Exactly Solvable And Integrable Systems

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus zero and have q --series with integral coefficients. Rational maps relating these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps relating the associated differential systems.

It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical invariants then define topological conserved quantities. Underlying evolution equations are shown to be associated with a triad of linear equations. Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov -Strachan (2+1) dimensional nonlinear Schrödinger equations respectively.

A connection is established between the soliton equations and curves moving in a three dimensional space V 3 . The sign of the self-interacting terms of the soliton equations are related to the signature of V 3 . It is shown that there corresponds a moving curve to each soliton equations.

We show that any multi-component matrix KP hierarchy is equivalent to the standard one-component (scalar) KP hierarchy endowed with a special infinite set of abelian additional symmetries, generated by squared eigenfunction potentials. This allows to employ a special version of the familiar Darboux-B"acklund transformation techniques within the ordinary scalar KP hierarchy in the Sato formulation for a systematic derivation of explicit multiple-Wronskian tau-function solutions of all multi-component matrix KP hierarchies.

Higher flows of the Heisenberg ferromagnet equation and the Wadati-Konno-Ichikawa equation are generalized into multi-component systems on the basis of the Lax formulation. It is shown that there is a correspondence between the multi-component systems through a gauge transformation. An integrable semi-discretization of the multi-component higher Heisenberg ferromagnet system is obtained.

By using `anyon like' representations of permutation algebra, which pick up nontrivial phase factors while interchanging the spins of two lattice sites, we construct some integrable variants of Haldane-Shastry (HS) spin chain. Lax equations for these spin chains allow us to find out the related conserved quantities. However, it turns out that such spin chains also possess a few additional conserved quantities which are apparently not derivable from the Lax equations. Identifying these additional conserved quantities, and the usual ones related to Lax equations, with different modes of a monodromy matrix, it is shown that the above mentioned HS like spin chains exhibit multi-parameter deformed and `nonstandard' variants of Y(g l M ) Yangian symmetry.

The general multi-soliton solution of the integrable coupled nonlinear Schrodinger equation (NLS) of Manakov type is investigated by using Zakharov-Shabat (ZS) scheme. We get the bright and dark multi-soliton solution using inverse scattering method of ZS scheme. Elastic and inelastic collision of N-solitons solution of the equation are also discussed.

The usual propetry of s<-->t duality for scattering amplitudes, e.g. for Veneziano amplitude, is deeply connected with the 2-dimensional geometry. In particular, a simple geometric construction of such amplitudes was proposed in a joint work by this author and S.Saito (solv-int/9812016). Here we propose analogs of one of those amplitudes associated with multidimensional euclidean spaces, paying most attention to the 3-dimensional case. Our results can be regarded as a variant of "Regge calculus" intimately connected with ideas of the theory of integrable models.

A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.

Classical results of Stieltjes are used to obtain explicit formulas for the peakon-antipeakon solutions of the Camassa-Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon-antipeakon pairs, and the details of the collisions are analyzed using results {}from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly