Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra

Abstract

The Kadomtsev–Petviashvili (KP) equation (ut+3uux/2+ 1/4 uxxx)x +3σuyy/4=0 allows an infinite-dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group depends on three arbitrary functions of time ‘‘t’’ and is shown to be related to a subalgebra of the loop algebra A(1)4. Low-dimensional subalgebras of the symmetry algebra are identified, specifically all those of dimension n≤3, and also a physically important six-dimensional Lie algebra containing translations, dilations, Galilei transformations, and ‘‘quasirotations.’’ New solutions of the KP equation are obtained by symmetry reduction, using the one-dimensional subalgebras of the symmetry algebra. These solutions contain up to three arbitrary functions of t.

Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation

“…That is to say in the case of transformation (3), v = θ (x , y ,t ) is a solution to (4) whenever u = θ (x, y,t) is a solution to (2). This condition implies if (2) and (4) have a unique solution, then…”

“…David, Kamran, Levi and Winternitz [2] studied the Lie point symmetry group of the KP equation via the traditional Lie group approach. Lou and Ma [3] developed the direct method presented by Clarkson and Kruskal (C-K) [4] to construct the finite symmetry group of the KP equation.…”

Group-invariant solutions of a nonlinear acoustics model

“…These commutation relations show that a Kac-Moody-Virasoro (kmv) structure can be associated with an infinite dimensional subalgebra of L [5], and the latter property tends to associate integrability with the (2 + 1)-dimensional ZK equation [13,14,15], which as is well-known can be linearized by a generalized hodograph transformation [16]. In order to classify subalgebras of L under the adjoint action of its Lie group G, we need to have an explicit expression for w(ǫ) = Ad(exp(ǫv))w 0 , for every pair of generators v, w 0 of L. However, using the commutation relations (2.2) such an expression can easily be obtained either by interpreting w(ǫ) as the flow of Ad through w 0 of the one-parameter subgroup generated by v, or again by rewriting w(ǫ) in terms of the Lie series (see [17, P. 205]).…”

“…In order to classify subalgebras of L under the adjoint action of its Lie group G, we need to have an explicit expression for w(ǫ) = Ad(exp(ǫv))w 0 , for every pair of generators v, w 0 of L. However, using the commutation relations (2.2) such an expression can easily be obtained either by interpreting w(ǫ) as the flow of Ad through w 0 of the one-parameter subgroup generated by v, or again by rewriting w(ǫ) in terms of the Lie series (see [17, P. 205]). The required classification of subalgebras of L under the adjoint action of G can henceforth be achieved by applying known techniques [13,17,18]. In this way, all one-dimensional and twodimensional subalgebras of L were classified in [5] and for the sake of completeness as already mentioned, we present again in this paper not only the list of canonical forms of non-equivalent two-dimensional subalgebras of L required for reductions to ODEs in which we are interested, but also the corresponding list for one-dimensional subalgebras.…”

Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

“…both of which are integrable soliton equations, possessing, respectively, infiniteand finite-dimensional Lie symmetry pseudo-groups, [1,5,6,7,18,19,20]. Let us recall how the classical Lie symmetry method, [22], works in the context of the KP equation.…”