VECTOR FORM OF KUNDU-ECKHAUS EQUATION AND ITS SIMPLEST SOLUTIONS Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 15. № 3 (2023). С. 151-166.

VECTOR FORM OF KUNDU-ECKHAUS EQUATION AND ITS SIMPLEST SOLUTIONS

A.O. SMIRNOV, A.A. CAPLIEVA

Abstract. Nowadays, new vector integrable models of nonlinear optics are actively investigated. This is motivated by a need to transmit more information per unit of time by using polarized waves. In our work we study one of such models and we construct an hierarchy of integrable vector nonlinear differential equations depending on the functional parameter r by using a monodromv matrix. The first equation of this hierarchy for r = a(piq) is a vector analogue of the Kundu-Eckhaus equation. As a = 0, the equations of this hierarchy turn into equations of the Manakov system hierarchy. Other values of the functional parameter r correspond to other integrable nonlinear equations. New elliptic solutions to the vector analogue of the Kundu-Eckhaus and Manakov system are presented. We also give an example of a two-gap solution of these equations in the form of a solitary wave. We show that there exist linear transformations of solutions to the vector integrable nonlinear equations into other solutions to the same equations. This statement is true for many vector integrable nonlinear equations. In particular, this is true for multicomponent derivative nonlinear Schrodinger equations and for the Kulish-Sklvanin equation. Therefore, the corresponding Baker-Akhiezer function can be constructed from a spectral curve only up to a linear transformation. In conclusion, we show that the spectral curves of the finite-gap solutions of the Manakov system and the Kundu-Eckhaus vector equation are trigonal curves whose genus is twice the number of phases of the finite-gap solution, that is, in the finite-gap solutions of the Manakov system and the vector analogue of the Kundu-Eckhaus equation, only half of the phases contain the variables t, z\,...,zn. The second half of the phases depends on the parameters of the solutions.

Keywords: Monodromv matrix, spectral curve, derivative nonlinear Schrodinger equation, vector integrable nonlinear equation.

Mathematics Subject Classification: 35Q51, 35Q55

1. Introduction

It is well known that the derived nonlinear Schrodinger equations [1], [2], [3], [4], [5], [6] have numerous applications in various fields of physics and mathematics. In this regard, studies of various types of solutions to these equations are constantly being carried out (see, for example, [7], [8], [9], [10], [11], [12]). At the same time, it should be noted that along with the Kaup-Newell [1] equation

i'Pz + Ptt + i(\p\2 p)t = 0, (1.1)

Chen-Lee-Liu equation [2]

i'Pz + Ptt + i \p\2Pt = 0, (1.2)

A.O. Smirnov, A.A. Caplieva, Vector form of Kundu-Eckhaus equation and its simplest

solutions.

© Smirnov A.O., Caplieva A.A. 2023.

The research was supported by the Russian Science Foundation (grant no. 22-11-00196), https: / / rscf.ru / project /22-11-00196/.

Submitted December 9, 2022.

and Gerdjikov-Ivanov equation [3], [4]

ipz + Ptt - ip2p*t + 1 \p\4p = 0, (1.3)

there exists the Kundu-Eekhaus equation [13], [14], [15], [16], [17]

ipz + Ptt - 2a \p\2p + a2 Ipl4p + 2iadt (\p\2) p = 0, a = ±1. (1.4)

Equation (1.4), as well as equations (1.1)—(1.3), contains first derivatives and also has numerous applications.

However, there is a significant difference between equations (1.1)—(1.3) and (1.4). The first three equations are consequences of compatibility conditions of Lax pairs with quadratic in spectral parameter Lax operators. Equation (1.4), in contrast to equations (1.1)—(1.3), is a result of the gauge transformation

p = , * = a /W2 M

of a solution p to the nonlinear Schrodinger equation

iPz + Ptt - 2a |p]2p = 0.

