Perturbation Expansions for Integrable PDE’s and the “Squared Eigenfunctions”†

University of Central Florida, Orlando FL, USA

Institute for Simulation & Training

Department of Mathematics

and

D.J. Kaup

† Research supported in part by NSF and AFOSR.

References

• V.E. Zakharov & A.B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971). The ZS eigenvalue problem. • M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Stud. Appl. Math. 53, 249-315 (1974). The AKNS Recursion Operator and Closure of AKNS eigenfunctions.• D.J. Kaup, SIAM J. Appl. Math. {\bf 31}, 121-133 (1976). Perturbation Expansion for AKNS.• D.J. Kaup, J. Math. Analysis and Applications 54, 849-864 (1976). Closure of the Squared Zakharov-Shabat Eigenstates.• V.S. Gerdjikov and E.Kh. Khristov, Bulg. J. Phys. 7, 28 (1980) Proof of closure of squared ZS eigenstates. • V.S. Gerdjikov and P.P. Kulish, Physica 3D, 549-564 (1981) The nxn problem, squared states and closure. • D.J. Kaup, J. Math. Phys. 25, 2467-71 (1984). Closure of the Sine-Gordon (Lab) Squared Eigenstates.

OUTLINE

• Purpose.• Direct and adjoint eigenvalue problems.• Inner products.• Analytical properties.• Time evolution. • Linear dispersion relations and the RH problem.• Proof of closure.• Perturbations of potentials and scattering data.• “Squared eigenfunctions” and their eigenvalue problem.• New differential form of recursion operator. • Summary.

Purpose

• To outline how this is done in general case. • Seeking to generalize the procedure.• Will only describe the actions needed.• To point out the key features and steps needed. • We do not do the “mechanics”; only outline. • This is work in progress. • This is more of a descriptive lecture than new work. • We summarize and give an overall view of these actions.

General System (nxn)

One formulates the adjoint problem by:

One then can take:

And it is easy to show that one may take:

Basis Eigenfunctions

Let’s take the basis eigenstates to be:

And define the scattering matrix by:

Then one has:

Inner Products

From the preceding, it follows that:

where:

Now one may integrate and obtain:

Complexities?Multi-sheeted?

Inner Products - 2

From the preceding, it follows that:

Then provided that no J = 0 and that and A are real,

Notes:• Nowhere have we had to use Trace(J) = 0. • What if we shifted the elements of J so that J was never zero?• Analytical properties only depends on the differences in the elements of J.• Above is general for any eigenvalue problem as given in first slide.• No symmetries need be imposed on the potentials.

Analytical Properties

• 2x2 case is simple – upper or lower half plane.• General nxn case is more complex (Gerdjikov). • One must use Fundament Analytical Solutions (FAS).• Construction of FAS requires Gauss Decomposition. • Then one is to solve a matrix “Riemann-Hilbert Problem.” • This gives one a set of “Linear Dispersion Relations.”• One can do the same by using Cauchy’s Contour Integral Theorem on each FAS.• For perturbations, one can bypass the Blue.

Time Evolution

Lax Operators:

They satisfy:

Whence:

Evolution Equation for Q follows from commutation relation.

Proof of Closure

• Developed by Gerdjikov and Khristov (1980).• AKNS proof of 1974 used Marchenko Eqs.• Will illustrate it on ZS “un-squared” problem. •Requires two functions: G(x,y) and \bar{G}(x,y).

Note:• Analytic in one region.• Green’s function-like.• Poles at bound states.

In addition, these must satisfy:

Note: Theta functions are gone.

Proof of Closure - 2

Now, one constructs:

Where h(y) is arbitrary, but L1. From which we can form

= sum of residues

Then from the asymptotics, analytical properties and some magic, the theta functions go and one obtains

Whence, if h(y) is integrable, we have closure.

Perturbations: Q(x;)

Returning to the RH problem:

We perturb it and obtain (Yang):

• This is a simple RH problem. Solve it for . • From the asymptotics of for large, one obtains Q(x).• One then has Q(x) in terms of T(). • The coefficients of such are the adjoint “squared eigenfunctions”.

Perturbations: ()

Return to the eigenvalue problem.

we perturb it and obtain, for any VA and any V:

Taking VA=A and V=, we find:

• The coefficients of Q are the “squared eigenfunctions”.• Needs to be put in form of T and (+-).• Those are “mechanics”. Those are not in this outline.• From above and previous, one has what the closure should be.

Squared Eigenstates

Let’s take this in component form.

They satisfy:

The squared states are of the form:

There are two types of squared states. The diagonal elements, which have no spectral parameter and can be integrated, and the off-diagonal elements which have such. Also:

Whence there are only N2 – 1 independent components. (UA = V-1)

Perturbed Q-Equations

• Off-diagonal W’s satisfy the perturbed Q-equations. • The recursion operator provides some insight into this. • However the integro-differential form is awkward to use.• Result should not depend of which states are used.• Example of perturbed NLS.

Now construct the squared eigenstates for ZS.

Perturbed Q-Equations - 2

W and D satisfy:

Solving the first equation for W, we have:

Whence:

Summary

• Discussed general eigenvalue problem and adjoint problem.

• Have not discussed “mechanics” required for different systems.

• Discussion also extends to “squared eigenfunction” problem.

• Evaluation of inner products. • Analytical properties and RH problem.• Time evolution of S and Q. • Perturbations of potentials and scattering data.• Discussed “squared eigenstates”.• Q are to satisfy perturbed Q – equations.• New purely differential form given for recursion operator. • Example given of same. That’s all for now folks.