D. H. Sattinger's research while affiliated with University of Minnesota Duluth and other places
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Publications (88)
The theory of completely integrable systems based on n × n first order isospectral problems is reviewed. The topics discussed are the inverse scattering theory for n × n systems; hierarchies of completely integrable Hamiltonian systems based on semi-simple Lie algebras; flat connections and gauge transformations; and the gauge theory of Bäcklund tr...
We describe the Hamiltonian structures, including the Poisson brackets and Hamiltonians, for free boundary problems for incompressible fluid flows with vorticity. The Hamiltonian structure is used to obtain variational principles for stationary gravity waves both for irrotational flows as well as flows with vorticity.
Russell’s velocity formula was at the center of the controversy over the existence of the solitary wave; but today the topic
is rarely mentioned. It is an immediate corollary of modern bifurcation theory; and it is fundamental to modelling waves in
deep water. A tsunami 60cm high in an ocean 4km deep is 377km long, travels with a velocity of 713km/...
Recently, the string density problem, considered in the pioneering work of M. G. Kreĭn, has arisen naturally in connection with the Camassa-Holm equation for shallow water waves. In this paper we review the forward and inverse string density problems, with some numerical examples, and relate it to the Camassa-Holm equation, with special reference t...
For free-surface water flows with a vorticity that is monotone with depth, we show that any critical point of a functional representing the total energy of the flow adjusted with a measure of the vorticity, subject to the constraints of fixed mass and horizontal momentum, is a steady water wave.
The analogy between the inverse scattering problem for the Schrdinger operator and an infinite dimensional factorization problem is discussed. The function introduced by Hirota is obtained as the Fredholm determinant of the lower minors of the scattering operator. The classical Darboux transformation is presented in the context of the dressing meth...
It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated...
Stieltjes’ solution of the classical moment problem is the forerunner of inverse spectral theory. In the modern theory of completely integrable systems, the method of inverse scattering is used to obtain explicit solutions of a class of Hamiltonian systems which arise as isospectral deformations of a linear operator. In this lecture we explain how...
We study solitary wave interactions in the Euler-Poisson equations modeling ion acoustic plasmas and their approximation by KdV n-solitons. Numerical experiments are performed and solutions compared to appropriately scaled KdV nn-solitons. While largely correct qualitatively the soliton solutions did not accurately capture the scattering shifts exp...
Recent results on inverse scattering problems associated to operators of the form (d/dx) 2 +k 2 m-q are described. Applications to nonlinear evolution equations including the equations of Camassa-Holm, Hunter-Saxton, Calogero-Françoise, and Jabobi flows such as the finite Toda flow and the Kac-van Moerbeke flow are outlined. In particular, the solu...
The completely integrable Hamiltonian systems discovered by Calogero and Franoise contain the finite-dimensional reductions of the Camassa–Holm and Hunter–Saxton equations. We show that the associated spectral problem has the same form as that of the periodic discrete Camassa–Holm equation. The flow is linearized by the Abel map on a hyperelliptic...
The nonlinear partial differential equation was proposed by Hunter and Saxton as an asymptotic model equation for nematic liquid crystals. Hunter and Zheng showed that it is a member of the Harry Dym hierarchy of integrable flows, and solved the equation explicitly for a family of finite dimensional, piecewise linear functions in the case when ux h...
Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa-- Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon.
As is well-known, the Toda lattice flow may be realized as an isospectral flow of a Jacobi matrix. A bijective map from a discrete string problem with positive weights to Jacobi matrices allows the pure peakon flow of the Camassa-Holm equation to be realized as an isospectral Jacobi flow as well. This gives a unified picture of the Toda, Jacobi, an...
Classical results of Stieltjes are used to obtain explicit formulas for the peakon–antipeakon solutions of the Camassa–Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon–antipeakon pairs, and the details of the collisio...
Numerical experiments involving the interaction of two solitary waves of the ion acoustic plasma equations are described. An exact 2-soliton solution of the relevant KdV equation was fitted to the initial data, and good agreement was maintained throughout the entire interaction. The data demonstrates that the soliton interactions are virtually elas...
A closed form of the multi-peakon solutions of the Camassa-Holm equation
is found using a theorem of Stieltjes on continued fractions. An
explicit formula is obtained for the scattering shifts.
The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transfor- mation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is p...
