五阶KdV方程

五阶KdV方程(Fifth order KdV equation)是一个非线性偏微分方程，简称fKdV方程：^[1] $u_{t}+\alpha *u^{2}*u_{x}+\beta *u_{x}*u_{xx}+\gamma *u*u_{xxx}+\delta *u_{xxxxx}=0$

解析解

u(x,t)=6*_{C}3^{2}*(-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\gamma ^{2}/\alpha +(60*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\delta *\beta ^{2})*JacobiND(_{C}2+_{C}3*x-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*_{C}3^{5}*t/\alpha ,{\sqrt {(}}2))^{2}/(\beta *(6*\beta ^{2}-120*\delta *\alpha +12*\gamma ^{2}+12*\beta *\gamma +6*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))

u(x,t)=6*_{C}3^{2}*(-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\gamma ^{2}/\alpha +(60*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\delta *\beta ^{2})*JacobiNS(_{C}2+_{C}3*x-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*_{C}3^{5}*t/\alpha ,I)^{2}/(\beta *(6*\beta ^{2}-120*\delta *\alpha +12*\gamma ^{2}+12*\beta *\gamma +6*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))

u(x,t)=-3*_{C}3^{2}*(-(3/2)*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2}))*\gamma ^{2}/\alpha +(15*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -18*\delta *\gamma ^{2}+180*\delta ^{2}*\alpha -30*\delta *\beta ^{2})*JacobiCN(_{C}2+_{C}3*x-(3/2)*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2}))*_{C}3^{5}*t/\alpha ,(1/2)*{\sqrt {(}}2))^{2}/(\beta *((3/2)*\beta ^{2}-30*\delta *\alpha +3*\gamma ^{2}+3*\beta *\gamma +(3/2)*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))

u(x,t)=-6*_{C}3^{2}*(-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma +{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\gamma ^{2}/\alpha +(60*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma +{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\delta *\beta ^{2})*JacobiDN(_{C}2+_{C}3*x-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma +{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*_{C}3^{5}*t/\alpha ,{\sqrt {(}}2))^{2}/(\beta *(6*\beta ^{2}-120*\delta *\alpha +12*\gamma ^{2}+12*\beta *\gamma -6*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))

