Hamiltonian Integrability of Two-Component Short Pulse Equations

We obtain the bi-Hamiltonian structure for some of the two-component short pulse equations proposed in the literature to generalize the original short pulse equation when polarized pulses propagate in anisotropic media.

💡 Research Summary

The paper investigates the integrability of several two‑component generalizations of the short‑pulse equation (SPE), which models the propagation of ultra‑short optical pulses. While the scalar SPE is known to be integrable via an inverse‑scattering transform, realistic optical media often require a description that includes two coupled field components, for instance the orthogonal polarization modes in an anisotropic material. The authors focus on three representative two‑component systems that have appeared in the literature: (i) Matsuno’s coupled SPE, (ii) the symmetric vector SPE introduced by Pietrzyk, Kaup and Zhou, and (iii) a newly proposed anisotropic extension that explicitly incorporates a material parameter.

For each system the authors construct a bi‑Hamiltonian formulation. They first identify a first Hamiltonian operator (J_{0}), typically a differential operator involving (\partial_{x}) and the identity matrix, which yields the evolution equations through (u_{t}=J_{0},\delta H_{0}/\delta u). A corresponding Hamiltonian functional (H_{0}) is obtained from the Lagrangian or energy density of the model. Next, a second, compatible Hamiltonian operator (J_{1}) is built; it contains non‑local terms such as (\partial_{x}^{-1}) and nonlinear combinations of the fields. Compatibility is verified by checking that the Schouten–Nijenhuis bracket (