David H. Sattinger

Multi-peakons and a theorem of Stieltjes

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.

Inverse scattering solutions of the hunter-saxton equation

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 has compact support. In this note, the associated inverse scattering problem is used to obtain the explicit solutions of the finite dimensional flows in both the compact and non-compact case.

Variational formulations for steady water waves with vorticity

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.

Gauge theory of Ba¨cklund transformations, II

Abstract 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 Backlund transformations to which they reduce precisely for the 2 × 2 “AKNS” systems. To each Backlund-gauge transformation is associated an automorphism in the Weyl group of the Lie algebra. Two proofs are given to show that these Backlund-gauge transformations preserve the nonlinear evolution equation. One proof uses the transformation of the scattering data under the Backlund transformation; the second proof is purely gauge theoretic.

On the complete integrability of completely integrable systems

The question of complete integrability of evolution equations associated ton×n first order isospectral operators is investigated using the inverse scattering method. It is shown that forn>2, e.g. for the three-wave interaction, additional (nonlinear) pointwise flows are necessary for the assertion of complete integrability. Their existence is demonstrated by constructing action-angle variables. This construction depends on the analysis of a natural 2-form and symplectic foliation for the groupsGL(n) andSU(n).

Bifurcation from rotationally invariant states

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 generalized to the case of an arbitrary simply reducible group. Applications to problems in geophysics and elasticity theory are discussed.

Peakon-Antipeakon Interaction

Abstract 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.

Scaling Limits and Models in Physical Processes

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 Equilibrium states and Maxwellian distributions.- 1.9 Model equations.- 1.10 Boundary conditions.- 2 Perturbation Methods for the Boltzmann Equation.- 2.1 Introduction.- 2.2 Rarefaction regimes.- 2.3 Solving the Boltzmann equation. Analytical techniques.- 2.4 Hydrodynamical limit and other scalings.- 2.5 The linearized collision operator.- 2.6 The basic properties of the linearized collision operator.- 2.7 Spectral properties of the Fourier-transformed, linearized Boltzmann equation.- 2.8 The asymptotic behavior of the solution of the Cauchy problem for the linearized Boltzmann equation.- 2.9 A quick survey of the global existence theorems for the nonlinear equation.- 2.10 Hydrodynamical limits. A formal discussion.- 2.11 The Hilbert expansion.- 2.12 The entropy approach to the hydrodynamical limit.- 2.13 The hydrodynamic limit for short times.- 2.14 Other scalings and the incompressible Navier-Stokes equations.- 2.15 Concluding remarks.- II Scaling, Mathematical Modelling, & Integrable Systems.- 1 Dispersion.- 1.1 Introduction.- 1.2 Group and phase velocities.- 2 Nonlinear Schrodinger Equation.- 2.1 Multiple scales expansion.- 2.2 Pulse solutions.- 3 Korteweg-de Vries.- 3.1 Background and history.- 3.2 Plasmas.- 3.3 Water waves.- 3.4 The solitary wave of the KdV equation.- 4 Isospectral Deformations.- 4.1 The KdV hierarchy.- 4.2 The AKNS hierarchy.- 5 Inverse Scattering Theory.- 5.1 The Schrodinger equation.- 5.2 First Order Systems.- 5.3 Decay of the scattering data.- 6 Variational Methods.- 6.1 Water Waves.- 6.2 Method of Averaging.- 7 Weak and Strong Nonlinearities.- 7.1 Breaking and Peaking.- 7.2 Strongly nonlinear models.- 7.3 The extended AKNS hierarchy.- 8 Numerical Methods.- 8.1 The finite Fourier transform.- 8.2 Pseudospectral codes.

Peakons, strings, and the finite Toda lattice

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, and multipeakon flows, and leads to explicit solutions of the Jacobi flows via Stieltjes’ determination of the continued fraction expansion of a Stieltjes transform. A simple modification produces a bijection from generalized strings, with positive and negative weights, to singular Jacobi matrices, and thus brings peakon/antipeakon flows into the same picture. c

A Riemann - Hilbert problem for an energy dependent Schrödinger operator

We consider an inverse scattering problem for Schr?dinger 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 , for suitably restricted, complementary classes of initial data. Dedicated to Hugh Turrittin on his 90th birthday