# Walter Oevel

List of Publications: PostScript , PDF ; last modification: August-2004

Numerical Computations in MuPAD 1.4
Walter Oevel, mathPAD 8(1) (1998), pp. 57-66
Some of the algorithms in the numeric package of \MuPAD~1.4 are discussed. In particular the routines for computing eigenvalues, for numerical quadrature and for solving ordinary differential equations are compared to the corresponding routines provided by Mathematica~3.0 and Maple~V.4.

MuPAD: the numeric package
Oliver Kluge, Walter Oevel, Stefan Wehmeier and Paul Zimmermann, Automath Technical Report No 22
This paper is a description of the MuPAD library package numeric containing various functions for numerical mathematics.

Symplectic Runge-Kutta-Schemes I: Order Conditions
Mark Sofroniou and Walter Oevel, SIAM Journal of Numerical Analysis 34 (1997), pp. 2063-2086
Much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating Hamiltonian systems of differential equations. Runge-Kutta schemes are symplectic subject to a simple algebraic condition. Starting with Butcher's formalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods; involving only $s(s+1)/2$ free parameters for a symplectic $s$ stage scheme. A graph theoretical process for determining the new order conditions is outlined. Furthermore, it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the parametrised coefficients. When coupled with the standard simplifying assumptions for implicit schemes the number of order conditions may be further reduced. In the new framework a simple symmetry of the parameter matrix yields (not necessarily symplectic) self-adjoint methods. In this case the order conditions associated with even trees become redundant.

Symplectic Runge-Kutta-Schemes II: Classification of Symmetric Methods
Walter Oevel and Mark Sofroniou, preprint, 1997
A complete classification of all symplectic self-adjoint Runge-Kutta methods with up to 6 stages is given and the derivation process used is outlined. This is made possible by the fact that most of these schemes are determined by linear systems of simplifying equations. Furthermore, the derivation process can be used for higher stage numbers but not without loss of generality. We also show how it is often straightforward to derive subclasses of methods possessing additional desirable properties, such as algebraic stability. Extensive use of computer algebra has proven invaluable in the classification process.

Symplectic Runge-Kutta-Schemes III: Canonical Elementary Differentials
Mark Sofroniou and Walter Oevel, preprint, December 1993
It has been shown that numerical methods for Hamiltonian systems may be characterised in terms of so-called canonical elementary differentials. Recent results by the authors demonstrate that the symplecticity condition for Runge-Kutta schemes may be eliminated by suitable parametrisation of the order conditions. These results are used to re-examine the relationship between order conditions for canonical Runge-Kutta methods and canonical elementary differentials.

1-Parameter Families of Maximal Order Runge-Kutta-Nystr\"om Methods.
Walter Oevel and Yuri Suris, preprint, 7 October 1996
Runge-Kutta-Nystr\"om methods with $s$ stages for the numerical solution of second order differential equations $d^2q/dt^2=f(t,q)$ are considered. It is shown that the Gauss-Legendre schemes of order $2s$ consist of 1-parameter families involving an arbitrary constant. These schemes are symplectic and self-adjoint.

Modified Order Theory for Partitioned Runge-Kutta and Runge-Kutta-Nystr\"om Methods
Walter Oevel and M. Sofroniou, preprint, 12 November 1996
A parametrization technique for deriving general symplectic and/or self-adjoint (symmetric) Runge-Kutta methods of standard type was discussed in a recent paper by the authors. Here these results are extended to the derivation of order conditions for partitioned and Nystr\"{o}m-Runge-Kutta methods. These methods are of particular interest for the symplectic time-integration of Hamiltonian systems in separable form.

Einf\"uhrung in die Numerische Mathematik (table of contents)
Walter Oevel, January 1996, introductory text book

Squared Eigenfunction Symmetries for Soliton Equations: Part I
Walter Oevel and Sandra Carillo, Journal of Mathematical Analysis and Its Applications, 217 (1998), pp. 161-178
Nonlinear integrable evolution equations in $1+1$ dimensions arise from constraints of the $2+1$ dimensional hierarchies associated with the Kadomtsev-Petviashvili (KP) equation, the modified KP equation and the Dym equation, respectively. Links of B\"acklund type and of reciprocal type are known to exist between the 2+1 dimensional systems. The corresponding links between the constrained flows are discussed in a general framework. In particular, squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered. The links between the soliton hierarchies are extended to these symmetries.

Squared Eigenfunction Symmetries for Soliton Equations: Part II
Walter Oevel and Sandra Carillo, Journal of Mathematical Analysis and Its Applications, 217 (1998), pp. 179-199

A $3\times3$ matrix spectral problem for the AKNS hierarchy and its binary nonlinearization
Wen-xiu Ma, Benno Fuchssteiner and Walter Oevel, Physica A 233 (1996), pp. 331-354
A three-by-three matrix spectral problem for the AKNS soliton hierarchy is proposed and the corresponding Bargmann symmetry constraint involved in Lax pairs and adjoint Lax pairs is discussed. The resulting nonlinearized Lax systems possess classical Hamiltonian structures, in which the nonlinearized spatial system is intimately related to stationary AKNS flows. These nonlinearized Lax systems also lead to a sort of involutive solutions to each AKNS soliton equation.

