The Gibbons–Tsarev equation is an integrable second order nonlinear partial differential equation. In its simplest form, in two dimensions, it may be written as follows:
u t u x t − u x u t t + u x x + 1 = 0 ( 1 ) {\displaystyle u_{t}u_{xt}-u_{x}u_{tt}+u_{xx}+1=0\qquad (1)}
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 1996, This system was also derived, as a condition that two quadratic Hamiltonians should have vanishing Poisson bracket.
Relationship to families of slit maps The theory of this equation was subsequently developed by Gibbons and Tsarev. In N {\displaystyle N} independent variables, one looks for solutions of the Benney hierarchy in which only N {\displaystyle N} of the moments A n {\displaystyle A^{n}} are independent. The resulting system may always be put in Riemann invariant form. Taking the characteristic speeds to be p i {\displaystyle p_{i}} and the corresponding Riemann invariants to be λ i {\displaystyle \lambda _{i}} , they are related to the zeroth moment A 0 {\displaystyle A^{0}} by:
∂ p i ∂ λ j = − ∂ A 0 ∂ λ j p i − p j , ( 2 a ) {\displaystyle {\frac {\partial p_{i}}{\partial \lambda _{j}}}=-{\frac {\frac {\partial A^{0}}{\partial \lambda _{j}}}{p_{i}-p_{j}}},\qquad (2a)}
∂ A 0 ∂ λ i ∂ λ j = 2 ∂ A 0 ∂ λ i ∂ A 0 λ j ( p i − p j ) 2 . ( 2 b ) {\displaystyle {\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}}}.\qquad (2b)}
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