Perturbation%20Expansions

1

• Perturbation Expansions
• for Integrable PDEs and the Squared
  Eigenfunctions

D.J. Kaup
University of Central Florida, Orlando FL, USA
Institute for Simulation Training
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.

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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.

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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.

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General System (nxn)
One formulates the adjoint problem by
One then can take
And it is easy to show that one may take
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Basis Eigenfunctions
Lets take the basis eigenstates to be
And define the scattering matrix by
Then one has
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Inner Products
From the preceding, it follows that
where
Complexities? Multi-sheeted?
Now one may integrate and obtain
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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.

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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 Cauchys Contour
• Integral Theorem on each FAS.
• For perturbations, one can bypass the Blue.

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Time Evolution
Lax Operators They satisfy Whence
Evolution Equation for Q follows from commutation
relation.
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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 \barG(x,y).

• Note
• Analytic in one region.
• Greens function-like.
• Poles at bound states.

In addition, these must satisfy
Note Theta functions are gone.
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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.
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Perturbations dQ(xz)
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.

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Perturbations dT(z)
Return to the eigenvalue problem. we perturb
it and obtain, for any VA and any V Taking
VAYA and VF, 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.

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Squared Eigenstates
Lets 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)
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Perturbed Q-Equations

• Off-diagonal Ws 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.
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Perturbed Q-Equations - 2
W and D satisfy
Solving the first equation for z W, we have
Whence
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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.

Thats all for now folks.