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), 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 KdV equation and the (1+1)-dimensional AKNS system (including nonlinear 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 is correct for almost all the known (1+1)-dimensional integrable local evolution systems 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.

arXiv comment9 pages 3 figures

References in corpus (16)

• Financial rogue waves
• Modes and states in Quantum Optics
• Nonlinear control of PT-symmetry and non-Hermitian topological states
• Rogue waves in the atmosphere
• Nonequilibrium Physics in Biology
• Soliton molecules and asymmetric solitons in fluid systems via velocity resonance
• Lattice tests of beyond Standard Model dynamics
• On Critical Velocities in Exciton Superfluidity
• Polarized Nucleon Structure Functions within a Chiral Soliton Model
• Generalized Gravitational Entropy for Warped Anti-de Sitter Space
• Tunable three-body interactions in driven two-component Bose-Einstein condensates
• Dynamical signatures of quasiparticle interactions in quantum spin chains
• Gauge Symmetry in Background Charge Conformal Field Theory
• On a class of integrable systems of Monge-Ampère type
• Duality of positive and negative integrable hierarchies via relativistically invariant fields
• Differential constraints for the Kaup -- Broer system as a reduction of the 1D Toda lattice

Cited by in corpus (4)

• Ren-integrable and ren-symmetric integrable systems
• Higher dimensional integrable deformations of the modified KdV equation
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• Residual Symmetry Reductions and Painlevé Solitons