On symmetries of the Gibbons-Tsarev equation

Abstract

We study the Gibbons-Tsarev equation $z_{yy}+z_x{z}_{xy}-z_y{z}_{xx}+1=0$ and, using the known Lax pair, we construct infinite series of conservation laws and the algebra of nonlocal symmetries in the covering associated with these conservation laws. We prove that the algebra is isomorphic to the Witt algebra. Finally, we show that the constructed symmetries are unique in the class of polynomial ones.

36 pages; minor corrections and improvements

References in corpus (4)

• Integrability of the Manakov--Santini hierarchy
• Hamiltonian structure of reductions of the Benney system
• Integrability in differential coverings
• 2D reductions of the equation $u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$ and their nonlocal symmetries

Cited by in corpus (1)

• Integrability of anti-self-dual vacuum Einstein equations with nonzero cosmological constant: an infinite hierarchy of nonlocal conservation laws