双曲型刘维方程

双曲型刘维方程(Hyperbolic Liouville equation)是一个非线性偏微分方程：^[1]

$u_{tt}=\alpha ^{2}*u_{xx}+\gamma *exp(\beta *u)$

作变换：

$u(x,t)=ln(v(x,t))/\beta$ 得：

$v_{tt}*v-(v_{t})^{2}-\alpha ^{2}*v_{xx}*v+\alpha ^{2}*(v_{x})^{2}-\beta ^{2}*v=0$

求得 v(x,t) 的行波解，作反代换得回 u(x,t)。

解析解

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }

{u(x,t)=ln((2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}3^{2}-_{C}4^{2}))*csc(_{C}2+_{C}3*x+_{C}4*t)^{2}/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}3^{2}-_{C}4^{2}))*sec(_{C}2+_{C}3*x+_{C}4*t)^{2}/(\gamma *\beta ))/\beta }

{u(x,t)=ln((2*(\alpha ^{2}*_{C}3^{2}-_{C}4^{2}))*sech(_{C}2+_{C}3*x+_{C}4*t)^{2}/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}4^{2}-_{C}5^{2}))*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,0,0)/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}+cot(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-cot(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}+tan(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-tan(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(-\alpha ^{2}*_{C}2^{2}+_{C}3^{2}+coth(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-coth(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }

{u(x,t)=ln(-(2*(-\alpha ^{2}*_{C}2^{2}+_{C}3^{2}+tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }

双曲型刘维方程行波图

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