特里忒蔡卡方程

特里忒蔡卡方程(Tritzeica equation)是一个最早由罗马尼亚数学家George Tritzeica在1907年在微分几何领域研究的非线性偏微分方程^[1]常见于微分几何学和物理学的非线性偏微分方程：^[2]

罗马尼亚数学家George Tritzeica

$u_{xy}=exp(u_{x,y})-exp(-2*u_{x,y})$

作变换 $w(x,y)=exp(u(x,y))$

得

$w(x,y)_{y,x}*w(x,y)-w(x,y)_{x}*w(x,y)_{y}-w(x,y)^{3}+1=0$

求得行波解，再用反代换 $u(x,y)=ln(w(x,y))$ 即得原方程的行波解。

解析解

u(x,y)=ln(-1/2-(1/2*I)*{\sqrt {(}}3)+(3/4+(3/4*I)*{\sqrt {(}}3))*csc(_{C}1+_{C}2*x+(3/4)*(1/2+(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})

u(x,y)=ln(-1/2-(1/2*I)*{\sqrt {(}}3)+(3/4+(3/4*I)*{\sqrt {(}}3))*sec(_{C}1+_{C}2*x+(3/4)*(1/2+(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})

u(x,y)=ln(-1/2+(1/2*I)*{\sqrt {(}}3)+(3/4-(3/4*I)*{\sqrt {(}}3))*csc(_{C}1+_{C}2*x+(3/4)*(1/2-(1/2*I*{\sqrt {(}}3))*y/_{C}2)^{2})

u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(-3/4+(3/4*I)*{\sqrt {(}}3))*coth(_{C}1+_{C}2*x+(3/4)*(-1/2+(1/2*I)*sqrt(3))*y/_{C}2)^{2})

u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(-3/4+(3/4*I)*{\sqrt {(}}3))*tanh(_{C}1+_{C}2*x+(3/4)*(-1/2+(1/2*I)*sqrt(3))*y/_{C}2)^{2})

u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(3/4-(3/4*I)*{\sqrt {(}}3))*cot(_{C}1+_{C}2*x+(3/4)*(1/2-(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})

u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(3/4-(3/4*I)*{\sqrt {(}}3))*tan(_{C}1+_{C}2*x+(3/4)*(1/2-(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})

w(x,y)=(8/3)*_{C}4^{2}-(1/3)*RootOf(64*_{C}4^{6}+27-24*_{C}4^{4}*_{Z}-6*_{C}4^{2}*_{Z}^{2}+_{Z}^{3})-4*_{C}4^{2}*JacobiDN(_{C}2+(1/2)*_{C}4*x+_{C}4*t,(1/2)*RootOf(-RootOf(64*_{C}4^{6}+27-24*_{C}4^{4}*_{Z}-6*_{C}4^{2}*_{Z}^{2}+_{Z}^{3})+_{Z}^{2})/_{C}4)^{2}

Tritzeica equation traveling wave plot

^ G. Tzitz´eica, “Geometric infinitesimale-sur une nouvelle classes de surfaces,”Comptes Rendus de l’Acad´emie des Sciences, vol. 144,pp. 1257–1259, 1907.
^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS

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