On the infinite‐dimensional symmetry group of the Davey–Stewartson equations

The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE’s) invariant, is obtained. The general element of this algebra depends on four arbitrary functions of time. The algebra is shown to have a loop structure, a property shared by the symmetry algebras of all known (2+1)‐dimensional integrable nonlinear equations. Subalgebras of the symmetry algebra are classified and used to reduce the DSE’s to various equations involving only two independent variables.

[1]  V. Kac.  Infinite dimensional Lie algebras: Frontmatter , 1990 .

[2]  Jiří Patera,et al.  Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group , 1975 .

[3]  M. Jimbo,et al.  Solitons and Infinite Dimensional Lie Algebras , 1983 .

[4]  Decio Levi,et al.  Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra , 1986 .

[5]  R. T. Sharp,et al.  Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups , 1977 .

[6]  Alfred Ramani,et al.  Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable? , 1986 .

[7]  Fritz Schwarz,et al.  A reduce package for determining lie symmetries of ordinary and partial differential equations , 1984 .

[8]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[9]  D. Levi,et al.  Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev-Petviashvili equation , 1986 .

[10]  K. Stewartson,et al.  On three-dimensional packets of surface waves , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[11]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[12]  D. Anker,et al.  On the soliton solutions of the Davey-Stewartson equation for long waves , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[13]  G. Soliani,et al.  Group analysis of the three‐wave resonant system in (2+1) dimensions , 1986 .