The ever-growing traffic in networks requires finding ways to increase bandwidth of optical fibers. Therefore, researchers are actively working on vector models of nonlinear optical waves propagation [18], [19], [20]. Many of these models have been known for a long time. One such model is the Manakov system [21], [22], [23], [24], [25], [26]

dzPi = id2pi - ^(M2 + \p2\2)pu ^

dzP2 = id2p2 - 2ia(\pi\2 + |p2|2)i>2.

We observe that vector nonlinear Schrodinger equations also have derivative forms, one of which is the equations [27], [28], [29], [30], [31]

idzPi = -d2pi - 2dt [(M2 + N2) Pi] ,

2 2 (L6)

idzP2 = -dtP2 - — dt[(M2 + N2) P2] .

In contrast to the above works, here we investigate a vector analogue of the Kundu-Eckhaus equation. We use a monodromv matrix to derive equations from the vector analogue of the Kundu-Eckhaus equation hierarchy and construct the simplest nontrivial solutions to first equation. We hope that the vector equation we have obtained, as well as the scalar one, will have numerous applications in physics and mathematics.

The work consists of Introduction, four sections, and concluding remarks. In Section 2 we define the Lax operator

i^t + U V = 0,

which depends on a functional parameter r E R, and investigate properties of corresponding monodromv matrix. Since the spectral curve equation is a characteristic equation of the monodromv matrix [32], it is not difficult to obtain properties of the spectral curves equations from properties of the monodromv matrix. As in the case of the Manakov system [26], the spectral curves equations are quite cumbersome and we do not provide them in this paper. We only note that, as it was shown in [26], a linear dependence of the functions pj leads to factorization of the spectral curve equation into separate components. Therefore, from our point of view, solutions with linearly independent pj are more interesting for studying and using in applications.

In Section 3 we derive stationary equations for multiphase solutions. These equations are analogs of the Novikov equations for the Korteweg-de Vries hierarchy. Also in this section, we define a hierarchy of the second operators of the Lax pair

№ zk + Wk tf = 0,

which depends on the functional parameter r^ dt= dZkr. The Lax pair compatibility conditions give an hierarchy of vector derivative nonlinear Schrodinger equations with an additional functional parameter far = dt<^ and = dZk As r = a(p*q), these equations are vector analogue of the Kundu-Eckhaus equation and its higher forms. Another choice of the functional parameter leads to other vector nonlinear equations. For 0 = 0 these equations turn into equations from the Manakov hierarchy [26], We note that an existence of a Lax pair makes it possible to use the Darboux transformation to construct new solutions to vector analogue of the Kundu-Eckhaus equation.

In Section 4, we construct one-phase solutions to vector analogue of the Kundu-Eckhaus equation. The first three solutions are expressed in terms of elliptic Jacobi functions, and for 0 = 0 they are new elliptic solutions to the Manakov system. Let us recall that elliptic solutions to the Manakov system obtained in [26] were expressed in terms of the Weierstrass functions. Here we construct solutions expressed in terms of the hyperbolic functions. In the end of the section we consider one-phase two-gap solutions. Despite the fact that in the last case the spectral curve has a genus equalling to 2, the corresponding solution is a traveling wave.

In Section 5 we show that there exist linear transformations of solutions to vector integrable nonlinear equations into other solutions to the same equations. Original and transformed solutions are associated with the same spectral curve, but they correspond to different Baker-Akhiezer functions. One of these Baker-Achiezer functions differs from the other by an orthogonal matrix factor. That is, the Baker-Achiezer function for considered vector nonlinear Schrodinger equation is determined up to an orthogonal transformation,

2. Monodromy matrix and its properties Let first equation of a Lax pair has the form

№t + U tf = 0, (2.1)

where

U = Uo + rJ, Uo = -XJ + Q,

>=3 (2 -;) ■ «=(-°qpo'>

p* = (p\,p2), q* = (qi,q2), I is identity matrix, r E R is a some function, and A is a spectral parameter.

