Papers by Elizabeth Mansfield
Mathematical and Computer Modelling, Apr 1, 1997
In this paper we study symmetry reductions of a class of nonlinear third order partial differenti... more In this paper we study symmetry reductions of a class of nonlinear third order partial differential equations where ǫ, κ, α and β are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters ǫ = 1, α = -1, β = 3 and κ = 1 2 , admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters ǫ = 0, α = 1, β = 3 and κ = 0, admits a "compacton" solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters ǫ = 1, α = -3 and β = 2, has a "peakon" solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.
Computer Physics Communications, Dec 1, 1998
An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif b... more An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif by Reid, is made of the 3+l-coupled nonlinear &hi-&linger (CNLS) system i~,+V21y+(~ly~2 + /@I') rY = 0, i@,+V2@+(~S(2 + ]@I*) @ = 0. This system describes transverse effects in nonlinear optical systems. It also arises in the study of the transmission of coupled wave packets and "optical solitons", in nonlinear optical fibres. First we apply Lie's method for calculating the classical Lie algebra of vector fields generating symmetries that leave invariant the set of solutions of the CNLS system. The large linear classical determining system of PDE for the Lie algebra is automatically generated and reduced to a standard form by the rif algorithm, then solved, yielding a l$dimensional classical Lie invariance algebra. A generalization of Lie's classical method, called the nonclassical method of Bluman and Cole, is applied to the CNLS system. This method involves identifying nonclassical vector fields which leave invariant the joint solution set of the CNLS system and a certain additional system, called the invariant surface condition. In the generic case the system of determining equations has 856 PDE, is nonlinear and considerably more complicated than the linear classical system of determining equations whose solutions it possesses as a subset. Very few calculations of this magnitude have been attempted due to the necessity to treat cases, expression explosion and until recent times the dearth of mathematically rigorous algorithms for nonlinear systems. The application of packages diffgrob2 and rif leads to the explicit solution of the nonclassical determining system in eleven cases. Action of the classical group on the nonclassical vector fields considerably simplifies one of these cases. We identify the reduced form of the CNLS system in each case. Many of the cases yield new results which apply equally to a generalized coupled nonlinear Schriidinger system in which Jly/* + /@I* may be replaced by an arbitrary function of Ipyi* + /@I'. Coupling matrices in 51(2, C) feature prominently in this family of reductions. @ 1998 Elsevier Science B.V.
arXiv (Cornell University), Jun 1, 2010
We study variational problems for curves approximated by B-spline curves. We show that one can ob... more We study variational problems for curves approximated by B-spline curves. We show that one can obtain discrete Euler-Lagrange equations for the data describing the approximated curves. Our main application is to the occluded curve problem in 2D and 3D. In this case, the aim is to find various aesthetically pleasing solutions as opposed to a solution of a physical problem. The Lagrangians of interest are invariant under the special Euclidean group action for which B-spline approximated curves are well suited. Smooth Lagrangians with special Euclidean symmetries involve curvature, torsion, and arc length. Expressions in these, in the original coordinates, are highly complex. We show that, by contrast, relatively simple discrete Lagrangians offer excellent results for the occluded curve problem. The methods we develop for the discrete occluded curve problem are general and can be used to solve other discrete variational problems for B-spline curves.
Теоретическая и математическая физика, 2000
Siam Journal on Applied Mathematics, Dec 1, 1994
In this article we present first an algorithm for calculating the determining equations associate... more In this article we present first an algorithm for calculating the determining equations associated with so-called "nonclassical method " of symmetry reductions (à la Bluman and Cole) for systems of partial differential equations. This algorithm requires significantly less computation time than that standardly used, and avoids many of the difficulties commonly encountered. The proof of correctness of the algorithm is a simple application of the theory of Gröbner bases. In the second part we demonstrate some algorithms which may be used to analyse, and often to solve, the resulting systems of overdetermined nonlinear pdes. We take as our principal example a generalised Boussinesq equation, which arises in shallow water theory. Although the equation appears to be non-integrable, we obtain an exact "two-soliton" solution from a nonclassical reduction.
Nonlinearity, May 1, 1994
In this paper we study a shallow water equation derivable using the Boussinesq approximation, whi... more In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et al [Stud. Appl. Math., 53 (1974) 249-315] and one by Hirota and Satsuma [J. Phys. Soc. Japan, 40 (1976) 611-612]. A catalogue of classical and nonclassical symmetry reductions, and a Painlevé analysis, are given. Of particular interest are families of solutions found containing a rich variety of qualitative behaviours. Indeed we exhibit and plot a wide variety of solutions all of which look like a two-soliton for t > 0 but differ radically for t < 0. These families arise as nonclassical symmetry reduction solutions and solutions found using the singular manifold method. This example shows that nonclassical symmetries and the singular manifold method do not, in general, yield the same solution set. We also obtain symmetry reductions of the shallow water equation solvable in terms of solutions of the first, third and fifth Painlevé equations. We give evidence that the variety of solutions found which exhibit "nonlinear superposition" is not an artefact of the equation being linearisable since the equation is solvable by inverse scattering. These solutions have important implications with regard to the numerical analysis for the shallow water equation we study, which would not be able to distinguish the solutions in an initial value problem since an exponentially small change in the initial conditions can result in completely different qualitative behaviours.
