Abstract
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa–Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators of specific forms is carried out, in order to obtain bi-Hamiltonian structures for the same systems of equations. Using reciprocal transformations, some exact solutions and Lax pairs are also constructed for the systems considered.
Highlights
In recent years there has been a growing interest in integrable non-evolutionary partial differential equations of the form (1 − Dx2)ut = F (u, ux, uxx, uxxx, . . .), u = u(x, t), ∂ Dx =, ∂x (1)where F is some function of u and its derivatives with respect to x
The perturbative symmetry approach has yielded a classification of integrable two-component systems of the form (3), producing two systems with quadratic nonlinearities (Theorem 2), two systems with cubic nonlinearities (Theorem 3), and two mixed quadratic/cubic systems (Theorem 4); the systems with mixed nonlinear terms include the others as limiting cases, by sending suitable parameters to zero
An alternative approach via compatible Hamiltonian operators has provided a different set of two-component systems, and has allowed us to find bi-Hamiltonian structures for all of the systems obtained from the symmetry approach
Summary
(see [15, 23] and [11, 30], respectively) All of the latter equations of Camassa-Holm type are integrable by the inverse scattering transform. We outline the perturbative symmetry approach in the context of non-evolutionary systems with two dependent variables, and explain how it leads to an integrability test for such systems. The fourth section is concerned with a different problem, namely that of classifying pairs of compatible Hamiltonian operators of specific forms in two dependent variables with the purpose of providing a bi-Hamiltonian structure for the systems in the aforementioned list. The paper ends with conclusions and suggestions for future work
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