We study a new hierarchy of equations containing the Short Pulse equation, which describes the evolution of very short pulses in nonlinear media, and the Elastic Beam equation, which describes nonlinear transverse oscillations of elastic beams under tension. We show that the hierarchy of equations is integrable. We obtain the two compatible Hamiltonian structures. We construct an infinite series of both local and nonlocal conserved charges. A Lax description is presented for both systems. For the Elastic Beam equations we also obtain a nonstandard Lax representation.

We construct the most general supersymmetric two boson system that is integrable.We obtain the Lax operator and the nonstandard Lax representation for this system. We show that, under appropriate redefinition of variables, this reduces to the supersymmetric nonlinear Schrödinger equation without any arbitrary parameter which is known to be integrable. We show that this supersymmetric system has three local Hamiltonian structures just like the bosonic counterpart and we show how the supersymmetric KdV equation can be embedded into this system.

We prove the integrability of the short pulse equation derived recently by Schäfer and Wayne from a hamiltonian point of view. We give its bi-hamiltonian structure and show how the recursion operator defined by the hamiltonian operators is connected with the one obtained by Sakovich andSakovich. An alternative zero-curvature formulation is also given.

We generalize the construction of Gelfand-Dikii brackets to the case of nonstandard Lax equations. We also discuss the possible origin of Kac-Moody algebras present in such systems.

We give a Lax description for the system of polytropic gas equations. The special structure of the Lax function naturally leads to the two infinite sets of conserved charges associated with this system. We obtain closed form expressions for the conserved charges as well as the generating functions for them. We show how the study of these generating functions can naturally lead to the recursion relation between the conserved quantities as well as the higher order Hamiltonian structures.