2D reductions of the equation $u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$ and their nonlocal symmetries

Abstract

We consider the 3D equation $u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$ and its 2D reductions: (1) $u_{yy}=(u_y+y)u_{xx}-u_x{u}_{xy}-2$ (which is equivalent to the Gibbons-Tsarev equation) and (2) $u_{yy}=(u_y+2x)u_{xx}+(y-u_x)u_{xy}-u_x$. Using reduction of the known Lax pair for the 3D equation, we describe nonlocal symmetries of~(1) and~(2) and show that the Lie algebras of these symmetries are isomorphic to the Witt algebra.

References in corpus (2)

• Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems
• Integrability of the Manakov--Santini hierarchy

Cited by in corpus (1)

• Reductions of the universal hierarchy and rdDym equations and their symmetry properties