Exactly Solvable And Integrable Systems

Integrable multistate or multiflavor/color models were recently introduced. They are generalizations of models corresponding to the defining representations of the U_q(sl(m)) quantum algebras. Here I show that a similar generalization is possible for all higher dimensional representations. The R-matrices and the Hamiltonians of these models are constructed by fusion. The sl(2) case is treated in some detail and the spin-0 and spin-1 matrices are obtained in explicit forms. This provides in particular a generalization of the Fateev-Zamolodchikov Hamiltonian.

The symmetry classification method is applied to the string-like scalar fields in two-dimensional space-time. When the configurational space is three-dimensional and reducible we present the complete list of the systems admiting higher polynomial symmetries of the 3rd, 4th and 5th-order.

The Lax pair for a derivative nonlinear Schrödinger equation proposed by Chen-Lee-Liu is generalized into matrix form. This gives new types of integrable coupled derivative nonlinear Schrödinger equations. By virtue of a gauge transformation, a new multi-component extension of a derivative nonlinear Schrödinger equation proposed by Kaup-Newell is also obtained.

In the framework of quantum groups and additive R-matrices, the fusion procedure allows to construct higher-dimensional solutions of the Yang-Baxter equation. These solutions lead to integrable one-dimensional spin-chain Hamiltonians. Here fusion is shown to generalize naturally to non-additive R-matrices, which therefore do not have a quantum group symmetry. This method is then applied to the generalized Hubbard models. Although the resulting integrable models are not as simple as the starting ones, the general structure is that of two spin-(s times s') sl(2) models coupled at the free-fermion point. An important issue is the probable lack of regular points which give local Hamiltonians. This problem is related to the existence of second order zeroes in the unitarity equation, and arises for the XX models of higher spins, the building blocks of the Hubbard models. A possible connection between some Lax operators L and R-matrices is noted.

Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable Hénon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivatives of the tau-functions (inherited from the original PDEs). The ODEs for the similarity solutions also have inherited Bäcklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlevé transcendent.

This paper discusses two equations with the conditional Painleve property. The usefulness of the singular manifold method as a tool for determining the non-classical symmetries that reduce the equations to ordinary differential equations with the Painleve property is confirmed once more

In this paper we prove the complete integrability of Toda flows on generic coadjoint orbits in simple Lie algebras.

Examples of the construction of Hamiltonian structures for dynamical systems in field theory (including one reputedly non-Hamiltonian problem) without using Lagrangians, are presented. The recently developed method used requires the knowledge of one constant of the motion of the system under consideration and one solution of the symmetry equation.

This paper aims to show that there exist non-symmetry constraints which yield integrable Hamiltonian systems through nonlinearization of spectral problems of soliton systems, like symmetry constraints. Taking the AKNS spectral problem as an illustrative example, a class of such non-symmetry constraints is introduced for the AKNS system, along with two-dimensional integrable Hamiltonian systems generated from the AKNS spectral problem.

We study characteristic Cauchy problems for the Korteweg-deVries (KdV) equation u t =u u x + u xxx , and the Kadomtsev-Petviashvili (KP) equation u yy =( u xxx +u u x + u t ) x with holomorphic initial data possessing nonnegative Taylor coefficients around the origin. For the KdV equation with initial value u(0,x)= u 0 (x) , we show that there is no solution holomorphic in any neighbourhood of (t,x)=(0,0) in C 2 unless u 0 (x)= a 0 + a 1 x . This also furnishes a nonexistence result for a class of y -independent solutions of the KP equation. We extend this to y -dependent cases by considering initial values given at y=0 , u(t,x,0)= u 0 (x,t) , u y (t,x,0)= u 1 (x,t) , where the Taylor coefficients of u 0 and u 1 around t=0 , x=0 are assumed nonnegative. We prove that there is no holomorphic solution around the origin in C 3 unless u 0 and u 1 are polynomials of degree 2 or lower.