Seventh order KdV equation

Seventh order KdV equation is a nonlinear partial differential equation

u_t + 6 * u * u_x + u_xxx − u_xxxxx + α * u_xxxxxxx = 0

Analytic solution

$$u(x, t) = 6*_C3^2*((60*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\delta-72*\delta*\gamma^2+720*\delta^2*\alpha-120*\beta^2*\delta-(6*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\gamma^2/\alpha)*JacobiND(_C2+_C3*x-(6*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*_C3^5*t/\alpha, \sqrt(2))^2/(\beta*(-120*\delta*\alpha+12*\gamma*\beta+12*\gamma^2+6*\beta^2+6*\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))$$

$$u(x, t) = -3*_C3^2*(-18*\delta*\gamma^2+180*\delta^2*\alpha+(15*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\delta-30*\beta^2*\delta-(3/2)*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2))*\gamma^2/\alpha)*JacobiCN(_C2+_C3*x-(3/2)*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2))*_C3^5*t/\alpha, (1/2)*\sqrt(2))^2/(\beta*(-30*\delta*\alpha+3*\gamma*\beta+3*\gamma^2+(3/2)*\beta^2+(3/2)*\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))$$

$$u(x, t) = -6*_C3^2*((60*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\delta-72*\delta*\gamma^2+720*\delta^2*\alpha-120*\beta^2*\delta-(6*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\gamma^2/\alpha)*JacobiDN(_C2+_C3*x-(6*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*_C3^5*t/\alpha, \sqrt(2))^2/(\beta*(-120*\delta*\alpha+12*\gamma*\beta+12*\gamma^2+6*\beta^2+6*\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))$$

$$u(x, t) = 3*_C3^2*(-18*\delta*\gamma^2+180*\delta^2*\alpha+(15*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\delta-30*\beta^2*\delta-(3/2)*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2))*\gamma^2/\alpha)*JacobiNC(_C2+_C3*x-(3/2)*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2))*_C3^5*t/\alpha, (1/2)*\sqrt(2))^2/(\beta*(-30*\delta*\alpha+3*\gamma*\beta+3*\gamma^2+(3/2)*\beta^2+(3/2)*\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))$$

$$u(x, t) = 6*_C3^2*(-(60*(-2*\gamma*\beta-\beta^2+12*\delta*\alpha+\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\delta-72*\delta*\gamma^2+720*\delta^2*\alpha-120*\beta^2*\delta+(6*(-2*\gamma*\beta-\beta^2+12*\delta*\alpha+\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\gamma^2/\alpha)*JacobiNS(_C2+_C3*x+(6*(-2*\gamma*\beta-\beta^2+12*\delta*\alpha+\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*_C3^5*t/\alpha, I)^2/(\beta*(-120*\delta*\alpha+12*\gamma*\beta+12*\gamma^2+6*\beta^2+6*\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))$$

$$u(x, t) = -6*_C3^2*(-(60*(-2*\gamma*\beta-\beta^2+12*\delta*\alpha+\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\delta-72*\delta*\gamma^2+720*\delta^2*\alpha-120*\beta^2*\delta+(6*(-2*\gamma*\beta-\beta^2+12*\delta*\alpha+\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\gamma^2/\alpha)*JacobiSN(_C2+_C3*x+(6*(-2*\gamma*\beta-\beta^2+12*\delta*\alpha+\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*_C3^5*t/\alpha, I)^2/(\beta*(-120*\delta*\alpha+12*\gamma*\beta+12*\gamma^2+6*\beta^2+6*\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))$$

$$u(x, t) = 2*_C2^2*(-(2*(2*\gamma*\beta+\beta^2-12*\delta*\alpha-\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2)))*\gamma/\alpha-24*\gamma*\delta+40*\beta*\delta)/(2*\beta^2+2*\sqrt(4*\gamma^2*\beta^2+4*\gamma*\beta^3+\beta^4-40*\delta*\alpha*\beta^2))$$

$$u(x,t)=(86625/591361)*sech(5*(x-(180000/591361)*t)/\sqrt(1538))^6$$

$$u(x,t)=(86625/591361)*csch(5*(x-(180000/591361)*t)/\sqrt(1538))^6$$

$$u(x,t)=(86625/591361)*sec(5*(x-(180000/591361)*t)/\sqrt(1538))^6$$

$$u(x,t)=(86625/591361)*csc(5*(x-(180000/591361)*t)/\sqrt(1538))^6$$

Traveling wave plot

Seventh order KdV equation traveling wave plot

Reference

Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential 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