Wronskian solutions of the constrained KP hierarchy
Walter Oevel and Walter Strampp, J. Math. Phys. 37 (1996), pp. 6213-6219
The integrable Kadomtsev-Petviashvili (KP) hierarchy is compatible with generalized $k$-constraints of the type $(L^k)_=sum_i q_i D^{-1}r_i$. A large class of solutions -- among them solitons -- can be represented by Wronskian determinants of functions satisfying a set of linear equations. In this paper we shall obtain additional conditions for these functions imposed by the constraints.

Gauge transformations of constrained KP flows: new integrable hierarchies
Anjan Kundu, Walter Strampp and Walter Oevel, J. Math. Phys. 36 (1995), pp. 2972-2984
Integrable systems in 1+1 dimensions arise from the KP hierarchy as symmetry reductions involving square eigenfunctions. Exploiting the residual gauge freedom in these constraints new integrable systems are derived. They include generalizations of the hierarchy of the Kundu-Eckhaus equation and higher order extensions of the Yajima-Oikawa and Melnikov hierarchies. Constrained modified KP flows yield further integrable equations such as the hierarchies of the derivative NLS equation, the Gerdjikov-Ivanov equation and the Chen-Lee-Liu equation.

Squared Eigenfunctions of the (Modified) KP Hierarchy and Scattering Problems of Loewner Type
Walter Oevel and Wolfgang Schief, Reviews in Mathematical Physics 6 (1994), pp. 1301-1338
It is shown that products of eigenfunctions and (integrated) adjoint eigenfunctions associated with the (modified) Kadomtsev-Petviashvili (KP) hierarchy form generators of a symmetry transformation. Linear integro-differential representations for these symmetries are found. For special cases the corresponding nonlinear equations are the compatibility conditions of linear scattering problems of Loewner type. The examples include the 2+1-dimensional sine-Gordon equation with space variables occuring on an equal footing introduced recently by Konopelchenko and Rogers. This equation represents a special squared eigenfunction symmetry of the Ishimori hierarchy.

A note on the Poisson brackets associated with Lax operators
W. Oevel, Physics Letters A 186 (1994), pp 79-86
Modifications of the standard Poisson brackets associated with differential scattering operators are considered. A linear bracket originates from a non-standard $r$-matrix on the algebra of pseudo-differential operators. Two quadratic brackets are investigated which provide Hamiltonian formulations for various reductions of the modified KP hierarchy.

Hamiltonian structures of the Melnikov system and its reductions
Walter Oevel, Jurij Sidorenko and Walter Strampp, Inverse Problems 9 (1993), pp. 737-747
The bi-Hamiltonian structure of an integrable dynamical system introduced by Melnikov is studied. This equation arises as a symmetry constraint of the KP hierarchy via squared eigenfunctions and can be understood as a Boussinesq-system with a source. The standard linear and quadratic Poisson brackets associated with the space of pseudo-differential symbols are used to derive two compatible Hamiltonian operators. Via reduction techniques a bi-Hamiltonian formulation for the Drinfeld-Sokolov system is derived.

An r-Matrix Approach to Nonstandard Classes of Integrable Equations
Boris G. Konopelchenko and Walter Oevel, Publ. RIMS, Kyoto Univ. 29 (1993), pp. 581-666
Three different decompositions of the algebra of pseudo-differential operators and the corresponding $r$-matrices are considered. Three associated classes of nonlinear integrable equations in 1+1 and 2+1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The 2+1-dimensional hierarchies are associated with the Kadomtsev-Petviashvili (KP) equation, the modified KP equation and a Dym equation, respectively. Reductions of the general hierarchies lead to other known integrable 2+1-dimensional equations as well as to a variety of integrable equations in 1+1 dimensions. It is shown, how the multi-Hamiltonian structure of the 1+1-dimensional equations can be obtained from the underlying $r$-matrices. Further, intimate relations between the equations associated with the three different $r$-matrices are revealed. The three classes are related by Darboux theorems originating from gauge transformations and reciprocal links of the Lax operators. These connections are discussed on a general level, leading to a unified picture on (reciprocal) B\"acklund and auto-B\"acklund transformations for large classes of integrable equations covered by the KP, the modified KP, and the Dym hierarchies.

Constrained KP Hierarchy and Bi-Hamiltonian Structures
Walter Oevel and Walter Strampp, Commun. Math. Phys. 157 (1993), pp. 51-81
The Kadomtsev-Petviashvili (KP) hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions. Constraining the KP flows in terms of squared eigenfunctions one obtains 1+1-dimensional integrable equations with scattering problems given by pseudo-differential Lax operators. The bi-Hamiltonian nature of these systems is shown by a systematic construction of two general Poisson brackets on the algebra of associated Lax-operators. Gauge transformations provide Miura links to modified equations. These systems are constrained flows of the modified KP hierarchy, for which again a general description of their bi-Hamiltonian nature is given. The gauge transformations are shown to be Poisson maps relating the bi-Hamiltonian structures of the constrained KP hierarchy and the modified KP hierarchy. The simplest realization of this scheme yields the AKNS hierarchy and its Miura link to the Kaup-Broer hierarchy.

