Exactly Solvable And Integrable Systems

The Korteweg de Vries (KdV) equation with small dispersion is a model for the formation and propagation of dispersive shock waves in one dimension. Dispersive shock waves in KdV are characterized by the appearance of zones of rapid modulated oscillations in the solution of the Cauchy problem with smooth initial data. The modulation in time and space of the amplitudes, the frequencies and the wave-numbers of these oscillations and their interactions is approximately described by the g -phase Whitham equations. We study the initial value problem for the Whitham equations for a one parameter family of monotone decreasing initial data. We obtain the bifurcation diagram of the number g of interacting oscillatory zones.

We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, to study the KP equations. In this approach they have the form of local conservation laws, and can be traded for a system of ordinary differential equations of Riccati type, which we call the Central System. We show that the latter can be linearized by means of a Darboux covering, and we use this procedure as an alternative technique to construct rational solutions of the KP equations.

We discuss the geometry of the Marsden-Ratiu reduction theorem for a bihamiltonian manifold. We consider the case of the manifolds associated with the Gel'fand-Dickey theory, i.e., loop algebras over sl(n+1). We provide an explicit identification, tailored on the MR reduction, of the Adler-Gel'fand-Dickey brackets with the Poisson brackets on the MR-reduced bihamiltonian manifold N. Such an identification relies on a suitable immersion of the space of sections of the cotangent bundle of N into the algebra of pseudo differential operators connected to geometrical features of the theory of (classical) W_n algebras.

Binary nonlinearization of AKNS spectral problem is extended to the cases of higher-order symmetry constraints. The Hamiltonian structures, Lax representations, r -matrices and integrals of motion in involution are explicitly proposed for the resulting constrained systems in the cases of the first four orders. The obtained integrals of motion are proved to be functionally independent and thus the constrained systems are completely integrable in the Liouville sense.

We algebraically construct the Fock space of the Sutherland model in terms of the eigenstates of the pseudomomenta as basis vectors. For this purpose, we derive the raising and lowering operators which increase and decrease eigenvalues of pseudomomenta. The operators exchanging eigenvalues of two pseudomomenta have been known. All the eigenstates are systematically produced by starting from the ground state and multiplying these operators to it.

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive auto-Bäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of P34 equations, and a little known Bäcklund transformation for P2 itself. We then use our results on Bäcklund transformations to obtain, for each member of the P2 hierarchy, a sequence of special integrals.

We show that the auto-Backlund transformations of the sine-Gordon, Korteweg-deVries, nonlinear Schrodinger, and Ernst equations are related to their respective CPT symmetries. This is shown by applying the CPT symmetries of these equations to the Riccati equations of the corresponding pseudopotential functions where the fields are allowed to transform into new solutions while the pseudopotential functions and the Backlund parameter are held fixed.

We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map completely integrable system into other completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and R-matrices. As an example, we consider canonical transformations of the extended phase space for the Toda lattices and the Stackel systems.

For the Toda lattice and the Holt system we consider properties of canonical transformations of the extended phase space, which preserve integrability. The separated variables are invariant under change of the time. On the other hand, mapping of the time induces transformations of the action-angles variables and a shift of the generating function of the Bäcklund transformation.

The KP equation has a large family of quasiperiodic multiphase solutions. These solutions can be expressed in terms of Riemann-theta functions. In this paper, a finite-dimensional canonical Hamiltonian system depending on a finite number of parameters is given for the description of each such solution. The Hamiltonian systems are completely integrable in the sense of Liouville. In effect, this provides a solution of the initial-value problem for the theta-function solutions. Some consequences of this approach are discussed.