Following [32], [26], we assume that there exists a monodromy matrix M such that the matrix function ^ = Mtf is also an eigenfunction of Lax operator (2.1). Then the matrix M satisfies the equation

iMt + UM - MU = 0. (2.2)

In the case of a finite-gap matrix potential Q, the monodromy matrix M is a polynomial in the spectral parameter A [32], [26]

n

M = £ m, (t)\j. (2.3)

j=o

Substituting (2.3) in (2.2) and simplifying, we get that the matrix M has the following structure

n— 1

M = K + £ ck Vn -k + CnUo + Jnt

k=1

where Vi = \UQ +

Vk+i = AVk + Vk%

K0

f—Tk H|\ \ Gk ^ y

fc > 1,

— Cn+1 — Cn+2 0 0 Jn I 0 Cn+1 Cn+3

0 cn+4 cn+2 )

Cj are some constants, Tk = TrFk.

It follows from equation (2.2) that the entries of the matrices Vk0 satisfy recurrence relations

Hi = idt p + rp, G1 = idtq — rq, Fk = — id- (Gk pt + qH

Hk+i = idtHk + rHk + [P'k + Ti) p, Gk+i = —idtGk + rGk — (Fk + TkI) q.

(2.4)

In particular, Fi = qpt, Ti

ptq

qtp,

H2 = — dt2p + 2irdtp + (2ptq + r2 + idtr) p, G2 = 5t2q + 2irdtq — (2ptq + r2 — idtr) q, F2 = 2 (qpt) r + i (q<V — 5tqpt) = qH — Gipt T2 = 2(ptq)r + ¿(q^tp — p^q) = Hiq — ptGi, H3 = — id3p — 3r5t2p + 3i (ptq + r2 + idtr) dtp

+ (3ï5tptq + r3 + 6rptq + 3irdtr — d2r) p, G3 = — id?q + 3rd?q + 3i (ptq + r2 — idtr) dtq

+ (3iptdtq — r3 — 6rptq + 3irdrr + 5t2r) q, F3 = 3(qpt)r2 + 3ï(q5tpt — 5tqpt)r + dt qdtpt — q^t2pt — 5t2qpt + 3(ptq)qpt

qH* — G2pt — GiHi — (ptq)qpt

T3 = 3(ptq)r2 + ^(q^p — ptdtq)r + 3(pt q)2 + dtpt dtq — pt d2q — qt d2tp = qtH2 — G2p — GiHi — (ptq)2.

For r = 0 all above equations turn into corresponding equations for the Manakov system [26].

3. Stationary and evolutionary equations Equation (2.2) implies the following stationary equations:

Hn+1 + ^ ck Hn+1-k + C1n p — 0

fc=i n

Gn+1 + ^^ CkGn+1-k — Cn q — 0

/ y ^kHn+1-k

k=1 (3.1)

k=1

where

q = I 2Cn+1 + Cn+2 Cn+3

\ Cn+4 Cn+1 + 2Cn+2/

All multiphase solutions to evolutionary integrable nonlinear equations are simultaneously solutions to some stationary equations.

In the case of reduction q = ap* (a = ±1), the identities

G* = -aHk, H* = -aGk, F* = F*, Tk E R

follow from recurrence relations (2.4). Therefore, the constants Cj in stationary equations (3.1) should satisfy the conditions ck E R (1 ^ k ^ n + 2), cn+4 = c*+3. Let a second operator of a Lax pair have the form

№ *fc + Wk tf = 0 (3.2)

where Wk = Vk + rk J. Then the compatibility condition of equations (2.1) and (3.2) as well as equation

idtWk - idZk U + UWk - WkU = 0 yield the evolutionary integrable nonlinear equations

idZkp = Hfc+i - rkp, idZkq = Gfc+i + rkq (3.3)

and an additional relation

dZk r = dtrk. (3.4)

It follows from (3.4) that there exists a function 0 such that

r = dt^, rk = dZk 4>. The first systems of integrable nonlinear equations from hierarchy (3.3) have the form

idZlp = -d2p + 2irdtp + (2p*q + r2 + idtr - r^ p,

2 / f 2 \ (3.5)

idZl q = dt q + 2irdtq - (2p q + r - idtr - nj q,

and

dZ2 p = - 53p + 3ird2p + 3 (p*q + r2 + idtr) dtp

+ (35ipiq - ir3 - 6irptq + 3rdtr + id2r + ir2) p, dZ2 q = -5t3q - 3ird2q + 3 (p*q + r2 - idtr) dtq + (3pi5iq + ir3 + 6irptq + 3rdrr - id2r - ir2) q.

Equation (3.5) is one of a vector derivative nonlinear Schrodinger equations and (3.6) is a vector modified Korteweg-de Vries equation. Both equations are parametrized by an arbitrary real function 0.

By (2.4) and (3.3) we get the identities

dZk Fi = dtFk+i and dZk Ti = dtTk+i.

Therefore, there exist functions $ and (f) such that

Fi = dt$, Fk+i = dZk $ and Ti = dt4>, Tk+1 = dZk

(3.6)

(3.8)

Therefore, if we put (f) = acj) or

r = aT\ and rk = aTk+1, (3.7)

then equations (3.3), (3.7) determine evolutionary integrable nonlinear equations from the hierarchy of a vector analogue of the Kundu-Eckhaus equation. In particular, for k = 1, or for

r = aT1 = a(p*q) and r1 = aT2 = 2a2(ptq)2 + ia(qtdtp — ptdtq),

equation (3.5) becomes

idZlp = —5t2p + 2ia(píq)5íp + (2p4q — a2(p4q)2 + 2iaptdtq) p,

idz1 q = d2tq + 2io;(piq)dtq — (2p*q — a2(p*q)2 — 2iadtptq) q.

As q = Sp*, S = diag(<71, u2), 7j = ±1, equations (3.8) transform into a vector analogue of the Kundu-Eckhaus equation. It is not difficult to see that for a = 0 equations (3.8) transform into Manakov system [26].

4. One-phase solutions 4.1. Solutions in elliptic Jacobi functions. For n =1 stationary equations have the form

H2 + C1H1 + CÍ p = 0, G2 + ci Gi — Ciq = 0 (4.1)

or (for c4 = c5 = 0 and r = aT1, r1 = aT2)

d?P1 = i( C1 + 2a(ptq))dtp 1 + (2 C2 + C3 + (2 + C1«)pí q + a2(p*q)2 + %adt(ptq))p 1, d?P2 = i( C1 + 2a(ptq))dtp 2 + (C2 + 2 C3 + (2 + da)pí q + a2(p*q)2 + iadt(ptq))p2, d^q1 = i (C1 + 2a(ptq))dt ^ + (2 C2 + C3 + (2 + da)piq + a2(pq)2 — xadt(ptq)) q1 dfa = — i (C1 + 2a(ptq))dt q2 + (C2 + 2 C3 + (2 + da)piq + a2(pq)2 — xadt(ptq)) q2 Replacing functions pj and qj by formulas

Pj = Pj^6, Qj = Qje~t&, dt6 = 1C1 + ap*q, we obtain the following identities

(4.2)

d?p1 = ( 2piq + 2C2 + C3 — 1 c2 ) P1,

5^2 = 2piq + C2 + 2C3 — cj) p2,

(4.3)

1

?

dt2p1 = ^2p*q + 2C2 + C3 - 1 C2) qi,

d]q2 = ^2piq + C2 + 2C3 - 1 c^ cfc.

It is easy to see that the functions pj and p are solutions of the same second order linear differential equations. Hence, their products Uj = pjp satisfy the corresponding Appel's equations ([33, Part II, Ch. 14, Ex. 10], [34])

di3u1 — (8u1 + 8u2 + 8 c2 + 4c3 — c21)dtu1 — 4 dt(u1 + u2)u1 = 0,

d?3U2 — (8U1 + 8U2 + 4 C2 + 8 C3 — c1)dtU2 — 4 dt(u1 + «2)^2 = 0. We denote u1 + u2 = u, u1 — u2 = v. In these notation, equations (4.4) become

d?u + (c2 — 6 c2 — 6 c3 — 12u) dtu = 2( c2 — c3)dtv, d^v + (ci — 6 C2 — 6 C3 — 8u)dtv = 2( C2 — C3 + 2v)dtu.

(4.4)

(4.5)

The simplest solutions of equations (4.5) can be obtained as v = (c3 - c2)/2. In this case, the function u satisfies the equation

d3u + (c2 - 6c2 - 6c3 - 12u)dtu = 0

or

d2u + (c2 - 6c2 - 6c3)u - 6u2 = c1, (4.6)

where c1 is an integration constant. Simplifying relation (4.6), we obtain the equation

(dtu)2 = 4u3 - (c2 - 6C2 - 6c3)u2 + 2ciu + c2, (4.7)

where c2 is a second integration constant. It is well known that solutions to equation (4.7) are elliptic functions or their degenerations.

It is easy to verify that one of the non-degenerate solutions to the equation (4.7) has the form

u = k2 sn2(i; k) + -2c\ - 1(C2 + C3) - 1(1 + k2), (4.8)

where sn(i; k) is an elliptic Jacobi function [35], [36], which satisies the equation

[sn'(i)]2 = (1 - sn2(i))(1 - k2 sn2(t)). The integration constant for solution (4.8) is equal to 1 1 3 2

Ci =24- ^2 + c3)ci + ^ + C3)2 - 3(1 - k2 + k4),

¿2 = - C1 + ^ + C3)C1 - ^ + C3)2C1 + ^i1 - k2 + k4^

1 2 4

+ ^2 + C3)3 - 22(c2 + C3)(1 - k2 + k4) - -(2 - 3k2 - 3k4 + 2k6).

Knowing functions u and v, we obtain functions u

1

K \ k2 2, ,, c1 c2 l + fc2 ui = -(u + v) = — sn (t; k) +-------,

1 2V ; 2 v ' ; 24 2 6 '

l , , k2 2, ,, cl c3 l + fc2 u2 = -(u — w) = — sn (i; k) +-------.

2 2V ; 2 v ' ; 24 2 6

Thus, functions g and g are solutions to the equations

<9t2gi = (2k2 sn2(i; k) - 2(l + k2) - -2c? + c^ qi,

5t2p2 = Î2fc2 sn2(i; k) - 2(l + fc2) - -2c2 + c^ P2.

(4.9)

(4.10)

Since functions g satisfy the same equations as g, and the Wronskian of these solutions are constant:

W [g,g ] = 2i Wi,

functions fij and g are equal to

g = /u] exp I —i Wj J —| , g = /u] exp | iWj J — (4.ll)

where uj are defined by formulas (4.9). Substituting expressions (4.ll) into equation (4.10) and simplifying, we get

W2 = 69-2(c? - 12ci+i - 4 - 4k2)(c? - 12Cj+i + 8 - 4k2)(c? - l2ci+i - 4 + 8k2). (4.12)

It follows from equation (4.12) that there are three cases when p = q and p1 = jp. If

1

1

C2 = T7i(4 + 8 — 4k2), C3 = -(c1 + 8 k2 — 4),

12

12

then

p1 = p1 = -—= dn(i; k), P2 = p2 = cn(i; fc).

In this case, the solution to equations (3.8) has the form

P1 = ip1(t — C1^1)e

ie

Q1 = —p1, P2 = ip2 (t — C1Z1)e

2 = — 2,

where

p1(T) = ^1|dn(T;fc), p2(T) = —2 cn(T;fc),

0=ft +(1 — f) ¿1 + a/ (fc2 sn2(i — C1Z1; fc) — ^^

dt.

