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ТЕОРЕТИЧЕСКАЯ ФИЗИКА
8
УДК 524.85
V. A. Yurov
GENERALIZED EVANS FUNCTION FOR CONTINUOUS SPECTRUM
Решается задача определения функции EH (X) такой, что если {Xn }
— точки непрерывного спектра оператора H и X Ф Xn, то EH (X) определена и не равна нулю.
The task is to define a function EH(X), such that if {Xn } are the points of the continuous spectrum of operator H and X Ф Xn, then EH(X) is defined and is non-zero.
Ключевые слова: функция Эванса, дзета-функция, непрерывный спектр.
Key words: Evans function, Zeta function, continuous spectrum.
Introduction
This article is dedicated to the task of defining a function EH (2), associated with a differential operator H, that is sensitive to the spectrum of this operator. If we restrict our attention to the discrete spectrum adjsc (H) and require
that equality EH (2) = 0 should only de satisfied for 2 e and require that equality EH (2) = 0 adjsc (H), we arrive to a familiar object, called the Evans function. Originally introduced by J.W. Evans [1] in attempt to analyze the stability of nerve pulses in the Fitzhugh — Nagumo system, it has since became pretty much a staple tool in mathematical physics (cf., [2—10]). However, it's usefulness proves to be severely limited when one takes the continuous spectrum (<7cont) into the picture; it is not uncommon to end up with the Evans function being exactly equal to zero if some special properties of continuous spectrum are not satisfied. In many parts this difficulty stems from the way the Evans function is commonly defined as a Wronskian of two solutions possessing the specific asymptotes at +» and -» respectively.
The purpose of this article therefore lies in the attempt to make a step in a different direction and take a look at different possibilities for the Evans function that would not heavily depend on the position of continuous spectrum, but would instead provide us with information about acont itself. Namely, we are trying to define such an object that if {Xn} are the points of the continuous spectrum of operator H and 2 Ф Xn, then EH (X) is defined and is non-zero. The requirement for this object to be equal to zero for the points belonging to the point spectrum remains similar to the requirements for the standard Evans function.
Вестник Балтийского государственного университета им. И. Канта. 2011. Вып. 5. С. 8—12.
Generalized Evans function for continuous spectrum
Main definitions
Let Hyn = Xnyn, and let {An } denote the set of all discrete and continuous spectra points. Let X be an arbitrary real parameter and define the operator: A = H-X Obviously, Ayn = (An-X) y/n, i.e. the eigenvalues of operator A have the form an = Xn - X. If the spectrum of A is the discrete one,
det A = det(H -1) = n an = n Qn -1 )•
n n
Note, that Evans function is usually defined on the class of rapidly decreasing potentials with the aid of a certain integral representation. If, however, we turn our attention to the class of reflectionless potentials, it is possible to show via Darboux transformations (cf. [2] for example), that the Evans function is a legible candidate for the factorization presented above. The rigorous proof of this important fact will be a subject of a separate article..
Now let us introduce gA (s) = ^ an . Using the definition of T-function we
n
can write (Among other applications, this representation proves to be extremely handy in calculating required by the one-loop quantum correction. [3]):
when A < An, and
'A (s) =
(-1)s
-\an\t
r(s) 0 n
Next, introduce the Green functions:
for A > A,.
G- (t; x, y) = S e-an T* (y) Tn (x); A <A*,
n
G+(t; x y) = Sea"‘T*(y) Tn(x); A>An ■
(1)
These functions satisfy the equations
— + AG = 0, dG+- AG,= 0, dt dt +
with initial conditions
■(D).
G±(0; x, y) = 5 (x - y).
On a final remark, note that
S e an = J d xG- (t; x, x), S ean = J d xG+ (t; x, x)
(2)
(3)
(4)
And, therefore
'A ( s) =
—1— J dt ts 1J dDx G- (t; x, x), А < А , Г ( s) і J
(- 1)s *
v ’ J dt ts-1 J dDx G+ (t; x, x), А > Аn.
Г ( s)
9
n
0
10
Continuous spectrum (a free particle case)
S2
Let H = -A =----------with u = 1,..., D. Then (2) transforms into
Sr MdxM
^-(AD + A )G-= 0• (5)
The Fourier integral of G- has the form
(2-f
where nß=xß-yß. Solution of (5), satisfying (3) is
G_ (t; x, y) = jA-Lelk^(Ä-k>
(2k)
or, in other words,
G-(,;x-y) - i4r*e~im- (6)
Alternatively, one can deduce
d°k Jkßt]ß+( Ä-kß)t
g+ <ny)=itwe
However, this integral diverges, thus the function G+ (t; x, y) doesn't exist. Upon substitution of (6) into (1) and (4) we get
V +»
(s) = D D D2-i dt ts-1-p/2ekt, (7)
2d nD/2r(s) i W
where VD = i dDx = +» — is a total volume of the configuration space. In order to incorporate this quantity into our considerations and receive a meaningful answer we will invoke a usual trick: assume it to be of finite value now and will take a limit VD ^ » at the very end.
