Two types of linear inhomogeneous integral equations, which yield solutions of a broad class of nonlinear evolution equations, are investigated. One type is characterized by a two-fold integration with an arbitrary measure and contour over a complex variable k, and thier complex conjugates,...

We derive a hierarchy of ibtegrable mappings (integrable ordinary difference equations) corresponding to solutions of the initial-value problem of an integrable partial difference equation with periodic initial data. For each n ϵ N this hierarchy contains at least one integrable mapping...

A new description in terms of one and the same linear inhomogeneous integral equation is proposed for the nonlinear Schrödinger equation (NLS), as well as for the equation of motion for the classical isotropic Heisenberg spin chain in the continuum limit (IHSC). From the integral equation which...

A systematic method for deriving Bäcklund transformations for singular (linear) integral equations is presented. The method leads in a natural way to the Bäcklund transformations for the corresponding (integrable) nonlinear partial differential equations. By repeated use of the method a...

In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from...

A systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations (PDE's) is developed. The method starts from a general type of linear integral equations, containing integrations over an arbitrary contour with an arbitrary measure in the...

We generalize the concept of symplectic maps to that of k- symplectic maps: maps whose kth iterates are symplectic. Similarly, k-symmetries and k-integrals are symmetries (resp. integrals) of the kth iterate of the map. It is shown that k-symmetries and k-integrals are related by the...

A systematic treatment is given of the equation of motion of the classical anisotropic Heisenberg spin chain, both in the discrete case and in the continuum limit in which the spins Sm(t) associated with the lattice sites m are replaced by a spin density S(x, t), which is a function of the time...

It is shown that the solutions of the continuous Anisotropic Heisenberg Spin Chain (AHSC) can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the Isotropic Heisenberg Spin Chain (IHSC) and the Nonlinear Schrödinger equation (NLS). An...

We give a treatment of the non-resonant bifurcations involving asymmetric fixed points with Jacobian J ≠ 1 in reversible mappings of the plane. These bifurcations include the saddle-node bifurcation not in the neighbourhood of a fixed point with J = 1, as well as the so-called transcritical...