Gibbons–Tsarev equation

The Gibbons–Tsarev equation is a second order 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

The equation arises in the theory of dispersionless integrable systems, as the condition that solutions of the Benney moment equations may be parametrised by only finitely many of their dependent variables. It was first discussed by John Gibbons and Serguei Tsarev in ^[2], and subsequently developed in ^[3].

In N independent variables, the equation has solutions parametrised by N functions of a single variable. A class of these may be constructed in terms of N-parameter families of conformal maps to a domain with N slits, and the equation can be understood as the consistency condition between the Loewner equations describing the growth of each slit.

Analytic solution

Some examples of analytic solutions of the 2-dimensional system are:

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 Academic 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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