Deformation conjecture: deforming lower dimensional integrable systems to higher dimensional ones by using conservation laws

Abstract

Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries and/or infinitely many conservation laws), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the usual (1 + 1)-dimensional Korteweg-de Vries (KdV) equation and the (1 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) system (including nonlinear Schrödinger (NLS) equation as a special example). It is interesting that the deformed (3+1)-dimensional KdV equation is also an extension of the (1 + 1)-dimensional Harry-Dym (HD) type equations which are reciprocal links of the (1+1)-dimensional KdV equation. The Lax pairs of the (3 + 1)-dimensional KdV-HD system and the (2 + 1)-dimensional AKNS system are explicitly given. The higher order symmetries, i.e., the whole (3 + 1)-dimensional KdV-HD hierarchy, are also explicitly obtained via the deformation algorithm. The single soliton solution of the (3 + 1)-dimensional KdV-HD equation is implicitly given. Because of the effects of the deformation, the symmetric soliton shape of the usual KdV equation is no longer conserved and deformed to be asymmetric and/or multi-valued. The deformation conjecture holds for all the known (1 +1)-dimensional integrable local evolution systems that have been checked, and we have not yet found any counter-example so far. The introduction of a large number of (D + 1)-dimensional integrable systems of this paper explores a serious challenge to all mathematicians and theoretical physicists because the traditional methods are no longer directly valid to solve these integrable equations.