35A05 General existence and uniqueness theorems
35J10 Schrödinger operator (See also 35Pxx)
35R30 Inverse problems (undetermined coefficients, etc.) for PDE
78A46 Inverse scattering problems
Uniform estimates for the decay structure of the $n$-soliton solution of the Korteweg-deVries equation are obtained. The KdV equation, linearized at the $n$-soliton solution is investigated in a class $\WW$ consisting of sums of travelling waves plus an exponentially decaying residual term. An analog of the kernel of the time-independent equation i...
The classical version of the three wave interaction models the creation and
destruction of waves; the quantized version models the creation and destruction
of particles. The quantum three wave interaction is described and the Bethe
Ansatz for the eigenfunctions is given in closed form. The Bethe equations are
derived in a rigorous fashion and are s...
Using the scattering transform for n th order linear scalar operators, the Poisson bracket found by Gel’fand and Dikii, which generalizes the Gardner Poisson bracket for the KdV hierarchy, is computed on the scattering side. Action-angle variables are then constructed. Using this, complete integrability is demonstrated in the strong sense. Real act...
The isospectral flows of an nth order linear scalar differential operator L under the hypothesis that it possesses a Baker-Akhiezer function were originally investigated by Segal and Wilson from the point of view of infinite dimensional Grassmanians, and the reduction of the KP hierarchy to the Gel'fand-Dikii hierarchy. The associated first order s...
We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the n × n AKNS hierarchies and the Gel'fand-Dikii hierarchies. The third arises in string theory as the representation of the Heisenberg group by where L is an nth order scalar differentia...
According to kinetic theory, a gas in normal conditions (no chemical reactions, no ionization phenomena, etc.) is formed of elastic molecules rushing hither and thither at large speeds, colliding and rebounding according to the laws of elementary mechanics. Within the scope of these notes (except in some remarks), the molecules of a gas will be ass...
The nonlinear Schrödinger equation arises in a multitude of contexts, but its best known application is to the self-focussing of one-dimensional electromagnetic waves, for example optical pulses along an optical fiber, in nonlinear media [76]. There are a number of formal derivations of this equation, one of which is based on multiple scales expans...
THE FORWARD PROBLEM Consider the eigenvalue problem for the Schrödinger equation
$$
\left( {{D^2} + {k^2} + \frac{1}{6}u} \right)\psi = 0
$$ (5.1.1)
where u is real and lies in S. S(ℝ) is the class of all C∞ functions on the real line for which
$$
\mathop {\sup }\limits_x \left| {{x^m}{D^n}u} \right| < + \infty
$$
for all non-negative integers m an...
I Scaling and Mathematical Models in Kinetic Theory.- 1 Boltzmann Equation and Gas Surface Interaction.- 1.1 Introduction.- 1.2 The Boltzmann equation.- 1.3 Molecules different from hard spheres.- 1.4 Collision invariants.- 1.5 The Boltzmann inequality and the Maxwell distributions.- 1.6 The macroscopic balance equations.- 1.7 The H-theorem.- 1.8 E...
The problem of modelling nonlinear wave phenomena, despite 150 years of progress, is still far from resolved. For the Korteweg-de Vries equation, though nonlinear, has regular solutions for all time, given regular initial data. That is, singularities do not form under the evolution governed by the KdV equation. This fact must be reconciled with G....
The Korteweg-de Vries equation,
$$
{u_t} + u{u_x} + {u_{xxx}} = 0
$$ (3.1.1)
is the simplest equation that includes both the effects of nonlinearity and dispersion. The equation appears in various forms in the literature, sometimes with a factor of 6 or -6 in front of the nonlinear term. We shall use the present normalization, since that is the uni...
The Korteweg-de Vries (KdV) equation,
$$
{u_t} + {u_{xxx}} + u{u_x} = 0
$$ (4.1.1)
has smooth solutions for all positive and negative time given initial data which is sufficiently smooth, say C3. We won't go into the existence theory of the KdV equation here since the existence and uniqueness issues are ultimately resolved very simply by the method...
The Boltzmann equation has an important application to the study of flight in the upper atmosphere, which occurs, e.g., in connection with the re-entry of a space shuttle.
In these lectures I shall describe the modelling of the competing effects of dispersion and nonlinearity in physical systems. In particular, I shall discuss in detail the derivation of the Korteweg-de Vries and nonlinear Schrödinger equations. The interplay of dispersion and nonlinearity is at the heart of the phenomena which these equations purpor...
In order to test the validity of various model equations for nonlinear dispersive systems, it is extremely handy to have some numerical codes to solve dispersive systems of partial differential equations and compare them with the various approximations proposed. To this end, Yi Li1 and I have written some simple codes to be run in Matlab. These sol...
