A classical integrable system and the involutive representation of solutions of theKdV equation

Under the constrained condition induced by the eigenfunction expression of the reflectionless potentialu=?2<q,q>≡f(q), the Schrödinger equation, the spatial part of the Lax pair of theKdV equation, is nonlinearized to be a completely integrable system (?^2N ,dpΛdq, H=F₀): $$F_0 = \tfrac{1}{2}\left\langle {p,p} \right\rangle + \tfrac{1}{2}\left\langle {q,q} \right\rangle ^2 - \tfrac{1}{8}\left\langle {Aq,q} \right\rangle $$ , while the nonlinearization of the time part leads to itsN-involutive system {F _m }. The involutive solutions of the compatible system (F ₀), (F _m ) are mapped byf into the solutions of them-thKdV equation.     

A class of completely integrable quantum systems associated with classical root systems

We classify the completely integrable systems associated with classical root systems whose potential functions are meromorphic at an infinite point.     

A new discrete integrable system and its discrete integrable coupling system

In terms of the properties of the known loop algebra and difference operators, a new algebraic system X is constructed, which is devote to working out the well-known generalized cubic Volterra lattice equations hierarchy. Then an extended algebraic system of X is presented, from which the integrable coupling system of generalized cubic Volterra lattice equations are obtained.     

Zero-curvature representation for a new fifth-order integrable system

We present a zero curvature representation for one of the new integrable systems found by Mikhailov, Novikov, and Wang in the preprint nlin.SI/0601046 at arXiv.org. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 227–229, 2006.     

A Gentzen system for involutive residuated lattices

We establish a cut-free Gentzen system for involutive residuated lattices and provide an algebraic proof of completeness. As a result we conclude that the equational theory of involutive residuated lattices is decidable. The connection to noncommutative linear logic is outlined. Received July 22, 2004; accepted in final form July 19, 2005.     

A new integrable generalized Neumann system related to soliton hierarchy

In the present paper, the nonlinearization approach is applied to the soliton hierarchy associated with 3 × 3 matrix spectral problems. A new finite-dimensional integrable generalized C. Neumann system is obtained. The involutive system of conserved integrals is constructed by a direct method. Moreover the involutive solution of the soliton hierarchy is also given.     

Lax pair, cyclic basis, and a new integrable system

A direct approach to zero-curvature representation, as introduced by Marvan is applied to generate a new class of integrable equations by starting with the generic Lax operator (x-part) for a coupled set of nonlinear equations. The class of equation so obtained is more general than those usually obtained with the help of standard recursion operator. Finally some particular type of equations are identified by the special choice of the arbitrary functions occurring in the final solution of the time part of the Lax equation. The methodology is specially useful when the x-part of the Lax operator does not contain any spectral parameter.     

Five-wave classical scattering matrix and integrable equations

We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u?u/?x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.     

Generalized Jacobians of spectral curves and completely integrable systems

Consider an ordinary differential equation which has a Lax pair representation , where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold of this Lax pair is an affine part of a non-compact commutative algebraic group – the generalized Jacobian of the spectral curve with its points at “infinity” identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pair on the generalized Jacobian is translation-invariant. Received April 29, 1997; in final form September 22, 1997     

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. IV. A completely integrable quantum system

The purpose of this article is to show that the ground state representation of the infinite one-dimensional spin $\text{1}\text{2}$ Heisenberg chain, with isotropic nearest neighbor interactions, provides an example of a completely integrable quantum system.     

Exact soliton solutions to a new coupled integrable short light-pulse system

In this paper, we investigate the soliton structure of a two-component vector short-pulse system as part of the new multi-component short-pulse system derived by Dimakis, Muller-Hoissen and Matsuno, separately, and describing the propagation of ultra short vector light-pulses in optical fibers. In the wake of the derivation of the Lax-pairs of the system above, we study deeply its inverse scattering transform within the viewpoint of the Wadati–Konno–Ichikawa approach. As a result, we unwrap some interesting scattering features of elastic interactions. Accordingly, we address some physical implications of the results.     

Completely integrable one-dimensional classical and relativistic time-dependent Hamiltonians

In this paper, we look for first integralsI(q;p;t) of time-dependent one-dimensional HamiltoniansH(q;p;t). We first present a formalism based on the use of canonical transformations, and it is seen thatI(q;p;t) can always be written in terms of two variablesI=P(u;v), whereu andv are functions ofq, p andt, without loss of generality. Moreover, it is shown that any Hamiltonian with first integralI(q;p;t) can be made autonomous in the space (u, v, T), whereT is a new time. On the other hand, the cases of a particle moving classically and relativistically in a time-dependent potentialV(q;t) are studied. In both cases, completely integrable potentials, together with the corresponding first integrals, are derived.CEA/CEV-M, BP7, 77181 Country, France, and CEA/CEN-S, SPEC, 91191, Gif sur Yvette Cédex, France. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 355–363, June, 1994.     

Jacobi polynomials and Lax representation for completely integrable dynamical systems

We consider a method aimed at the investigation of completely integrable dynamical systems by using the Lax representation of their equations of motion. The Lax representations are found for the integrable case of the Henon-Heiles system and for an anisotropic oscillator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1088–1091, August, 1994.