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Infinitely many Lax pairs and symmetry constraints of the KP equation. (English) Zbl 0898.58029

Summary: Starting from a known Lax pair, one can get infinitely many coupled Lax pairs, infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways.

In this paper, taking the well-known Kadomtsev-Petviashvili (KP) equation as a special example, we show that infinitely many nonhomogeneous linear Lax pairs can be obtained by using infinitely many symmetries, differentiating the spectral functions with respect to the inner parameters. Using a known Lax pair and the Darboux transformations (DT), infinitely many nonhomogeneous nonlinear Lax pairs can also be obtained. By means of the infinitely many Lax pairs, DT, and the conformal invariance of the Schwartz form of the KP equation, infinitely many new nonlocal symmetries can be obtained naturally. Infinitely many integrable models in $\left(1+1\right)$-dimensions, $\left(2+1\right)$-dimensions, $\left(3+1\right)$-dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the KP equation related to the infinitely many Lax pairs.

##### MSC:
 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q53 KdV-like (Korteweg-de Vries) equations 37C80 Symmetries, equivariant dynamical systems
##### Keywords:
Lax pairs; symmetries; Darboux transformations; KP equation