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Infinite-dimensional homogeneous manifolds with translational symmetry and nonlinear partial differential equations. (English) Zbl 0644.58024
A construction relating scale-invariant, nonlinear partial differential equations and the orbits of the group of translations on infinite- dimensional homogeneous manifolds is proposed. It presupposes that the homogeneous manifold M has translations and scale transformations among its automorphisms. This assumption is made to ensure that the set of orbits of the group of translations is invariant under scale transformations. What is important is that the proposed construction provides a way of inducing scale covariance of derived nonlinear equations for which the set of orbits can be identified with a set of solutions of these equations. Application to the derivation of the potential Korteweg-de Vries (KdV) and the sine-Gordon equation yields as an important intermediate result the construction of their respective Lax pairs out of the commuting vectors tangent to the orbits.
In yet another application, an infinite-dimensional, scale-invariant Riccati equation is derived. The latter is known to be related to the potential Kadomtsev-Petviashvili (KP) equation. The orbit leading to the Riccati equation is then computed in different charts. The transition functions between charts are shown to generate the nonlinear term in the potential KP equation. Also, the relation to the Zakharov-Shabat dressing method is briefly discussed.

MSC:
58J70 Invariance and symmetry properties for PDEs on manifolds
35Q99 Partial differential equations of mathematical physics and other areas of application
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