库普-库珀施密特方程(Kaup-Kupershmidt Equation)是一个非线性偏微分方程：

\frac{\partial^4 u(x,t)}{\partial x^4} +\frac{\partial u(x,t)}{\partial x} +45(\frac{\partial u(x,t)}{\partial x}u(x,t)^2 -(75/2)\frac{\partial^2 u(x,t)}{\partial x^2}\frac{\partial u(x,t)}{\partial x} -15u(x,t)*\frac{\partial^3 u(x,t)}{\partial x^3}

行波解
利用Maple软件包TWSolution，随所选定展开函数不同，可得多种行波解
;tanh 展开
g[2] := {u(x, t) = -(2/3)(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2+(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2tanh(_C1+(-(1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[3] := {u(x, t) = -(2/3)(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2+(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2tanh(_C1+(-(1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}
g[4] := {u(x, t) = -(2/3)((1/2)sqrt(2)-(1/2I)sqrt(2))^2+((1/2)sqrt(2)-(1/2I)sqrt(2))^2tanh(_C1+((1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[5] := {u(x, t) = -(2/3)((1/2)sqrt(2)+(1/2I)sqrt(2))^2+((1/2)sqrt(2)+(1/2I)sqrt(2))^2tanh(_C1+((1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}
g[6] := {u(x, t) = -(4/3)(-(1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2+2(-(1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2tanh(_C1+(-(1/44)sqrt(2)11^(3/4)-(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[7] := {u(x, t) = -(4/3)(-(1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2+2(-(1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2tanh(_C1+(-(1/44)sqrt(2)11^(3/4)+(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[8] := {u(x, t) = -(4/3)((1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2+2((1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2tanh(_C1+((1/44)sqrt(2)11^(3/4)-(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[9] := {u(x, t) = -(4/3)((1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2+2((1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2tanh(_C1+((1/44)sqrt(2)11^(3/4)+(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}

;JacobiSN 展开
g[2] := {u(x, t) = -(1/2)_C3^2-(1/6)sqrt(-3_C3^4-4)+((1/2)_C3^2+(1/2)sqrt(-3_C3^4-4))JacobiSN(_C2+_C3x+_C4t, (1/2)sqrt(2_C3^2+2sqrt(-3*_C3^4-4))/_C3)^2}
g[3] := {u(x, t) = -(1/2)_C3^2+(1/6)sqrt(-3_C3^4-4)+((1/2)_C3^2-(1/2)sqrt(-3_C3^4-4))JacobiSN(_C2+_C3x+_C4t, (1/2)sqrt(2_C3^2-2sqrt(-3*_C3^4-4))/_C3)^2}
g[4] := {u(x, t) = -4_C3^2-(2/33)sqrt(-1452_C3^4-11)+(4_C3^2+(2/11)sqrt(-1452_C3^4-11))JacobiSN(_C2+_C3x+_C4t, (1/22)sqrt(242_C3^2+11sqrt(-1452*_C3^4-11))/_C3)^2}
g[5] := {u(x, t) = -4_C3^2+(2/33)sqrt(-1452_C3^4-11)+(4_C3^2-(2/11)sqrt(-1452_C3^4-11))JacobiSN(_C2+_C3x+_C4t, (1/22)sqrt(242_C3^2-11sqrt(-1452*_C3^4-11))/_C3)^2}

;sech 展开
g[2] := {u(x, t) = (1/3)(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2-(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2sech(_C1+(-(1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[3] := {u(x, t) = (1/3)(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2-(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2sech(_C1+(-(1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}
g[4] := {u(x, t) = (1/3)((1/2)sqrt(2)-(1/2I)sqrt(2))^2-((1/2)sqrt(2)-(1/2I)sqrt(2))^2sech(_C1+((1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[5] := {u(x, t) = (1/3)((1/2)sqrt(2)+(1/2I)sqrt(2))^2-((1/2)sqrt(2)+(1/2I)sqrt(2))^2sech(_C1+((1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}
g[6] := {u(x, t) = (2/3)(-(1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2-2(-(1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2sech(_C1+(-(1/44)sqrt(2)11^(3/4)-(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[7] := {u(x, t) = (2/3)(-(1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2-2(-(1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2sech(_C1+(-(1/44)sqrt(2)11^(3/4)+(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[8] := {u(x, t) = (2/3)((1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2-2((1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2sech(_C1+((1/44)sqrt(2)11^(3/4)-(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[9] := {u(x, t) = (2/3)((1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2-2((1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2sech(_C1+((1/44)sqrt(2)11^(3/4)+(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}

;sec、coth 展开
g[2] := {u(x, t) = -(1/3)(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2+(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2sec(_C1+(-(1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[3] := {u(x, t) = -(1/3)(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2+(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2sec(_C1+(-(1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}
g[4] := {u(x, t) = -(1/3)((1/2)sqrt(2)-(1/2I)sqrt(2))^2+((1/2)sqrt(2)-(1/2I)sqrt(2))^2sec(_C1+((1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[5] := {u(x, t) = -(1/3)((1/2)sqrt(2)+(1/2I)sqrt(2))^2+((1/2)sqrt(2)+(1/2I)sqrt(2))^2sec(_C1+((1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}
g[6] := {u(x, t) = -(2/3)(-(1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2+2(-(1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2sec(_C1+(-(1/44)sqrt(2)11^(3/4)-(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[7] := {u(x, t) = -(2/3)(-(1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2+2(-(1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2sec(_C1+(-(1/44)sqrt(2)11^(3/4)+(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[8] := {u(x, t) = -(2/3)((1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2+2((1/22)sqrt(2)11^(3/4)-(1/22I)sqrt(2)11^(3/4))^2sec(_C1+((1/44)sqrt(2)11^(3/4)-(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[9] := {u(x, t) = -(2/3)((1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2+2((1/22)sqrt(2)11^(3/4)+(1/22I)sqrt(2)11^(3/4))^2sec(_C1+((1/44)sqrt(2)11^(3/4)+(1/44I)sqrt(2)11^(3/4))x+_C3t)^2}
g[10] := {u(x, t) = -(2/3)(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2+(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2coth(_C1+(-(1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}

;csch 展开
{u(x, t) = _C4}
g[2] := {u(x, t) = (1/3)(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2+(-(1/2)sqrt(2)-(1/2I)sqrt(2))^2csch(_C1+(-(1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[3] := {u(x, t) = (1/3)(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2+(-(1/2)sqrt(2)+(1/2I)sqrt(2))^2csch(_C1+(-(1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}
g[4] := {u(x, t) = (1/3)((1/2)sqrt(2)-(1/2I)sqrt(2))^2+((1/2)sqrt(2)-(1/2I)sqrt(2))^2csch(_C1+((1/2)sqrt(2)-(1/2I)sqrt(2))x+_C3*t)^2}
g[5] := {u(x, t) = (1/3)((1/2)sqrt(2)+(1/2I)sqrt(2))^2+((1/2)sqrt(2)+(1/2I)sqrt(2))^2csch(_C1+((1/2)sqrt(2)+(1/2I)sqrt(2))x+_C3*t)^2}

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