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Lou, Sen-Yue; Hu, Xing-Biao

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

J. Math. Phys. 38, No. 12, 6401-6427 (1997).
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 \((1+1)\)-dimensions, \((2+1)\)-dimensions, \((3+1)\)-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.

Cited in 1 Review
Cited in 92 Documents

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
37C80 Symmetries, equivariant dynamical systems (MSC2010)

Keywords:

Lax pairs; symmetries; Darboux transformations; KP equation
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\textit{S.-Y. Lou} and \textit{X.-B. Hu}, J. Math. Phys. 38, No. 12, 6401--6427 (1997; Zbl 0898.58029)
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