We consider the scattering transform for Schrödinger operators with energy dependent
potentials. We prove unique invertibility of the transform when there are no bound states
and find a simplified recovery formula. We construct as a special case a one-soliton
"breather". As an application we prove a global existence theorem for a class of
non-linea...
The question of complete integrability of evolution equations associated to
$n\times n$ first order isospectral operators is investigated using the inverse
scattering method. It is shown that for $n>2$, e.g. for the three-wave
interaction, additional (nonlinear) pointwise flows are necessary for the
assertion of complete integrability. Their existe...
The scattering theory of n×n first-order systems on the line is formulated in terms of a flat connection on a vector bundle over R×P 1 (C). The relation of the scattering data to a set of transition matrices is discussed. The scattering transform is obtained as a sectionally holomorphic gauge transformation. The winding number constraints of D. Bar...
The solution of the Yang–Baxter equation for integrable systems is shown to be equivalent to the existence of a differential identity. Quantum integration formulas for the calculation of commutators of monodromy matrices are given. Based on the integration formulas and the systematic use of differential identities, the Yang–Baxter equations for the...
Partial Contents: Markov random fields and their applications; Proceedings of the conference on integration, topology, and geometry in linear spaces; Problems of elastic stability and vibrations; Rational constructions of modules for simple Lie algebras; Umbral calculus and Hopf algebras; Complex contour integral representation of cardinal spline f...
The “dressing method” of Zakharov and Shabat is applied to the theory of the τ function, vertex operators, and the bilinear identity obtained by Sato and his co-workers. The vertex operator identity relating the τ function to the Baker-Akhiezer function is obtained from their representations in terms of the Fredholm determinants and minors of the s...
A class of gauge transformations is constructed for Hamiltonian hierarchies of completely integrable systems on semi-simple Lie algebras. These transformations create or annihilate poles in the scattering data, hence create or annihilate solitons in the potential Q. In that sense they generalize Bäcklund transformations to which they reduce precise...
A recursion formula is described which generates infinite hierarchies of completely integrable Hamiltonian systems of nonlinear partial differential equations. These equations govern the evolution of a function u of x, t which takes its values in a semisimple Lie algebra. A Hamiltonian for the hierarchy is given in terms of a meromorphic connection...
This chapter examines the gauge theories for soliton problems, focusing on lax equations and isospectral deformations. Some of the striking properties of the K d V equation are its Hamiltonian structure, an infinite number of conservation laws, an associated invariant (isospectral) scattering problem, and multi-soliton solutions. The first step in...
In analyzing the dynamics of a physical system governed by nonlinear equations the following questions occur: Are there equilibrium states of the system? How many are there? Are they stable or unstable? What happens as external parameters are varied? In particular, what happens when a known solution becomes unstable as some parameter passes through...
This chapter discusses the application of group theoretic methods in the bifurcation theory. The subject of bifurcation is an important topic for applied mathematics as it arises naturally in any physical system described by a nonlinear set of equations depending on a set of parameters. The chapter presents an equation that represents a nonlinear s...
This chapter reviews recent progress in bifurcation theory. To begin with, bifurcation theory deals with the analysis of branch points of nonlinear functional equations in a vector space, usually a Banach space. The subject of bifurcation is an important topic for applied mathematics in as much as it arises naturally in any physical system describe...
Bifurcation in the presence of the rotation group is investigated. The covariant bifurcation equations are derived using the familiar angular momentum operators of quantum mechanics. Variational methods are also discussed. It is shown that the quadratic terms either vanish for odd l or possess a gradient structure for even l. This result is general...
Earlier results on stability of traveling waves (TW) reported by the
author (1975, 1976) are refined. A sharper analysis is provided of the
weighted norms introduced by the author in earlier work on TW stability.
TW solutions of argument (x + ct), c constant, are studied. Nonlinear
parabolic equations or systems of such equations and their correspo...
In many physical applications the equations describing a system are invariant under some transformation group. When bifurcation problems arise in such a situation, the group invariance may lead to multiplicities of the branch point. The main goal of the present paper is to demonstrate in a precise way the application of group representation theory...
If a layer of fluid is heated from below, convective instabilities set in when the temperature drop exceeds a certain critical value, and the convective motions which evolve often display a striking cellular structure. (See the experimental accounts by E. L. KOSCHMIEDER [11]. ) For a historical and mathematical account of these problems and the rel...
