Gibbons–Tsarev equation

The Gibbons–Tsarev equation is a nonlinear partial differential equation^[1]. In its simplest form, in two dimensions, it may be written as follows:

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 1996^[2], and subsequently developed in ^[3]

Gibbons-Tsarev equation has analytic solutions.

Analytic solution

u(x,t)=-{\frac {1}{2}}x^{2}+{_{C}1}x+{_{C}2}

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

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

Traveling wave plot

Reference

^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p764 CRC PRESS
^ J. Gibbons and S.P. Tsarev, Reductions of the Benney Equations, Physics Letters A, Vol. 211, Issue 1, Pages 19–24, 1996
^ J. Gibbons and S.P. Tsarev, Conformal Maps and the reduction of Benney equations,Phys Letters A, vol 258,No4-6, pp 263–271, 1999

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 p764 CRC PRESS

[2] J. Gibbons and S.P. Tsarev, Reductions of the Benney Equations, Physics Letters A, Vol. 211, Issue 1, Pages 19–24, 1996

[3] J. Gibbons and S.P. Tsarev, Conformal Maps and the reduction of Benney equations,Phys Letters A, vol 258,No4-6, pp 263–271, 1999

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