Darboux Theorems and Wronskian Formulas for Integrable Systems I: Constrained KP Flows
Walter Oevel, Physica A 195 (1993), pp. 533-576
Generalizations of the classical Darboux theorem are established for pseudo-differential scattering operators of the form $L=sum u_i D^i+sum_i Phi.i D^{-1} Psi.i^*.$ Iteration of the Darboux transformation leads to a gauge transformed operator with coefficients given by Wronskian formulas involving a set of eigenfunctions of $L$. Nonlinear integrable partial differential equations are associated with the scattering operator $L$ which arise as a symmetry reduction of the multicomponent KP hierarchy. With a suitable linear time evolution for the eigenfunctions the Darboux transformation is used to obtain solutions of the integrable equations in terms of Wronskian determinants.

Gauge Transformations and Reciprocal Links in 2+1 Dimensions
Walter Oevel and Colin Rogers, Rev. Math. Phys. 5 (1993), pp. 299-330
Generalized Lax equations are considered in the spirit of Sato theory. Three decompositions of an underlying algebra of pseudo-differential operators lead, in turn, to three different classes of integrable nonlinear hierarchies. These are associated with Kadomtsev-Petviashvili, modified Kadomtsev-Petviashvili and Dym hierarchies in 2+1 dimensions. Miura- and auto-B\"acklund transformations are shown to originate naturally from gauge transformations of the Lax operators. General statements on reciprocal links between these hierarchies are established, which, in particular, give rise to novel reciprocal auto-B\"acklund transformations for the Dym hierarchy. These links are formulated as Darboux theorems for the associated Lax operators.

The Bi-Hamiltonian Structure of Fully Sypersymmetric Korteweg-de Vries Systems
Walter Oevel and Ziemowit Popowicz, Commun. Math. Phys. 139 (1991), pp. 441-460
The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to the N=1 and the N=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to the N=2 case additional symmetries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.

Poisson Brackets for Integrable Lattice Systems
Walter Oevel, in Algebraic Aspects of Integrable Systems, (eds. A.S. Fokas and I.M. Gelfand), Birkhaeuser 1996, pp. 261-283
Poisson brackets associated with Lax operators of lattice systems are considered. Linear brackets originate from various $r$-matrices on the algebra of (pseudo-) shift symbols. Quadratic brackets are investigated which provide Hamiltonian formulations for various reductions of the (modified) Lattice KP hierarchy.

Darboux Transformations for Integrable Lattice Systems
Walter Oevel, in Nonlinear Physics (eds. E. Alfinito et al.), World Scientific (1996), pp. 233-240
A framework for a general description of Darboux transformations for Lax representations of discrete integrable systems is presented. The Lax equations are regarded as dynamical systems in the algebra of shift operators which is embedded in an algebra of pseudo-difference symbols. Gauge transformations are given by operators satisfying a dressing equation in this space. Special dressing operators are found which are parametrized by (adjoint) eigenfunctions of the Lax system. They give rise to Darboux like transformations as well as adjoint and binary versions. Reductions to finite operators are discussed.

Darboux Theorems and the KP Hierarchy
Walter Oevel and Wolfgang Schief, in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, (ed. P.A. Clarkson), Kluwer 1993, pp. 193-205
Generalizations of the classical Darboux theorem are established for arbitrary ordinary matrix differential operators. Darboux transformations may be regarded as gauge transformations, where the gauge operator is a first order differential operator parametrized by an eigenfunction. Adjoint Darboux transformations triggered by adjoint eigenfunctions are introduced via transposition of the operators. The composition of a Darboux and an adjoint Darboux transformation leads to the notion of binary transformations, which are triggered by pairs of eigenfunctions and adjoint eigenfunctions. A formalism involving pseudo-differential symbols is used to give a general formulation of these transformations for arbitrary scattering operators. It is shown how exact solutions of the multicomponent KP hierarchy are generated from these transformations.

Matrix Sato Theory and Integrable Equations in 2+1 Dimensions
Boris G. Konopelchenko and Walter Oevel, in: Proceedings of the 7th Workshop on Nonlinear Evolution Equations and Dynamical Systems (NEEDS'91), Baia Verde, Italy, 19-29 June 1991
Two different decompositions of the algebra of matrix valued pseudo-differential symbols are considered. The two associated classes of nonlinear integrable equations in 2+1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The two classes represent the multi-component KP hierarchy and a modified" multi-component KP hierarchy. It is shown that general Darboux theorems provide Miura transformations between these two classes as well as auto-B\"acklund transformations within the classes.