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Jacek Szmigielski

Publications and source records attributed to Jacek Szmigielski.

34 records · Page 2Linked to original sources
arXiv2013

An inverse spectral problem related to the Geng-Xue two-component peakon equation

We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a 'discrete cubic string' type -- a nonselfadjoint generalization of a classical inhomogeneous string -- but presents some interesting novel features: there are two Lax pairs, both of which contribute to the correct complete spectral data, and the solution to the inverse problem can be expressed using quantities related to Cauchy biorthogonal polynomials with two different spectral measures. The latter extends the range of previous applications of Cauchy biorthogonal polynomials to peakons, which featured either two identical, or two closely related, measures. The method used to solve the spectral problem hinges on the hidden presence of oscillatory kernels of Gantmacher-Krein type implying that the spectrum of the boundary value problem is positive and simple. The inverse spectral problem is solved by a method which generalizes, to a nonselfadjoint case, M. G. Krein's solution of the inverse problem for the Stieltjes string.

math.SP↗
arXiv2013

Shock creation and Painlevé property of colliding peakons in the Degasperis-Procesi Equation

The Degasperis-Procesi equation (DP) is one of several equations known to model important nonlinear effects such as wave breaking and shock creation. It is, however, a special property of the DP equation that these two effects can be studied in an explicit way with the help of the multipeakon ansatz. In essence this ansatz allows one to model wave breaking as a collision of hypothetical particles (peakons and antipeakons), called henceforth collectively multipeakons. It is shown that DP multipeakons have Painlevé property which implies a universal wave breaking behaviour, that multipeakons can collide only in pairs, and that there are no multiple collisions other than, possibly simultaneous, collisions of peakon-antipeakon pairs at different locations. Moreover, it is demonstrated that each peakon-antipeakon collision results in creation of a shock thus making possible a multi shock phenomenon.

nlin.SI↗
arXiv2013

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 $K_{n,n}$.

math.CO↗
arXiv2013

Peakon-antipeakon interactions in the Degasperis-Procesi Equation

Peakons are singular, soliton-like solutions to nonlinear wave equations whose dynamics can be studied using ordinary differential equations (ODEs). The Degasperis-Procesi equation (DP) is an important example of an integrable PDE exhibiting wave breaking in the peakon sector thus affording an interpretation of wave breaking as a mechanical collision of particles. In this paper we set up a general formalism in which to study collisions of DP peakons and apply it, as an illustration, to a detailed study of three colliding peakons. It is shown that peakons can collide only in pairs, no triple collisions are allowed and at the collision a shockpeakon is created. We also show that the initial configuration of peakon-antipeakon pairs is nontrivially correlated with the spectral properties of an accompanying non-selfadjoint boundary value problem. In particular if peakons or antipeakons are bunched up on one side relative to a remaining antipeakon or peakon then the spectrum is real and simple. Even though the spectrum is in general complex the existence of a global solution in either time direction dynamics is shown to imply the reality of the spectrum of the boundary value problem.

math-ph↗
arXiv2012

Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two--matrix model

We apply the nonlinear steepest descent method to a class of 3x3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polynomials.

nlin.SI↗
arXiv2009

Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation

Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation, admitting peaked soliton (peakon) solutions, which has nonlinear terms that are cubic, rather than quadratic. In this paper, the explicit formulas for multipeakon solutions of Novikov's cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation.

nlin.SI↗
arXiv2008

Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for continuous solutions, and then for more general shock wave solutions with discontinuities. We show that triple collisions of solitons cannot take place for continuous solutions, but give an example of a triple collision in the presence of a shock.

nlin.SI↗
arXiv2006

The inverse spectral problem for the discrete cubic string

Given a measure $m$ on the real line or a finite interval, the "cubic string" is the third order ODE $-ϕ'''=zmϕ$ where $z$ is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a nonselfadjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis--Procesi equation. We solve the spectral and inverse spectral problem for the case of $m$ being a finite positive discrete measure. In particular, explicit determinantal formulas for the measure $m$ are given. These formulas generalize Stieltjes' formulas used by Krein in his study of the corresponding second order ODE $-ϕ''=zmϕ$.

math.SP↗
arXiv2005

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.

nlin.SI↗
arXiv2005

Multi-peakon solutions of the Degasperis-Procesi equation

We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation.

nlin.SI↗
arXiv1998

Representations of the Heisenberg algebra by difference operators

We construct a class of representations of the Heisenberg algebra in terms of the complex shift operators subject to the proper continuous limit imposed by the correspondence principle. We find a suitable Hilbert space formulation of our construction for two types of shifts: (1) real shifts, (2) purely imaginary shifts. The representations involving imaginary shifts are free of spectrum doubling. We determine the corresponding coordinate and momentum operators satisfying the canonical commutation relations. The eigenvalues of the coordinate operator are in both cases discrete.

hep-th↗
arXiv1997

A Riemann-Hilbert Problem for an Energy Dependent Schrödinger Operator

\We consider an inverse scattering problem for Schrödinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for two distinct symmetry classes. As an application we prove global existence theorems for the two distinct systems of partial differential equations $u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted, complementary classes of initial data.

solv-int↗
arXiv1997

On Pairs of Difference Operators Satisfying: [P,Q] = Id

Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider $P$ and $X$ to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of $[P,X]=Id$.

hep-th↗
arXiv1993

On Soliton Content of Self Dual Yang-Mills Equations

Exploiting the formulation of the Self Dual Yang-Mills equations as a Riemann-Hilbert factorization problem, we present a theory of pulling back soliton hierarchies to the Self Dual Yang-Mills equations. We show that for each map $ \C^4 \to \C^{\infty } $ satisfying a simple system of linear equations formulated below one can pull back the (generalized) Drinfeld-Sokolov hierarchies to the Self Dual Yang-Mills equations. This indicates that there is a class of solutions to the Self Dual Yang-Mills equations which can be constructed using the soliton techniques like the $τ$ function method. In particular this class contains the solutions obtained via the symmetry reductions of the Self Dual Yang-Mills equations. It also contains genuine 4 dimensional solutions . The method can be used to study the symmetry reductions and as an example of that we get an equation exibiting breaking solitons, formulated by O. Bogoyavlenskii, as one of the $2 + 1 $ dimensional reductions of the Self Dual Yang-Mills equations.

hep-th↗