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 .

Relationship to families of slit maps

The theory of this equation was subsequently developed in . In N independent variables, one looks for solutions of the Benney hierarchy in which only N of the moments A^n are independent. The resulting system may always be put in Riemann invariant form. Taking the characteristic speeds to be $p_{i}$ and the corresponding Riemann invariants to be $\lambda _{i}$, they are related to the zeroth moment $A^{0}$ by:

${\frac {\partial p_{i}}{\partial \lambda _{j}}}=-{\frac {\frac {\partial A^{0}}{\partial \lambda _{j}}}{p_{i}-p_{j}}},$

${\frac {\partial A^{0}}{\partial \lambda _{i}\partial \lambda _{j}}}=2{\frac {{\frac {\partial A^{0}}{\partial \lambda _{i}}}{\frac {\partial A^{0}}{\lambda _{j}}}}{(p_{i}-p_{j})^{2}}}.$

This system 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 D, normally the complex half-plane, to a domain with N slits. Each slit is taken along a fixed curve with one end fixed on the boundary of D and one variable end point; the system can then 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:

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

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

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