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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.