解析失败 (转换错误。服务器（“https://wikimedia.org/api/rest_”）报告：“Cannot get mml. upstream request timeout”): {\displaystyle u(x,t)=_{C}5-(3*(-4*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\beta ^{2}*_{C}5^{2}-8*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\gamma ^{2}*_{C}5^{2}-(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\beta *_{C}5^{3}*\alpha -(3/2)*\gamma *(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{3}*\alpha +(1/4)*\gamma *(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{3}*\beta ^{2}/\delta +(2/5)*\gamma ^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{3}*\beta /\delta -\beta ^{2}*_{C}5^{4}*\alpha +4800*_{C}3^{6}*\beta *\delta ^{2}*_{C}5-160*_{C}3^{4}*\beta ^{2}*_{C}5^{2}*\delta -800*\delta ^{2}*\alpha *_{C}3^{4}*_{C}5^{2}+16*\gamma ^{2}*_{C}5^{3}*\beta *_{C}3^{2}-320*\gamma ^{2}*_{C}3^{4}*_{C}5^{2}*\delta +7200*\gamma *_{C}3^{6}*\delta ^{2}*_{C}5+10*\gamma *\beta ^{2}*_{C}5^{3}*_{C}3^{2}-3*\gamma *_{C}5^{4}*\beta *\alpha -48000*_{C}3^{8}*\delta ^{3}+2*\beta ^{3}*_{C}5^{3}*_{C}3^{2}+8*\gamma ^{3}*_{C}5^{3}*_{C}3^{2}-2*\gamma ^{2}*_{C}5^{4}*\alpha +10*_{C}5^{4}*\delta *\alpha ^{2}+20*\gamma *_{C}5^{3}*\delta *\alpha *_{C}3^{2}-480*\gamma *_{C}3^{4}*_{C}5^{2}*\beta *\delta -12*\gamma *_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{2}*\beta +40*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *\alpha *_{C}5^{2}+120*_{C}3^{4}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *\gamma *_{C}5+120*_{C}3^{4}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *\beta *_{C}5-1200*_{C}3^{6}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta ^{2}+40*_{C}5^{3}*\delta *\alpha *\beta *_{C}3^{2}+(1/20)*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\beta ^{3}*_{C}5^{3}/\delta +(1/5)*\gamma ^{3}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{3}/\delta ))*JacobiSN(_{C}2+_{C}3*x+(1/25)*_{C}3*(3*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5*\beta *\gamma +45*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *_{C}5*\alpha +(1/2*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2})))*\alpha *_{C}5^{2}*\beta -150*_{C}3^{4}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *\beta -9*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\gamma ^{2}*_{C}5-(9/4*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2})))*\gamma *\alpha *_{C}5^{2}+90*_{C}3^{4}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *\gamma +(1/20)*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{2}*\beta *\gamma ^{2}/\delta -(1/20)*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{2}*\beta ^{2}*\gamma /\delta +(3/10)*\gamma ^{3}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{2}/\delta +420*_{C}3^{4}*\delta *_{C}5*\beta *\gamma +3600*_{C}3^{6}*\delta ^{2}*\gamma +300*_{C}3^{4}*\delta *_{C}5*\beta ^{2}-6000*_{C}3^{6}*\delta ^{2}*\beta +2*_{C}5^{2}*\beta *\gamma ^{2}*_{C}3^{2}-2*_{C}5^{2}*\beta ^{2}*\gamma *_{C}3^{2}+_{C}5^{3}*\beta *\gamma *\alpha -130*_{C}5^{2}*\delta *\alpha *\beta *_{C}3^{2}+15*_{C}5^{3}*\delta *\alpha ^{2}+12*\gamma ^{3}*_{C}5^{2}*_{C}3^{2}-3*\gamma ^{2}*_{C}5^{3}*\alpha -360*_{C}3^{4}*\gamma ^{2}*\delta *_{C}5)*t/(\delta *(-\alpha *_{C}5+2*\gamma *_{C}3^{2}+(1/20)*\gamma *(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))/\delta )),(1/20)*{\sqrt {(}}10)*{\sqrt {(}}(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))/\delta )/_{C}3)^{2}/(60*_{C}3^{4}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *\beta -6*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\gamma ^{2}*_{C}5+60*_{C}3^{4}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *\gamma -3*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5*\beta ^{2}-(3/2*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2})))*\gamma *\alpha *_{C}5^{2}-(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\alpha *_{C}5^{2}*\beta +(1/20)*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\beta ^{3}*_{C}5^{2}/\delta -120*_{C}3^{4}*\delta *_{C}5*\beta ^{2}+10*_{C}5^{2}*\beta ^{2}*\gamma *_{C}3^{2}+16*_{C}5^{2}*\beta *\gamma ^{2}*_{C}3^{2}-240*_{C}3^{4}*\gamma ^{2}*\delta *_{C}5-3*_{C}5^{3}*\beta *\gamma *\alpha -\beta ^{2}*_{C}5^{3}*\alpha +2400*_{C}3^{6}*\delta ^{2}*\beta +2400*_{C}3^{6}*\delta ^{2}*\gamma +10*_{C}5^{3}*\delta *\alpha ^{2}+8*\gamma ^{3}*_{C}5^{2}*_{C}3^{2}-2*\gamma ^{2}*_{C}5^{3}*\alpha +2*\beta ^{3}*_{C}5^{2}*_{C}3^{2}-360*_{C}3^{4}*\delta *_{C}5*\beta *\gamma +20*_{C}5^{2}*\delta *\alpha *\beta *_{C}3^{2}+(2/5)*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{2}*\beta *\gamma ^{2}/\delta +(1/4)*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{2}*\beta ^{2}*\gamma /\delta -9*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5*\beta *\gamma +30*_{C}3^{2}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*\delta *_{C}5*\alpha +(1/5)*\gamma ^{3}*(2*\gamma *_{C}5+_{C}5*\beta -40*\delta *_{C}3^{2}+{\sqrt {(}}4*\gamma ^{2}*_{C}5^{2}+4*\gamma *_{C}5^{2}*\beta +_{C}5^{2}*\beta ^{2}-40*\delta *\alpha *_{C}5^{2}))*_{C}5^{2}/\delta )}

行波图

General Fifth order KdV equation traveling wave plot

参考文献

^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p1034 CRC PRESS

*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

[1] Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p1034 CRC PRESS

[1]