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Partial differential equations possessing Frobenius integrable decompositions. (English) Zbl 1203.35059
Summary: Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Bäcklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.
MSC:
35C05Solutions of PDE in closed form
35Q53KdV-like (Korteweg-de Vries) equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35A30Geometric theory for PDE, characteristics, transformations