Benno Fuchssteiner

Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation

Abstract Some explicit traveling wave solutions to a Kolmogorov-Petrovskii-Piskunov equation are presented through two ansatze. By a Cole-Hopf transformation, this Kolmogorov-Petrov-skii-Piskunov equation is also written as a bilinear equation and two solutions to describe nonlinear interaction of traveling waves are further generated. Backlund transformations of the linear form and some special cases are considered.

Integrable theory of the perturbation equations

Abstract An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.

Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation

Abstract The main subject of the paper is to give a survey and to present new methods on how integrability results (i.e. results for symmetry groups, inverse scattering formulations, action-angle transformations and the like) can be transferred from one equation to others in case the equations are NOT related by Backlund transformations. As a main example the so-called Camassa-Holm equation is chosen for which the relevant results are obtained by having a look on the Korteweg de vries (KdV) equation. The Camassa-Holm equation turns out to be a different-factorization equation of the KdV, it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations. We follow here an old method already marginally mentioned in Fuchssteiner and Fokas (1981), and Fuchssteiner (1983) and recently applied by others (Olver and Rosenau, 1995). The method allows an immediate recovery of the recursion operator for the Camassa-Holm equation from the invariance structure of the KdV, although both equations are not related by Backlund transformations. However, in addition, and different from other approaches, from there by use of the squared eigenfunction relation for the KdV equation, the Lax pair formulation for the different-factorization equation is derived. For the example under consideration it is, of course, the one obtained in Camassa and Holm (1993). Since the methods proposed here can be transferred to any compatible factorization of recursion operators its application, even in the special case which was chosen for illustration, leads to a large class of integrable equations among which the Camassa-Holm equation can be found as well as a three-parameter family of generalizations of the equation. The advantage of the general approach to the Lax pair presentation is that direct transformations between action and angle variables are obtained. So, using this Lax pair formulation, as a novel result, a direct transformation between action- and angle-variables for the Camassa-Holm equation is derived. Further novel results in the paper are: a hodograph link back from a Backlund transformation of the Camassa-Holm equation to a particular member of the KdV-hierarchy, additional symmetries, and the construction of the conformal algebra for the hierarchy of the Camassa-Holm equation. The methods involved include: hereditary operators, bi-hamiltonian formulations, nilpotent flows and scaling symmetries.

Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations

An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows (λt=λl, l⩾0) from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.

The Hierarchy of the Benjamin-ono Equation

Abstract The Benjamin-Ono (BO) equation is shown to posses two non-local linear operators, which generate its infinitely many commuting symmetries and constants of the motion in involution. These symmetries define the hierarchy of the BO equation, each number of which is a hamiltonian system. The above operators are the nonlocal analogues of the Lenard operator and its adjoint for the Korteweg-de Vries equation.

The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy

Abstract The bi-Hamiltonian structure is established for the perturbation equations of the KdV hierarchy and the perturbation equations themselves also provide examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions.

A 3 sx 3 matrix spectral problem for AKNS hierarchy and its binary nonlinearization

A 3 × 3 matrix spectral problem for AKNS soliton hierarchy is introduced and the corresponding Bargmann symmetry constraint involving Lax pairs and adjoint Lax pairs is discussed. An explicit new Poisson algebra is proposed and thus the Liouville integrability is established for the nonlinearized spatial system ind a hierarchy of nonlinearized temporal systems under the control of the nonlinearized spatial system. The obtained nonlinearized Lax systems, in which the nonlinearized spatial system is intimately related to stationary AKNS flows, lead to a sort of new involutive solutions to each AKNS soliton equation. Therefore, the binary nonlinearization theory is successfully extended to a case of 3 × 3 matrix spectral problem for AKNS hierarchy.

Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation

Abstract Two nonlocal recursion operators are given, which yield explicit formulas for infinite hierarchies of symmetry generators and conservation laws for the two-dimensional Korteweg-de Vries equation. It is shown that the constants of the motion are in involution and that the symmetries commute.

Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure

Basic invariants, such as conserved quantities, symmetries, mastersymmetries, and recursion operators are explicitly constructed for the following nonlinear lattice systems: The modified Korteweg–de Vries lattice, the Ablowitz–Ladik lattice, the Brusci–Ragnisco lattice, the Ragnisco–Tu lattice and some cases of the class of integrable systems introduced by Bogoyavlensky. The algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the Lax pair analysis is discussed. By explicit presentation of the higher‐order members of the corresponding hierarchies new completely integrable lattice flows are found. For all systems, multi‐Hamiltonian formulations are given.

The bi‐Hamiltonian structure of some nonlinear fifth‐ and seventh‐order differential equations and recursion formulas for their symmetries and conserved covariants

Using a bi‐Hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth‐ and seventh‐order nonlinear partial differential equations; among them, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the Kupershmidt equation. We show that the Lie algebras of the symmetry groups of these equations are of a very special form: Among the C∞ vector fields they are generated from two given commuting vector fields by a recursive application of a single operator. Furthermore, for some higher order equations, those multisoliton solutions, which for ‖t‖→∞ asymptotically decompose into traveling wave solutions, are characterized as eigenvector decompositions of certain operators.