The periodic solutions for coupled integrable dispersionless equations

By using the Jacobi elliptic-function method,this paper obtains the periodic solutions for coupled integrable dispersionless equations. The periodic solutions include some kink and anti-kink solitons.     

Lagrangian foliations and Lax equations

If X is a bihamiltonian vector field tangent to a foliation which is Lagrangian with respect to both symplectic structures, the dynamical system x=X(x) implies a local Lax equation =[L, B], but in canonical adapted coordinates, this equation reduces to the trivial equation =0.     

Integrable boundary conditions and modified Lax equations

We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding “transfer” matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix and the boundary matrix.     

On generalized Lax equations of the Lax triple of the BKP and CKP hierarchies

Based on the Lax triple (B_m, B_n, L) of the BKP and CKP hierarchies, we derive the nonlinear evolution equations from the generalized Lax equation. The solutions of some evolution equations are presented, such as soliton and rational solutions.     

Bilagrangian connections and the reduction of Lax equations

We obtain the Lax equations associated with a dynamical system endowed with a bilagrangian connection and a closed two-form parallel along the dynamical field . The case of Lagrangian dynamical systems is analysed and the nonnoether constants of motion found by Hojman and Harleston are recovered as being associated to a reduced Lax equation. Completely integrable dynamical systems are also shown to be a particular case of these systems.     

On Lax equations arising from Lagrangian foliations

It is shown that Lax equations associated with dynamical systems on T ^*Q of the same dimension as Q arise as local expressions of parallelism of a (1,1)-tensor field along the dynamical vector field if the partial connection defined by the symplectic form admissible for a Lagrangian foliation is considered.     

Integrable systems of partial differential equations determined by structure equations and Lax pair

It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.     

The matrix Lax representation of the generalized Riemann equations and its conservation laws

It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter-Saxton equation. New matrix and scalar Lax representation are presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.     

On the Lax representation for the nonlinear evolution equations

By the classical differential geometry techniques it is shown that a general partial differential equation of the second order with two independent variables can be represented in the Lax operator form [X ₁ X ₂]=0, whereX _i =/x _i – ⁱ ,i=1,2 and ⁱ are the 3×3 matrices. The problem of the introduction of the spectral parameter in this representation is shortly discussed.Presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.The author is pleased to thank V. K. Mel'nikov for the discussion of this work.     

Surfaces specified by integrable systems of partial differential equations determined by structure equations and Lax pair

A system of evolution equations can be developed from the structure equations for a submanifold embedded in a three-dimensional space. It is seen how these same equations can be obtained from a generalized matrix Lax pair provided a single constraint equation is imposed. This can be done in Euclidean space as well as in Minkowski space. The integrable systems which result from this process can be thought of as generalizing the SO(3)$SO\left(3\right)$ and SO(2,1)$SO(2,1)$ Lax pairs which have been studied previously.     

A geometrical setting for Lax equations associated to dynamical systems

We develop a geometrical framework for dealing with Lax equations associated to dynamical systems over a manifold M. We also show that this theory reproduces the global versions of Lax equations given before as well as the usual theory of reduced systems obtained from systems defined on Lie groups and with such group as a symmetry group.     

On Nth-order rogue wave solution to nonlinear coupled dispersionless evolution equations

The generalized Darboux transformation is applied to obtain Nth-order rogue wave solution of nonlinear coupled dispersionless evolution equations (NLCDEE). In particular, the interesting structures for these solutions from first to third order rogue wave are shown.     

On exact N-loop soliton solution to nonlinear coupled dispersionless evolution equations

We investigate some nonlinear coupled dispersionless evolution equations (NLCDEE) modelling the dynamics of a current-fed string within an external magnetic field in 2D-space. Using a blend of transformations of independent variables, we derive from the previous equations a Schäfer-Wayne short pulse equation (SWSPE). By means of a transformation back to the original independent variables, we find the N-loop soliton solution to the coupled equations. We give some detail on the scattering behavior of two-loop solitons.     

Two types of loop algebras and their expanding Lax integrable models

Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation U_t-V_x+[U,V]=0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator \widetilde{J} is presented by constructing a subalgebra \widetilde{G } of the loop algebra \widetilde A₂. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their (2+1)-dimensional forms are derived from a (2+1)-dimensional zero-curvature equation.     

Infinite dimensional symmetric spaces and Lax equations compatible with the infinite Toda chain

In this paper we present a natural embedding of the infinite Toda chain in a set of Lax equations in the algebra $LT$ consisting of $\mathbb{Z}\times \mathbb{Z}$-matrices that possess only a finite number of nonzero diagonals above the main central diagonal. This hierarchy of Lax equations describes the evolution of deformations of a set of commuting anti-symmetric matrices and corresponds to splitting this algebra into its anti-symmetric part and the subalgebra of matrices in $LT$ that have no component above the main diagonal. We show that the projections of these deformations satisfy a set of zero curvature relations, which demonstrates the compatibility of the system. Further we introduce a suitable $LT$-module in which we can distinguish elements, the so-called wave matrices, that will lead you to solutions of the hierarchy. We conclude by showing how wave matrices of the infinite Toda chain hierarchy can be constructed starting from an infinite dimensional symmetric space.     

Kac-moody lie algebras and soliton equations: II. Lax equations associated with A1(1)

The soliton equations associated with sl(2) eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment; they are shown to be generated by a single family of commuting Hamiltonians on a subalgebra of the loop algebra of sl(2). The conserved densities and fluxes of the usual ANKS hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems. The Hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment.