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]。

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

解析解

扎哈罗夫-库兹涅佐夫方程有许多解析解，包括^[2]

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

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

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

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

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

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

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

其中

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

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

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

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；

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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

参考文献

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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