Let us assume that A < 0, i.e. that A = -p. Then
r rs) = VD r(s-D2)
2DnD/2^2s-D r(s) '
Now it is natural to introduce a function that will serve as a density of gA(s) and as such, will not depend on the configurational volume VD :
c(D)s) = d^A(s) = 1 r(s - D/2)
A dVD 2DnD/2^2s-D r(s) '
Finally, rewrite the determinant of H - A as:
det(H -A) = {e-gA'(0) fD, and one can introduce what we'll call the «geometric» Evans function
Generalized Evans function for continuous spectrum
Before we move further on let us show a couple of examples of the «geometric» Evans function:
The geometric Evans function sD(A) is defined for A < 0 and A<An.
Note, that the case X > Xn doesn't allow for a Green function (i.e. G+ doesn't exist). The case A = An cannot be analyzed since in this case the corresponding equation turns into an identity. Finally, when A = 0, s ^ 0 s ^ 0, the equation (7) diverges at the lower limit, hence ^-function doesn't exist either.
In conclusion: eD(A) is defined and non-zero only for: A< 0,A>An, and, therefore, the continuous spectrum exist at An > 0. To put it in other terms, sD(A) is defined and finite only when An e continuous spectrum (i. e., An > 0) and when A = An for any An (i. e. for A < 0).
Thus, eD(A) satisfies all the conditions specified in the statement of the problem.
1. Evans J. W. Nerve axon equations. Stability of the nerve impulse // Indiana Univ. Math. J. 1972/73. 22. P. 577— 593.
2. Leble S. B., Salle M.A., Yurov A. V. Darboux transforms for Davey — Stewartson equations and solitons in multidimensions // Inverse Problems. 1992. Vol. 8, № 2. P. 207 — 218.
3. Bordag M., Yurov A. Spontaneous symmetry breaking and reflectionless scattering data / / Physical Review D — Particles, Fields, Gravitation and Cosmology. 2003. Vol. 67, № 2.
4. Jones C. K. R. T. Stability of the traveling wave solution of the Fitzhugh — Na-gumo system // Transactions of the AMS. 1984. 286 (2). P. 431—469.
5. Alexander J., Gardner R., Jones C. A topological invariant arising in the stability analysis of traveling waves // J. Reine Angew. Math. 1990. 410. P. 167—212.
6. Pego R. L., Weinstein M. I. Eigenvalues, and instabilities of solitary waves // Philos. Trans. Roy. Soc. London Ser. A, 1992. 340 (1656). P. 47 — 94.
7. Gardner R.A., Zumbrun K. The gap lemma and geometric criteria for instability of viscous shock profiles // Comm. Pure Appl. Math. 1998. 51 (7). P. 797 — 855.
8. Sandstede B. Stability of travelling waves // Handbook of dynamical systems. Amsterdam, 2002. Vol. 2. P. 983—1055.
(we remind here, that A = -^2).
It is also rather helpful to keep in mind that
11
ln ¡л2 -w(s - D/2) + Y(s ) Г(s - D/2)
Г( )
References
12
9. Benzoni-Gavage S. Stability of semi-discrete shock profiles by means of an Evans function in infinite dimensions // J. Dynam. Differential Equations. 2002. 14(3). P. 613 - 674.
10. Coombes S., Owen M. R. Evans functions for integral neural field equations with Heaviside firing rate function // SIAM J. Appl. Dyn. Syst. 2004. 3 (4). P. 574 — 600.
11. Gesztesy F., Latushkin Y., Makarov K.A. Evans functions, Jost functions, and Fredholm determinants // Arch. Ration. Mech. Anal. 2007. 186 (3). P. 361—421.
12. Deng J., Nii S. An infinite-dimensional Evans function theory for elliptic boundary value problems // J. Differential Equations. 2008. 244 (4). P. 753 — 765.
Об авторе
Валериан Артемович Юров — асп., Колумбийский университет, Миссури, США, e-mail: vayt37@mail.missouri.edu
Author
Valerian Yurov — PhD Student, Columbia University, Missouri, USA, e-mail: vayt37@mail.missouri.edu
УДК 517.956
А. А. Юрова
ДИНАМИКА ЛОКАЛИЗОВАННОГО ИМПУЛЬСА, ОПИСЫВАЕМОГО УРАВНЕНИЕМ ДЭВИ - СТЮАРТСОНА II
Изучен класс точных локализованных решений уравнения Дэви — Стюартсона II типа; показано, что с течением времени такие решения теряют пространственную локализацию с характерным временным масштабом, совпадающим с характерным пространственным масштабом начальной локализации. Потеря локализации выражается в появлении резонансных пиков, число которых определяется характером поведения опорной функции на бесконечности. В частности, экспоненциально локализованные возмущения распадаются на бесконечное число резонансов.
A class of spatially localized solutions of Davey — Stewartson II equation is examined; it is shown that such solutions tend to lose the locality properties with time scale corresponding to a characteristic space scale of initial localization. The locality loss manifests itself with emergence of resonance spikes, whose total number is determined by the asymptotic behavior of support function on infinity. In particular, the exponentially localized perturbations split into an infinite number of the resonances.
Ключевые слова: уравнение Дэви — Стюартсона, преобразование Дарбу, солитоны, интегрируемые системы, пары Лакса.
Key words: Davey — Stewartson equation, Darboux transform, solitons, integrable systems, Lax pairs.
Вестник Балтийского государственного университета им. И. Канта. 2011. Вып. 5. С. 12 — 17.
