R. L. Anderson

On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems

We present an algorithmic method for obtaining an hereditary symmetry (the generalized squared‐eigenfunction operator) from a given isospectral eigenvalue problem. This method is applied to the n×n eigenvalue problem considered by Ablowitz and Haberman and to the eigenvalue problem considered by Alonso. The relevant Hamiltonian formulations are also determined. Finally, an alternative method is presented in the case two evolution equations are related by a Miura type transformation and their Hamiltonian formulations are known.

Superposition principles for matrix Riccati equations

A superposition rule is obtained for the matrix Riccati equation (MRE) W=A+WB+CW+WDW [where W(t), A(t), B(t), C(t), and D(t) are real n×n matrices], expressing the general solution in terms of five known solutions for all n≥2. The symplectic MRE (W=WT, A=AT, D=DT, C=BT) is treated separately, and a superposition rule is derived involving only four known solutions. For the ‘‘unitary’’ and GL(n,R) subcases (with D=A and C=BT, or D=−A and C=BT, respectively), superposition rules are obtained involving only two solutions. The derivation of these results is based upon an interpretation of the MRE in terms of the action of the groups SL(2n,R), SP(2n,R), U(n), and GL(n,R) on the Grassman manifold Gn(R2n).

Systems of ordinary differential equations with nonlinear superposition principles

Abstract This work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations. x = η(x,t). (x, η ϵ R n , t ϵ R ) . in terms of a finite number of particular solutions. The point of departure is Lies criterion according to which such a rule exists if and only if the vector fields η ( x , t ). ∇ generate a finite dimensional Lie algebra. We provide three different constructive methods for deriving superposition rules and apply them to systems of coupled Riccati equations of the projective and conformal types based, respectively, on the Lie algebra sl ( n + 1, R ) and o ( p + 1, n − p + 1).

A nonlinear superposition principle admitted by coupled Riccati equations of the projective type

A definite theorem due to Lie which group theoretically characterizes those systems of ordinary differential equations which possess nonlinear superposition principles is employed along with an observation by Lie on the exponentiated form of a fibered Lie algebra to obtain an explicit expression for the Vessiot-Guldberg-Lie nonlinear superposition principle admitted by n-coupled Riccati equations of the projective type. This also, immediately, yields an explicit expression for the generalized cross-ratio for the projective group in n-dimensions.

Group theoretical approach to superposition rules for systems of Riccati equations

The group theoretical structure shown by Lie to underlie systems of ordinary differential equations having a superposition rule, is used to explicitly derive such rules for Riccati equations associated to projective and conformal group actions.

Bäcklund transformations and new solutions of nonlinear wave equations in four-dimensional space-time

Bäcklund transformations for several nonlinear field equations in four-dimensional space-time relating two solutions of the same equation (symmetry), or two different equations (dynamical), are given. These transformations can be used to generate new families of solutions and infinitely many conservation laws for nonlinear equations.Bäcklund transformations and solutions of nonlinear equations have been studied extensively in one-space and one-time dimension. We give here a fairly general method for a class of equations in four-dimensional space-time which paves the way for many further generalizations.

An invertible linearization map for the quartic oscillator

The set of world lines for the nonrelativistic quartic oscillator satisfying Newtons equation of motion for all space and time in 1-1 dimensions with no constraints other than the “spring” restoring force is shown to be equivalent (1-1-onto) to the corresponding set for the harmonic oscillator. This is established via an energy preserving invertible linearization map which consists of an explicit nonlinear algebraic deformation of coordinates and a nonlinear deformation of time coordinates involving a quadrature. In the context stated, the map also explicitly solves Newtons equation for the quartic oscillator for arbitrary initial data on the real line. This map is extended to all attractive potentials given by even powers of the space coordinate. It thus provides classes of new solutions to the initial value problem for all these potentials.

The Benjamin-Ono equation — Recursivity of linearization maps — Lax pairs

Analytic linearization maps (resp., inverse linearization maps) in the sense of Flato, Pinczon and Simon are constructed for the Benjamin-Ono equation. These maps have a simple structure, referred to as their recursivity. This recursivity leads to the construction of Lax pairs and Gelfand-Levitan-type equations.

Explicit nonsoliton solutions of the Benjamin-Ono equation

We linearize the nonlinear space-time translation Lie algebra for the Benjamin-Ono equation. This permits the construction of global nonsoliton solutions.

A nonlinear superposition principle for Riccati equations of the conformal type

In this communication, we sketch a proof that n-coupled Riccati equations of the conformal type admit a Vessiot-Guldberg-Lie nonlinear superposition principle. The method of proof yields directly an explicit expression for a nonlinear superposition law. By such a law we mean an expression for the general solution of these equations in terms of a finite number (here n+2) of particular solutions and n arbitrary constants, where the particular solutions are arbitrary up to certain independence conditions, The terminology Riccati equations of the conformal type will be defined in the next section. The method of derivation of this nonlinear superposition law is of interest in its own right because it utilizes conformal group operations exclusively and hence exhibits the group morphology of the law.