arXiv (Cornell University), Mar 7, 2019
We study variational systems for space curves, for which the Lagrangian or action principle has a... more We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimizing frame, also known as the Normal, Parallel or Bishop frame (see [1], [36]). Such systems have previously been studied using the Frenet-Serret frame. However, the Rotation Minimizing frame has many advantages, and can be used to study a wider class of examples. We achieve our results by extending the powerful symbolic invariant calculus for Lie group based moving frames, to the Rotation Minimizing frame case. To date, the invariant calculus has been developed for frames defined by algebraic equations. By contrast, the Rotation Minimizing frame is defined by a differential equation. In this paper, we derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants. We then derive the syzygy operator needed to obtain Noether's conservation laws as well as the Euler-Lagrange equations directly in terms of the invariants, for variational problems with a Euclidean symmetry. We show how to use the six Noether laws to ease the integration problem for the minimizing curve, once the Euler-Lagrange equations have been solved for the generating differential invariants. Our applications include variational problems used in the study of strands of proteins, nucleid acids and polymers.
arXiv (Cornell University), Jul 31, 2012
In this work we prove a weak Noether type theorem for a class of variational problems which inclu... more In this work we prove a weak Noether type theorem for a class of variational problems which include broken extremals. We then use this result to prove discrete Noether type conservation laws for certain classes of finite element discretisation of a model elliptic problem. In addition we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether's 1st Theorem (E. Noether 1918). We summarise extensive numerical tests, illustrating the conservativity of the discrete Noether law using the p-Laplacian as an example.
In the seminal paper “Invariante Variationsprobleme” by Emmy Noether, she showed that for systems... more In the seminal paper “Invariante Variationsprobleme” by Emmy Noether, she showed that for systems derived from a variational principle, the associated conservation laws could be obtained from Lie group actions that left the variational problem unchanged. Recently, we proved that these conservation laws could be rewritten as the divergence of the product of a moving frame and a vector of invariants. The aim of this talk is to illustrate how the knowledge of the conservation laws structure helps reduce the extremising problem, in particular for variational problems that are invariant under the special Euclidean group SE(3).
Foundations of Computational Mathematics, Santander 2005
This volume is a collection of articles based on the plenary talks presented at the 2005 meeting ... more This volume is a collection of articles based on the plenary talks presented at the 2005 meeting in Santander of the Society for the Foundations of Computational Mathematics. The talks were given by some of the foremost world authorities in computational mathematics. The topics covered reflect the breadth of research within the area as well as the richness and fertility of interactions between seemingly unrelated branches of pure and applied mathematics. As a result this volume will be of interest to researchers in the field of computational mathematics and also to non-experts who wish to gain some insight into the state of the art in this active and significant field.
arXiv (Cornell University), Aug 10, 2018
In this second part of the paper, we consider finite difference Lagrangians which are invariant u... more In this second part of the paper, we consider finite difference Lagrangians which are invariant under linear and projective actions of SL(2), and the linear equi-affine action which preserves area in the plane. We first find the generating invariants, and then use the results of the first part of the paper to write the Euler-Lagrange difference equations and Noether's difference conservation laws for any invariant Lagrangian, in terms of the invariants and a difference moving frame. We then give the details of the final integration step, assuming the Euler Lagrange equations have been solved for the invariants. This last step relies on understanding the Adjoint action of the Lie group on its Lie algebra. We also use methods to integrate Lie group invariant difference equations developed in Part I. Effectively, for all three actions, we show that solutions to the Euler-Lagrange equations, in terms of the original dependent variables, share a common structure for the whole set of Lagrangians invariant under each given group action, once the invariants are known as functions on the lattice.
arXiv (Cornell University), Jun 26, 2019
We investigate some infinite dimensional Lie algebras and their associated Poisson structures whi... more We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If G is a Lie group, g its Lie algebra and M is a manifold on which G acts, then the set of smooth maps from M to g has at least two Lie algebra structures, both satisfying the required property to be a Lie algebroid. We may then apply a construction by Marle to obtain a Poisson bracket on the set of smooth real or complex valued functions on M × g *. In this paper, we investigate these Poisson brackets. We show that the set of examples include the standard Darboux symplectic structure and the classical Lie Poisson brackets, but is a strictly larger class of Poisson brackets than these. Our study includes the associated Hamiltonian flows and their invariants, canonical maps induced by the Lie group action, and compatible Poisson structures. Our approach is mainly computational and we detail numerous examples. The Lie brackets from which our results derive, arose from the consideration of connections on bundles with zero curvature and constant torsion. We give an alternate derivation of the Lie bracket which will be suited to applications to Lie group actions for applications not involving a Riemannian metric. We also begin a study of the infinite dimensional Poisson brackets which may be obtained by considering a central extension of the Lie algebras.
Foundations of Computational Mathematics, 2016
In this paper, we develop the theory of the discrete moving frame in two different ways. In the f... more In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver's one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, Communicated by Peter Olver.
Proceedings of the 2001 international symposium on Symbolic and algebraic computation, 2001
Acta Applicandae Mathematicae, 1995
In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) eq... more In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation u xxxt + αu x u xt + βu t u xx − u xt − u xx = 0, (1)
Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry grou... more Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the authors showed the mathematical structure behind both the Euler-Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. In this paper we demonstrate that the knowledge of this structure considerably eases finding the extremal curves for variational problems invariant under the special Euclidean groups SE(2) and SE(3).
Foundations of Computational Mathematics, 2013
Group based moving frames have a wide range of applications, from the classical equivalence probl... more Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer-Cartan invariants, for which there are recursion formulae. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices respectively under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.
In recent works [1, 2], the authors considered various Lagrangians, which are invariant under a L... more In recent works [1, 2], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems
In recent works, the authors considered various Lagrangians, which are invariant under a Lie grou... more In recent works, the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.
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Papers by Elizabeth Mansfield