Gibbons–Tsarev equation

The Gibbons–Tsarev equation is a second order nonlinear partial differential equation. 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, in this case 2 of them. It was first introduced by John Gibbons and Serguei Tsarev in .

The theory of this equation was subsequently developed in . 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 from a fixed domain, normally the complex half-plane, to a domain with N slits. Each slit is taken along a fixed curve with a variable end point, 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:

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

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

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