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

**html**.

### Research grants

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

### 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).