KdV-mKdV方程是一个非线性偏微分方程：

u_{t}+6\alphauu_{x}+6\betau^2u_{x}+\gamma*U_{xxx}=0

解析解
:u(x, t) = -1/(2\beta)-\sqrt(\beta\gamma(-1+_C1^2))_C3JacobiNC(-_C2-_C3x+(1/2)_C3(-2\beta\gamma_C3^2+4\beta_C3^2\gamma_C1^2-3)t/\beta, _C1)/\beta
:u(x, t) = -1/(2\beta)+\sqrt(\beta\gamma(-1+_C1^2))_C3JacobiNC(-_C2-_C3x+(1/2)_C3(-2\beta\gamma_C3^2+4\beta_C3^2\gamma_C1^2-3)t/\beta, _C1)/\beta
:u(x, t) = -1/(2\beta)-\sqrt(-\beta\gamma(-1+_C1^2))_C3JacobiND(_C2+_C3x+(1/2)_C3(2\beta_C3^2\gamma_C1^2-4\beta\gamma_C3^2+3)t/\beta, _C1)/\beta
:u(x, t) = -1/(2\beta)+\sqrt(-\beta\gamma(-1+_C1^2))_C3JacobiND(_C2+_C3x+(1/2)_C3(2\beta_C3^2\gamma_C1^2-4\beta\gamma_C3^2+3)t/\beta, _C1)/\beta

:u(x, t) = -1/(2\beta)-\gamma_C2sech(_C1+_C2x-(1/2)_C2(2\beta\gamma_C2^2-3)t/\beta)/\sqrt(\beta*\gamma)

:u(x, t) = -1/(2\beta)-\gamma_C3JacobiDN(_C2+_C3x+(1/2)_C3(2\beta_C3^2\gamma_C1^2-4\beta\gamma_C3^2+3)t/\beta, _C1)/\sqrt(\beta*\gamma)

:u(x, t) = -1/(2\beta)-\gamma_C2cot(_C1+_C2x-(1/2)_C2(4\beta\gamma_C2^2-3)t/\beta)/\sqrt(-\beta*\gamma)

:u(x, t) = -1/(2\beta)-\gamma_C2coth(_C1+_C2x+(1/2)_C2(4\beta\gamma_C2^2+3)t/\beta)/\sqrt(-\beta*\gamma)

:u(x, t) = -1/(2\beta)-\gamma_C2csch(_C1+_C2x-(1/2)_C2(2\beta\gamma_C2^2-3)t/\beta)/\sqrt(-\beta*\gamma)

:u(x, t) = -1/(2\beta)-\gamma_C2tan(_C1+_C2x-(1/2)_C2(4\beta\gamma_C2^2-3)t/\beta)/\sqrt(-\beta*\gamma)

:u(x, t) = -1/(2\beta)-\gamma_C2tanh(_C1+_C2x+(1/2)_C2(4\beta\gamma_C2^2+3)t/\beta)/\sqrt(-\beta*\gamma)

:u(x, t) = -1/(2\beta)-\gamma_C3JacobiNS(_C2+_C3x+(1/2)_C3(2\beta\gamma_C3^2+2\beta_C3^2\gamma_C1^2+3)t/\beta, _C1)/\sqrt(-\beta*\gamma)

行波图
参考文献

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