Nonlinear Sciences

A dressing method is applied to a matrix Lax pair for the Camassa-Holm equation, thereby allowing for the construction of several global solutions of the system. In particular solutions of system of soliton and cuspon type are constructed explicitly. The interactions between soliton and cuspon solutions of the system are investigated. The geometric aspects of the Camassa-Holm equation ar re-examined in terms of quantities which can be explicitly constructed via the inverse scattering method.

The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward the following hypothesis. Any integrable equation of the type under consideration admits an infinite sequence of finite-field Darboux-integrable reductions. The property of Darboux integrability of a finite-field system is formalized as finite-dimensionality condition of its characteristic Lie-Rinehart algebras. That allows one to derive effective integrability conditions in the form of differential equations on the right hand side of the equation under study. To test the hypothesis, we use known integrable equations from the class under consideration. In this article, we show that all known examples do have this property.

A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are found.

In this paper, we consider polynomials associated with faces and internal quadrilaterals of a cuboctahedron and classify them under the requirement that they are consistent. These polynomials give rise to a system of partial difference equations on a face-centred cubic lattice. Our results were motivated by τ -functions related to discrete Painlevé equations.

We extend Painlevé IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlevé IV and II equations for special limits of the underlying parameters. We derive the underlying Bäcklund transformations, symmetry structure and requirements to satisfy Painlevé property.

We derive a set of identities for the theta functions on compact Riemann surfaces which generalize the famous trisecant Fay identity. Using these identities we obtain quasiperiodic solutions for a multidimensional generalization of the Hirota bilinear difference equation and for a multidimensional Toda-type system.

We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) ≅ GL(2, R ) ≅ Möbius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the N -soliton solutions through the KdV type Bäcklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the Bäcklund transformations of the standard form into the same common Hirota form and the same common Bäcklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the N -soliton solutions has same common form for KdV/mKdV/sinh-Gordon systems. Then the N -soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV N -soliton solutions. We also give general addition formulae coming from the KdV type Bäcklund transformation which plays not only an important role to construct the trigonometric/hyperbolic N -soliton solutions but also an essential role to construct the elliptic N -soliton solutions. In contrast to the KdV type Bäcklund transformation, the well-known mKdV/sinh-Gordon type Bäcklund transformation gives the non-cyclic symmetric N -soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.

We study integrability --in the sense of admitting recursion operators-- of two nonlinear equations which are known to possess compacton solutions: the K(m,n) equation introduced by Rosenau and Hyman D t (u)+ D x ( u m )+ D 3 x ( u n )=0, and the CSS equation introduced by Coooper, Shepard, and Sodano, D t (u)+ u l−2 D x (u)+αp D x ( u p−1 u 2 x )+2α D 2 x ( u p u x )=0. We obtain a full classification of {\em integrable K(m,n) and CSS equations}; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of CSS .

In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most quadratic growth. Cubic growth may reflect the existence of non-elliptic fibrations of invariants, whereas we conjecture that the exponentially growing cases lack the necessary conditions for the applicability of the discrete Liouville theorem.

At a very informal but practically convenient level, we discuss the step-by-step computation of nonlocal recursions for symmetry algebras of nonlinear coupled boson-fermion N=1 supersymmetric systems by using the SsTools environment.