LAX REPRESENTATION DOES NOT MEAN COMPLETE ...

IC/91/274 INTERNAL REPORT (Limited Distribution)

' International Atomic Energy Agency

1 Introduction Our aim is to study the inlLomogeneous nonlinear Schrodinger equations (INLS) (see [l], [-]> ["'Ji ['']» [•'•Ji [*>] an< l *-n(1 references therein):

and Nations Educational Scientific and Cultural Organization n

' 1 _ r ' - INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS and their gauge equivalent inhomogeneous Heisenbcrg ferromagnet equations (IHF) iSt + \S,SII] = F(x,t). LAX REPRESENTATION DOES NOT MEAN COMPLETE INTECRABILITY

V.S. Gerdjikov *

(2)

The gauge equivalence between the homogeneous versions of (1) and (2) is well known, see [7| Our aim is to study the interrelations between those two classes of inhomogeneous nonlinear evolution equations (INLEE) and to establish the relation between G(x,t) and F(x,t) which ensures their gauge equivalence. Next we show that all INLS and IHF equations allow Lax representations with the operators L and L being chosen as the Zakharov - Shabat system:

International Centre for Theoretical Physics, Trieste, Italy. x, t) -

i , t, A) = 0,

(3)

and its gauge equivalent: ABSTRACT (4) The gauge equivalence between the inhomogeneous versions of the nonlinear Schrfldinger and Ihe Heisenberg fenromagnet equations is studied. An unexplicit criterion for integrability is proposed. It is shown ihat in the nonintegrable cases the M-operators in their Lax representations possess pole singularities lying on the spectrum of the L-operators.

MIRAMARE - TRIESTE September 1991

Permanent address: hisiitute for Nuclear Research and Nuclear Energy, 1784 Solia, Bulgaria.

igT

q{x,t)g(j,t} =

S[x,t) — g ±00. Just for brevity in the summations below by Yi -V* we mean Y, \X% + A'^j,

2

We. used above the following notations:

Expansions Over the Symplectic Bases

The symplectic bases for (3) and (4) are certain bilinear combinations of llioir J™t solutions \j>, and V', 4>\ which are introduced by: lim

31

=

lim

z

=

lim

P{x,i,X) = - ~ { / * + -p-ir)(T,t,X),

lf{x,l} = 2iC?

Q(x,t,X) =

(6)

These two pairs of Jost solutions are related by: rf>{x,t,X) = ( T ' V f s . M ) ,

4>{x,t,X) =

l

S-

(Tj,X)T-'(0). ±_

The spectral data of (3) and (4) are determined by the scattering matrices:

da±

(H)

where (7t = ^ («r, ±i>j) with ff* being the Pauli matrices, * J ( x , t ) = * * ( T , < , A*), ^ * e t c - a n d n 0 is the projector:

r((, A) = 0"'(M, *)#(*,*, A) = ( J* ~ =T(t,X)T~l(t,0).

(15)

In order to preserve the analyticity properties of the first and the second columns of the Jost solutions J/' and ^ we have to impose on q(x, t) an additional unexplir.it condition:

onto the off-diagonal part of X. Analogously any function G(x,i) which possesses Fourier transform and can be represented in the form: (16)

This condition (see, e.g. [7]) ensures also the equivalence of the NLS and HF equations as Hamiltonian systems in the sense that they have the same set of integrals of motion. Now we state one of the main results in [10], namely that any off-diagonal matrixvalued function (7(x, () which allows Fourier decomposition can be expanded also over the symplectic basis {/', Q] of (3):

G(x,t) = i I"

allows a decomposition over the symptectic basis {P(x,t, X),Q(x, t,X)} of (4) [8]:

(7(1,0 = ij°° dX(V{G;t,\)Q(x,l,X)-)C(G;t,X)P{x,t,\)) + i Z ( W i t)Qt[x, t) - £J(G; ()/'**(*, ()) •

(17)

t=i

dX(V(G;t,X)Q(^t,X)-)C[G;t,X)P(xlt1X))

The inversion formulae in this case are given by: (11)

f(G-t,x) = .:[P(-,U),G(.,i)I, (18)

The corresponding inversion formulae are given by: P{G\t,X) = i\P{;t,\),G{;i)\, K[G;t,X) = ifcK-,l,A),G(.,t)l.

Vt(G;t)=i[I £ f ( « ; 0 = i[

where [, J denotes the skew .scalar product: ,S the standard expression that gives rise to the NLS eq. (see e.g. [7]); Vo commutes with L and therefore has the generic form 2

+ Vo) =

..j.

I (KjA)-K(X)

K(K)~K(0)\

(58)

Our considerations can be generalized to a number of INLEE related to Lax operators L for which the completeness of the squared solutions is proved. These include some polynomial {in A) generalizations of the Zakharov-Shabat system [12], the difference version of (1) and its Range equivalent [I'A], the Zakharov Shabat systems related to the semisimple Lie algebras [14] and others. 10

ACKNOWLEDGMENTS This paper was finished during the author's visits at Salerno University and at ICTP, Trieste and reported at the NEEDS 91 Conference in Galiipoli, Italy. The author is deeply grateful to Professors C. Bertocci, M. Boiti, S. de Filippo, F. Pempinelli, A. Verjovsky and G. Vilasi for their warm hospitality during his stay in Italy. Fruitful discussions with Professors M. Ablowitz, A. Fokas, J. Leon, M. Salerno and G. Vilasi are appreciated. Financial support from INFN, Section of Salerno, is acknowledged. The author would also like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO lor hospitality at the International Centre for Theoretical Physics, Trieste.

References [1] D.J. Kaup and A. Newell, Proc. Roy. Soc, London 361 A (1978) 413. [2] J. Leon, Phys. Lett. A 144A (1990) 444. [3] A.S. Fokas and M.J. Ablowitz, Stud. Appl. Math. 80 (1989) 253. [4] V.S. Gerdjikov, M.I. Ivanov. Preprint of INRNE, Sofia (1990) (submitted to Inverse Problems). [5] V.K. Metnikov, Reports on NEEDS 90, Dubna, July 1990 and NEEDS 91, Galiipoli, June 1991. [6] A. Latifi and J.Leon, Phys Lett. A 152A (1991) 171. [7] L.D. Faddi-ev and L.A. Takhtadjan, Hamiltonian Approach to Soliton Theory, (Springer Verbig, Berlin 1986). [8] V.S. Gerdjikov and A.B. Yanovski. Commun. Math. Phys 103 (1986) 549. [9] V.S. Gerdjikov Lett. Math. Pkys. 6 (1982) 315. [1O| V.S. Gerdjikov and E.Kh. Khristov, Bulgarian J. Pkys7 (1980) 28, 119. (In Russian). [11] F. Calogero and C. Nucci, J. Math. Phys. 32 (1991) 72. [12] V.S. Gerdjikov and M.I.Ivanov, Bulgarian J. Phys. 10 (1983) 13, 130. (In Russian). |13] V.S. Gerdjikov, M.I.Ivanov and Y.S. Vaklev, Inverse Problems 2 (1986) 413. [H] V.S. Gerdjikov, Inverse Problems 2 (1986) 51.

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