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

1. D.J. Kaup University of Central Florida, Orlando FL, USA Institute for Simulation & Training Perturbation Expansions for Integrable PDE’s and the “Squared Eigenfunctions”† and Department of Mathematics † Research supported in part by NSF and AFOSR.

2. 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.

3. 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.

4. 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.

5. General System (nxn) One formulates the adjoint problem by: One then can take: And it is easy to show that one may take:

6. Basis Eigenfunctions Let’s take the basis eigenstates to be: And define the scattering matrix by: Then one has:

7. Inner Products From the preceding, it follows that: where: Complexities? Multi-sheeted? Now one may integrate and obtain:

8. Inner Products - 2 From the preceding, it follows that: Then provided that no Ja = 0 and that z and zA are real, • Notes: • Nowhere have we had to use Trace(J) = 0. • What if we shifted the elements of J so that Ja 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.

9. 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.

10. Time Evolution Lax Operators: They satisfy: Whence: Evolution Equation for Q follows from commutation relation.

11. 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.

12. 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.

13. Perturbations: dQ(x;z) Returning to the RH problem: We perturb it and obtain (Yang): • This is a simple RH problem. Solve it for dc. • From the asymptotics of dc for large z, one obtains dQ(x). • One then has dQ(x) in terms of dT(z). • The coefficients of such are the adjoint “squared eigenfunctions”.

14. Perturbations: dT(z) Return to the eigenvalue problem. we perturb it and obtain, for any VA and any V: Taking VA=YA and V=F, we find: • The coefficients of dQ are the “squared eigenfunctions”. • Needs to be put in form of dT and c(+-). • Those are “mechanics”. Those are not in this outline. • From above and previous, one has what the closure should be.

15. Squared Eigenstates Let’s take this in component form. The squared states are of the form: They satisfy: 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)

16. 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.

17. Perturbed Q-Equations - 2 W and D satisfy: Solving the first equation for z W, we have: Whence:

18. 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”. • dQ are to satisfy perturbed Q – equations. • New purely differential form given for recursion operator. • Example given of same. That’s all for now folks.