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FROBENIUS INTEGRABLE DECOMPOSITIONS FOR TWO CLASSES OF NONLINEAR EVOLUTION EQUATIONS WITH VARIABLE COEFFICIENTS

Frobenius integrable decompositions are introduced for partial differential equations with variable coefficients. Two classes of partial differential equations with variable coefficients are transformed into Frobenius integrable ordinary differential equations. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations, the Boussinesq equation and the Camassa–Holm equation with variable coefficients.

The multi-component dispersive long wave equation hierarchy, its integrable couplings and their Hamiltonian structures

Abstract An isospectral problem is established based on loop algebra X ∼ , by making use of the generalized Tu scheme the multi-component Liouville integrable dispersive long wave (DLW) equation hierarchy is obtained. Then, two expanding loop algebra F ∼ M and Y ∼ are presented, which devoted to working out two integrable couplings of the multi-component DLW equation hierarchy. Finally, the Hamiltonian structures of the multi-component DLW equation hierarchy and the second integrable couplings are obtained by employing the trace variational identity.