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The method, developed by Gardner, Greene, Kruskal and Miura, for solving nonlinear evolution equations is looked at from three perspectives; (i) as a nonlinear Fourier transform, (ii) as a completely integrable Hamiltonian system, and (iii) from the operator theoretic standpoint. The interrelation between the three points of view is discussed and, in particular, the ubiquitous role of the “transmission coefficient” is brought out. The spectral representation of Lax's equation is studied in an abstract setting.

The characterization of di?usion of orbits in Hamiltonian quasiintegrable systems is a relevant topic in dynamics. For quasi-integrable Hamiltonian systems a possible model for global di?usion, valid for perturbation larger than a critical value, was given by Chirikov; while for smaller perturbation the Nekhoroshev theorem leave the possibility of exponentially slow di?usion along a peculiar the Arnold’s web. We have studied this problem using a numerical approach. The aim of this chapter is to give the state of the art concerning the detection of slow Arnold’s di?usion in quasi-integrable Hamiltonian systems.

In this paper, a difference scheme satisfying two conservative laws for the periodic Camassa-Holm equation is given. Convergence is proved. The numerical simulation exhibits the time evolution and the interaction of solitary wave solutions.

By using an iterative algebraic method, we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation. It is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure. Under a Bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional Hamiltonian system. By introducing the Abel-Jacobi coordinates, an algebro-geometric solution for the dispersive long wave equation is derived by resorting to the Riemann theta function.

This chapter is entirely devoted to the construction and a geometric study of a big family of integrable Hamiltonian systems. The phase space is C2d but equipped with many different Poisson structures: for each non-zero φ ∈ C[x, y] we construct (in Paragraph 2.2) a Poisson bracket {·, ·} d φ which makes (C2d , {·, ·} d φ ) into an affine Poisson variety. Each of these brackets has maximal rank 2d (in particular the algebra of Casimirs is trivial) and they are all compatible. An explicit formula for all these brackets is given; they grow in complexity (i.e., degree) with φ so that only the first members are (modified) Lie-Poisson structures.

A general theory for determining Hamiltonian model equations from noncanonical perturbation expansions of Hamiltonian systems is applied to the Boussinesq expansion for long, small amplitude waves in shallow water, leading to the Korteweg-de-Vries equation.New Hamiltonian model equations, including a natural “Hamiltonian version” of the KdV equation, are proposed. The method also provides a direct explanation of the complete integrability (soliton property) of the KdV equation. Depth dependence in both the Hamiltonian models and the second order standard perturbation models is discussed as a possible mechanism for wave breaking.

I began this lecture by posing the question of what the best way is to define integrability. The most promising definition seems to be one based on an associated overdetermined linear system which is “allowable”. The linear systems described in sections 3 and 4 are certainly allowable, and most known integrable equations can be obtained in this way. But there are exceptions, such as the KP equation, so the question is not yet settled. With this sort of definition, it would be difficult to establish whether or not a given equation was integrable. But one could try to classify all the integrable equations which arose from a certain type of linear system.