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

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

作变换：

u(x, t) = ln(v(x, t))/\beta 得：

v_{tt}v-(v_{t})^2-\alpha^2v_{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_C2^2-_C3^2))csc(_C1+_C2x+_C3t)^2/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(\alpha^2_C2^2-_C3^2))csch(_C1+_C2x+_C3t)^2/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(\alpha^2_C2^2-_C3^2))sec(_C1+_C2x+_C3t)^2/(\gamma\beta))/\beta}
: {u(x, t) = ln((2(\alpha^2_C2^2-_C3^2))sech(_C1+_C2x+_C3t)^2/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(\alpha^2_C3^2-_C4^2))csc(_C2+_C3x+_C4t)^2/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(\alpha^2_C3^2-_C4^2))sec(_C2+_C3x+_C4t)^2/(\gamma\beta))/\beta}
: {u(x, t) = ln((2(\alpha^2_C3^2-_C4^2))sech(_C2+_C3x+_C4t)^2/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(\alpha^2_C4^2-_C5^2))WeierstrassP(_C3+_C4x+_C5t, 0, 0)/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(\alpha^2_C2^2-_C3^2+cot(_C1+_C2x+_C3t)^2\alpha^2_C2^2-cot(_C1+_C2x+_C3t)^2_C3^2))/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(\alpha^2_C2^2-_C3^2+tan(_C1+_C2x+_C3t)^2\alpha^2_C2^2-tan(_C1+_C2x+_C3t)^2_C3^2))/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(-\alpha^2_C2^2+_C3^2+coth(_C1+_C2x+_C3t)^2\alpha^2_C2^2-coth(_C1+_C2x+_C3t)^2_C3^2))/(\gamma\beta))/\beta}
: {u(x, t) = ln(-(2(-\alpha^2_C2^2+_C3^2+tanh(_C1+_C2x+_C3t)^2\alpha^2_C2^2-tanh(_C1+_C2x+_C3t)^2_C3^2))/(\gamma\beta))/\beta}

行波图
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