3,179 research outputs found

### Multiplication of solutions for linear overdetermined systems of partial differential equations

A large family of linear, usually overdetermined, systems of partial
differential equations that admit a multiplication of solutions, i.e, a
bi-linear and commutative mapping on the solution space, is studied. This
family of PDE's contains the Cauchy-Riemann equations and the cofactor pair
systems, included as special cases. The multiplication provides a method for
generating, in a pure algebraic way, large classes of non-trivial solutions
that can be constructed by forming convergent power series of trivial
solutions.Comment: 27 page

### The methodology of using precedents

This paper elucidates the common law doctrine of stare decisis and the methodology of using precedents, including the practice of distinguishing and overruling them

### Dynamics of interlacing peakons (and shockpeakons) in the Geng-Xue equation

We consider multipeakon solutions, and to some extent also multishockpeakon
solutions, of a coupled two-component integrable PDE found by Geng and Xue as a
generalization of Novikov's cubically nonlinear Camassa-Holm type equation. In
order to make sense of such solutions, we find it necessary to assume that
there are no overlaps, meaning that a peakon or shockpeakon in one component is
not allowed to occupy the same position as a peakon or shockpeakon in the other
component. Therefore one can distinguish many inequivalent configurations,
depending on the order in which the peakons or shockpeakons in the two
components appear relative to each other. Here we are in particular interested
in the case of interlacing peakon solutions, where the peakons alternatingly
occur in one component and in the other. Based on explicit expressions for
these solutions in terms of elementary functions, we describe the general
features of the dynamics, and in particular the asymptotic large-time
behaviour. As far as the positions are concerned, interlacing Geng-Xue peakons
display the usual scattering phenomenon where the peakons asymptotically travel
with constant velocities, which are all distinct, except that the two fastest
peakons will have the same velocity. However, in contrast to many other peakon
equations, the amplitudes of the peakons will not in general tend to constant
values; instead they grow or decay exponentially. Thus the logarithms of the
amplitudes (as functions of time) will asymptotically behave like straight
lines, and comparing these lines for large positive and negative times, one
observes phase shifts similar to those seen for the positions of the peakons.
In addition to these K+K interlacing pure peakon solutions, we also investigate
1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2
peakon-antipeakon solution.Comment: 59 pages, 6 figures. pdfLaTeX + AMS packages + hyperref + TikZ.
Changes in v2: minor typos corrected, reference list updated and enhanced
with hyperlink

### Non-interlacing peakon solutions of the Geng-Xue equation

The aim of the present paper is to derive explicit formulas for arbitrary
peakon solutions of the Geng-Xue equation, a two-component generalization of
Novikov's cubically nonlinear Camassa-Holm type equation. By performing
limiting procedures on the previosly known formulas for so-called interlacing
peakon solutions, where the peakons in the two component occur alternatingly,
we turn some of the peakons into zero-amplitude "ghostpeakons", in such a way
that the remaining ordinary peakons occur in any desired configuration. We also
study the large-time asymptotics of these solutions.Comment: 133 pages, 25 figures. pdfLaTeX + AMS packages + hyperref + Tik

### Degasperis-Procesi peakons and the discrete cubic string

We use an inverse scattering approach to study multi-peakon solutions of the
Degasperis-Procesi (DP) equation, an integrable PDE similar to the Camassa-Holm
shallow water equation. The spectral problem associated to the DP equation is
equivalent under a change of variables to what we call the cubic string
problem, which is a third order non-selfadjoint generalization of the
well-known equation describing the vibrational modes of an inhomogeneous string
attached at its ends. We give two proofs that the eigenvalues of the cubic
string are positive and simple; one using scattering properties of DP peakons,
and another using the Gantmacher-Krein theory of oscillatory kernels.
For the discrete cubic string (analogous to a string consisting of n point
masses) we solve explicitly the inverse spectral problem of reconstructing the
mass distribution from suitable spectral data, and this leads to explicit
formulas for the general n-peakon solution of the DP equation. Central to our
study of the inverse problem is a peculiar type of simultaneous rational
approximation of the two Weyl functions of the cubic string, similar to
classical Pade-Hermite approximation but with lower order of approximation and
an additional symmetry condition instead. The results obtained are intriguing
and nontrivial generalizations of classical facts from the theory of Stieltjes
continued fractions and orthogonal polynomials.Comment: 58 pages, LaTeX with AMS packages, to appear in International
Mathematics Research Paper

### Separation of variables in quasi-potential systems of bi-cofactor form

We perform variable separation in the quasi-potential systems of equations of
the form $\ddot{q}=-A^{-1}\nabla k=-\tilde{A}^{-1}\nabla\tilde{k}${}, where $A$
and $\tilde{A}$ are Killing tensors, by embedding these systems into a
bi-Hamiltonian chain and by calculating the corresponding Darboux-Nijenhuis
coordinates on the symplectic leaves of one of the Hamiltonian structures of
the system. We also present examples of the corresponding separation
coordinates in two and three dimensions.Comment: LaTex, 30 pages, to appear in J. Phys. A: Math. Ge

### The Canada Day Theorem

The Canada Day Theorem is an identity involving sums of $k \times k$ minors
of an arbitrary $n \times n$ symmetric matrix. It was discovered as a
by-product of the work on so-called peakon solutions of an integrable nonlinear
partial differential equation proposed by V. Novikov. Here we present another
proof of this theorem, which explains the underlying mechanism in terms of the
orbits of a certain abelian group action on the set of all $k$-edge matchings
of the complete bipartite graph $K_{n,n}$.Comment: 16 pages. pdfLaTeX + AMS packages + TikZ. Fixed a hyperlink problem
and a few typo

### Quasi-Lagrangian Systems of Newton Equations

Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$
with an integral of motion quadratic in velocities are studied. These equations
generalize the potential case (when A=I, the identity matrix) and they admit a
curious quasi-Lagrangian formulation which differs from the standard Lagrange
equations by the plus sign between terms. A theory of such quasi-Lagrangian
Newton (qLN) systems having two functionally independent integrals of motion is
developed with focus on two-dimensional systems. Such systems admit a
bi-Hamiltonian formulation and are proved to be completely integrable by
embedding into five-dimensional integrable systems. They are characterized by a
linear, second-order PDE which we call the fundamental equation. Fundamental
equations are classified through linear pencils of matrices associated with qLN
systems. The theory is illustrated by two classes of systems: separable
potential systems and driven systems. New separation variables for driven
systems are found. These variables are based on sets of non-confocal conics. An
effective criterion for existence of a qLN formulation of a given system is
formulated and applied to dynamical systems of the Henon-Heiles type.Comment: 50 pages including 9 figures. Uses epsfig package. To appear in J.
Math. Phy

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