When a layer of fluid is heated from below convection sets in when the temperature drop across the layer exceeds a certain critical value. Then, under certain circumstances, the convective motions which take place are organized in cellular patterns - rolls or hexagonal solutions being the most common. [1], [3], [6]
Bifurcation in the presence of the rotation group is investigated. The covariant bifurcation equations are derived using the familiar angular momentum operators of quantum mechanics. Variational methods are also discussed. It is shown that the quadratic terms either vanish for odd l or possess a gradient structure for even l. This result is general...
We consider a container of fluid with a rod inserted in the centre. As
the rod rotates the surface of the fluid forms a curved surface whose
shape correctly balances the forces of gravity, internal stress,
atmospheric pressure, and surface tension. The surface of the fluid is
depressed in the neighbourhood of the rod in the case of a Newtonian
flui...
A number of authors have investigated conditions under which weak solutions of the initial-boundary value problem for the
nonlinear wave equation will blow up in a finite time. For certain classes of nonlinearities sharp results are derived in
this paper. Extensions to parabolic and to abstract operator equations are also given.
A system of differential equations is considered that represents a simple model governing the combustion of a material. The emphasis is not on the final state which the system eventually reaches, but rather the manner in which that state is attained.
1. Topological degree arguments show that bifurcation must take place at eigenvalues of odd multiplicity, while examples show bifurcation may not take place at eigenvalues of even multiplicity. The general problem of bifurcation at multiple eigenvalues is one which does not readily submit to a complete solution, so the approach must proceed by spec...
Proceedings of a Conference on Inverse Scattering on Line, held June 7-13, 1990 at the University of Massachusetts, Amherst, with support from the National Science Foundation, the National Security Agency, and the Office of Naval Research Incluye bibliografía
Citations
... Physical foundations and general description Anderson 21 Bhaskar and Nigam 55 Bridgman 56 Gattus and Karamitsos 20 Gibbings 57 Goldreich et al. 58 Ipsen 59 Lemons 10 Simon et al. 60 Szirtes 61 Weisskopf 62 Zohuri 63,64 Production processes Miragliotta 65 Scaling, asymptotic analysis, and renormalization Badii 66 Barenblatt 9 Barenblatt et al. 11 Batterman 67 Cercignani and Sattinger 68 Goldenfeld 22 Henriksen 69 Lesne and Laguës 70 ...
... Eq. (A.4) has an invariance under rotation, i.e., z ~ eiaz, ot E R. To exploit this we shall rewrite (A.4) in scaled polar coordinates, also introducing a scaled time r = Xals. See ref. [31] sin( o, -02), rz /~, = col + ql( 1 -r2) + ~[where, ...
Reference: Amplitude response of coupled oscillators
... However, the cases of even and odd l have been proved to be quite different [12, 13, 48]. In fact, for odd l there are no quadratic terms in the bifurcation equations whereas for even l quadratic terms are present but, because of the spherical symmetry, a unique quadratic term is allowed [48, 61]. It follows that in the case of even l, the behavior on the center manifold is uniquely determined up to the quadratic order. ...
... Examples are the water wave problem or equations from plasma physics, cf. [3]. For the Boussinesq equation ...
... With such an approach one finds again, although the calculations are now more involved, that the peakon ansatz (1.7) is a weak solution if and only if the quantities and satisfy the system of ODEs (1.10). We remark that one may of course also study periodic weak solutions, in particular periodic peakon solutions, and then the inverse of 1 − 2 will be different, but we will not consider that case here (see however Beals et al. [16,17]). ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/362391482191875-1463412270983_Q64/Jacek-Szmigielski.jpg)
... Here we set up notations and review the method of constructing soliton hierarchies from splittings of Lie algebras. For more details, see [2,4,[9][10][11]13,14,20]. ...
... When two solitons attract with each other then the trailing and leading edge of the nearby solitons overlap and the slope decreases [10]. It results in pulse broadening which can be cancelled by the SPM by careful design of the fiber [11]. ...
... We had at first started from a question whether soliton equations exist in the TDHF/TDHB manifold or do not, in spite of the difference that the solitons are described in terms of infinite degrees of freedom and the RPA in terms of finite ones. We had met with the inverse-scattering-transform method by AKNS [28] and the differential geometrical approaches on group manifolds [29]. An integrable system is explained by the zero-curvature, i.e., integrability condition of connection on the corresponding Lie group. ...
... When p = 2, these results were discussed in [4]. Their study was motivated by the applications in chemical reaction theory (see [2]) and in combustion theory (see [11,14]). ...
... Using the Riemann-Hilbert problem approach the inverse scattering problem with purely imaginary or real potential p was considered by Sattinger and Szmigielski [56,57]. In general, they discussed the case when there are no bound states, but they also considered the case when there is one bound states. ...
