Gibbons–Tsarev equation

Gibbons-Tsarev equation is a nonlinear partial differential equation

$u_{t}*u_{xt}-u_{x}*u_{tt}+u_{xx}+1=0$

This equation arises in hydrodynamics, in the theory of Benney moment theory, first discussed by John Gibbons and Serguei Tsarev in 1999

Gibbon-Tsarev equation has analytic solutions.

Analytic solution

$u(x,t)=-(1/2)*x^{2}+_{C}1*x+_{C}2$

$u(x,t)=exp(_{c}[2]*x)*_{C}1/_{c}[2]+x/_{c}[2]+_{C}2+(1/2)*_{c}[2]*t^{2}+_{C}3*t+_{C}4$

$u(x,t)=-(1/2)*(_{C}3+_{C}1*x+_{C}2*t)^{2}/_{C}1^{2}+(_{C}3+_{C}1*x+_{C}2*t)*_{C}2+_{C}3$

Traveling wave plot

Gibbons-Tsarev equation traveling wave plot

Gibbons-Tsarev equation traveling wave plot

Gibbons-Tsarev equation traveling wave plot

Gibbons-Tsarev equation traveling wave plot 4

Gibbons-Tsarev equation traveling wave plot 5

Reference

1. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
2. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
3. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
4. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
5. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
6. Dongming Wang, Elimination Practice,Imperial College Press 2004
7. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
8. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759