a Generalized Zakharov Kuznetsov Equation,Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 29–32

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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
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Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
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[1] Touqeer Nawaza, Ahmet Yıldırımb, Syed Tauseef Mohyud-Din：Analytical solutions Zakharov–Kuznetsov equations,Advanced Powder Technology,Volume 24, Issue 1, January 2013, Pages 252–256

[2] YAN Zhi-Lian and LIU Xi-Qiang Symmetry Reductions and Explicit Solutions for a Generalized Zakharov Kuznetsov Equation,Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 29–32

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