非线性扩散方程（Nonlinear Diffusion equation）是一个非线性偏微分方程：

u_{t}=\alphau_{xx}-\betau^3-\gamma*u^2

解析解
: u(x, t) = -(1/2)\gamma/\beta-(1/2)\gammacoth(_C1-(1/2)\gammax/\sqrt(-2\beta\alpha)-(1/4)(-2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta
: {u(x, t) = -(1/2)\gamma/\beta+(1/2)\gammacoth(_C1-(1/2)\gammax/\sqrt(-2\beta\alpha)+(1/4)(2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta}
: {u(x, t) = -(1/2)\gamma/\beta+(1/2)\gammacoth(_C1+(1/2)\gammax/\sqrt(-2\beta\alpha)+(1/4)(-2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta}
:{u(x, t) = -(1/2)\gamma/\beta-(1/2)\gammacoth(_C1+(1/2)\gammax/\sqrt(-2\beta\alpha)-(1/4)(2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta}
: {u(x, t) = -(1/2)\gamma/\beta-(1/2)\gammatanh(_C1-(1/2)\gammax/\sqrt(-2\beta\alpha)-(1/4)(-2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta}
: {u(x, t) = -(1/2)\gamma/\beta+(1/2)\gammatanh(_C1-(1/2)\gammax/\sqrt(-2\beta\alpha)+(1/4)(2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta}
: {u(x, t) = -(1/2)\gamma/\beta+(1/2)\gammatanh(_C1+(1/2)\gammax/\sqrt(-2\beta\alpha)+(1/4)(-2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta}
: {u(x, t) = -(1/2)\gamma/\beta-(1/2)\gammatanh(_C1+(1/2)\gammax/\sqrt(-2\beta\alpha)-(1/4)(2\gamma\beta/\sqrt(-2\beta\alpha)+\gamma^2)*t/\beta)/\beta}
: {u(x, t) = -(1/2)\gamma/\beta+(1/2I)\gammatan(_C1+(1/4)\gamma\sqrt(2)x/\sqrt(\beta\alpha)-(1/4)(I\gamma^2+\gamma\sqrt(2)\beta/\sqrt(\beta\alpha))t/\beta)/\beta}
: {u(x, t) = -(1/2)\gamma/\beta+(1/2I)\gammacot(_C1+(1/4)\gamma\sqrt(2)x/\sqrt(\beta\alpha)+(1/4)(I\gamma^2-\gamma\sqrt(2)\beta/\sqrt(\beta\alpha))t/\beta)/\beta}

行波图
参考文献

*谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社

*阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年

李志斌编著 《非线性数学物理方程的行波解》 科学出版社

#王东明著 《消去法及其应用》 科学出版社 2002

*何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445

#Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press

Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997

#Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
#Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
#Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
#Dongming Wang, Elimination Practice,Imperial College Press 2004

David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004

George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759