Johanna Michor

Department of Mathematics


Here you find a description of my current and former research interests. All articles are available for download. A list of publications is available as pdf or html.

Research grants

My research was partly supported by the following grants (PI Johanna Michor).

FWF Elise Richter Grant "Long time asymptotics of Soliton equations"
Duration: 2011 - 2017, FWF Grant No. V120
Publications:
FWF Erwin Schrödinger Fellowship "Scattering theory for CMV operators and applications to completely integrable systems"
Duration: 03/2007 - 03/2009, FWF Grant No. J2655
Publications:
ÖAW DOC research grant "Asymptotic analysis of Jacobi operators and applications to completely integrable systems"
Duration: 10/2003 - 10/2004, DOC Grant No. 21388
Publications:

Algebro-geometric solutions of Soliton equations

Completely integrable wave equations, also known as soliton equations, are a very important topic used to describe many physical phenomena. For example, the nonlinear Schroedinger equation (NLS) describes the evolution of small amplitude, slowly varying wave packets in nonlinear media and has been derived in such diverse fields as deep water waves, plasma physics, and nonlinear optical fibers. Algebro-geometric solutions are a natural extension of the class of soliton solutions. Remarkably, algebro-geometric solutions can still be constructed explicitly using methods from algebraic geometry.

In [GHMT07] we have determined all complex-valued quasi-periodic finite-gap solutions of the Ablowitz-Ladik (AL) hierarchy, a completely integrable sequence of systems of nonlinear evolution equations on the lattice ℤ. The first nonlinear equation in this hierarchy is the Ablowitz-Ladik system, a discretized version of the nonlinear Schroedinger equation. For the construction of the AL hierarchy via a zero-curvature and Lax approach see [GHMT08a]. In [GHMT08b] we derive the Lax operator for the AL hierarchy, which is a five-diagonal finite difference operator. Moreover, we give a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the AL hierarchy, using asymptotic expansions of the Green's function of the AL Lax operator. In [GHMT09] we solve the algebro-geometric initial value problem for the AL hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. This results are summarized in our monograph [GHMT2008] and in my Habilitation thesis [M12].
This is joint work with Fritz Gesztesy (Columbia), Helge Holden (NTNU), and Gerald Teschl (Vienna).

Inverse scattering transform for the Toda Hierarchy

Since the classical work of Gardner, Greene, Kruskal, and Miura [1] in 1967 on the Korteweg-de Vries (KdV) equation, the inverse scattering transform, a nonlinear analogue of the Fourier transform, has been one of the main tools for solving completely integrable nonlinear evolution equations. This method consists of three parts: The scattering problem for the associated Lax operator, the time evolution of the scattering data, and the reconstruction of the potential function from the (time dependent) scattering data via the inverse scattering problem.

In my PhD thesis [M05] I investigated the inverse scattering transform for the Toda hierarchy relative to a periodic background. More precisely, we consider short-range perturbations of quasi-periodic Jacobi operators (the Toda Lax operators) and find minimal scattering data which determine the perturbed operator uniquely (see also [EMT06]). Then we solve the associated initial value problem for the Toda hierarchy in [EMT07a]. The connection between the transmission coefficient and Krein's spectral shift theory is established in [MT07] and used to compute the conserved quantities for the Toda hierarchy relative to a quasi-periodic background.
In the case of a steplike situation, when the coefficients are asymptotically close to two different finite-gap operators as n→±∞, the scattering problem is solved in [EMT07b] under the restriction that the two background operators are isospectral. This restriction was later removed in [EMT08]. The inverse scattering transform with steplike finite-gap background is treated in [EMT09].
For an introduction to algebro-geometric quasi-periodic solutions of the Toda hierarchy see [GHMT2008]. There we also consider the Kac-van Moerbeke hierarchy, which appears as a special case in the Toda hierarchy, [MT09].
This is joint work with Iryna Egorova (ILT, Ukraine) and Gerald Teschl (Vienna).

Stability of Soliton equations

The classical result by Zabusky and Kruskal [2] states that a small initial perturbation of the constant solution of a soliton equation eventually splits into a number of stable solitons and a small oscillatory tail which decays. The oscillations arise from the continuous spectrum of the underlying Lax operator and the soliton part stems from the discrete spectrum. In particular, the solitons constitute the persistent part of solutions arising from arbitrary short range perturbations of the initial conditions.

The stability of the periodic Toda lattice has attracted a lot of interest recently. Kamvissis and Teschl [3], [4] investigated the case of short-range perturbations of an algebro-geometric quasi-periodic background operator and showed that the oscillation part does not decay as in the constant background case. Instead, it appears as a modulation of the quasi-periodic solution which undergoes a continuous phase transition in the isospectral class of the quasi-periodic background solution. The soliton part can be understood by adding or removing the solitons using the commutation method (Darboux-type transformations) for the underlying Jacobi operator, see [EMT09]. We explicitly computed the phase shift in the Jacobian variety caused by a soliton relative to a quasi-periodic finite-gap background and gave a full description of the effect of one commutation step on the scattering data. This is joint work with Iryna Egorova (ILT, Ukraine) and Gerald Teschl (Vienna).

For the Ablowitz-Ladik system, I show in [M10] that the leading asymptotic term of arbitrary decaying solutions is time independent. Moreover, if two bounded solutions of the Ablowitz-Ladik system are asymptotically close at the initial time they stay close.

Spectral theory for Jacobi matrices

A well-known theorem of Hochstadt [5] shows how to reconstruct a Jacobi matrix J from its spectrum plus the spectrum of the submatrix obtained by removing the first row and column of J. The generalisation of this theorem, i.e. if one removes the n-th row and column of J, is given in [MT04] and was part of my diploma thesis [M02].
This is joint work with Gerald Teschl (Vienna).

References

[1] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, A method for solving the Korteweg-de Vries equation, Phys. Rev. Letters 19, 1095-1097 (1967).
[2] N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15, 240-243 (1963).
[3] S. Kamvissis and G. Teschl, Stability of periodic soliton equations under short range perturbations, Phys. Lett. A 364-6, 480-483 (2007).
[4] S. Kamvissis and G. Teschl, Stability of the periodic Toda lattice under short range perturbations, arXiv:0705.0346.
[5] H. Hochstadt, On the construction of a Jacobi matrix from spectral data, Lin. Algebra Appl. 8, 435-446 (1974).