Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems

Abstract

We present a complete description of $2$-dimensional equations that arise as symmetry reductions of fourf $3$-dimensional Lax-integrable equations: (1) the universal hierarchy equation~$u_{yy}=u_z{u}_{xy}-u_y{u}_{xz}$; (2) the 3D rdDym equation $u_{ty}=u_x{u}_{xy}-u_y{u}_{xx}$; (3) The basic Veronese web equation $u_{ty}=u_t{u}_{xy}-u_y{u}_{tx}$; (4) Pavlov's equation $u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$.

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Cited by in corpus (9)

• Coverings over Lax integrable equations and their nonlocal symmetries
• Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
• Computing with Hamiltonian operators
• Nonlocal symmetries of Lax integrable equations: a comparative study
• 2D reductions of the equation $u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$ and their nonlocal symmetries
• Integrability properties of some symmetry reductions
• Reductions of the universal hierarchy and rdDym equations and their symmetry properties
• A note on the Hyper--CR equation, and gauged $N=2$ supergravity
• The Gibbons--Tsarev equation: symmetries, invariant solutions, and applications