The magnitudes of solutions (4.13) are shown on Figure 1. For

C2 = ^( c2 + 8 — 4 k2), C3 = ^( c2 — 4 k2 — 4),

we have

i , / , s ^ ^ k

p1 = p1 = ^i^;^ p2 = p2 = sn(i; k).

The corresponding solution to equations (3.8) reads as

P1 = ip1(i — C1Z1) e10, q1 P2 = p2(i — C1Z1) e 10, q2 = p2,

* 1,

where

p1(T ) = -1= dn(T ;fc), p2 (T ) = -fc= sn(T ;fc), 0 = yi + — fc2 — ^ z1 + a j ^fc2 sn2(i — c1z1; k) — 0 dt.

If

we have

C2 = ^( c2 + 8 fc2 — 4),

^3 — ( c2 — 4 k2 — 4),

i fc fc p1 = p1 = —=cn(t;k), p2 = p2 = -—^sn(t;k).

In this case, the solution to equations (3.8) has the form

where

P1 = i p1 (i — C1Z1) e10, q1 P2 = P2 (t — C1Z1) e 10, q2 = p2,

P1(T) = cn(T;fc), p2(T) = -fci sn(T;fc),

* 1,

* = +(fc2 — 1 — f) *+a/

fc sn (i — c1z1; k) —— ) dt

(4.13)

(4.14)

(4.15)

The dependency of solutions (4.13)-(4.15) on the variable z1 was found from equations (3.8).

The magnitude pi(t - c\Z\)

The magnitude p2(t - C\Z\)

Figure 1. The magnitudes of solutions (4.13) for k = 0.7, c\ = 1

4.2. Solutions in hyperbolic functions. Equation (4.7) is well-studied. In particular, it has the following solution

1

u — a2 tanh2(ai) + — ((— 6c2 — 6c3) — 8a2) Integration constants for a given function u are equal to

ci — 24 ((c2 — 6C2 — 6ca)2 — 16a4) ,

1

C2 = 432 (4 a2 + (c2 - 6C2 - 6C3)) (8 a2 - (c2 - 6C2 - 6C3)) .

- 6 02 - 6 C3)J ^ 8 a - (C1 - 6 L-2

In this case, functions pj and g satisfy the following equations:

_ / o„2

2

dtPi = ( 2a2 tanh2(a^ - 43" - 12 + ci+i ) Pi

i

and

u*

a2 1 PiQi = -¡r tanh2(ai) + ^T (ci - 8 a2 - 12Ci+i)

2 ------1 (a) + 24

We recall that the functions g also satisfy equations (4.16). Solving equations (4.16) with conditions (4.17), we get

g = kj + iatanh(ai))etfc?g = /p(kj - iatanh(ai))e —fc? 22

where

fc2 = ^ (c2 - 8a2 - 12Ci+i)

or

Ci+i = -2 (ci - 8 a2 - 12 k2) .

The corresponding solution to equations (3.8) has the form

(4.16)

(4.17)

(4.18)

Pi = ( ki + % a tanh[a(i - cizi)]) e% j, qj = p*, V2

9i = ^y + k^J t + mizi + a J ^atanh2(ai - acizi) + 1 + ^ ¿t,

where

mi = -2 a2 - - (ci + 2 fc )2 - k2 - k^ - a(h + fc2)( a2 + - kxk2 + ).

The magnitudes of solutions (4.19) are shown on Figure 2.

(4.19)

The magnitude \pi(t, zi)| The magnitude \p2(t, zi)|

Figure 2. The magnitudes of solutions (4.19) for a = 1, ki = 2, k2 = 3, ci = 1.

4.3. Two-gap one-phase solutions. If n = 1, v = const, and c3 = c2, then from (4.5) we have

v = ^^-r (utt — 6u2 + (ci — 6C2 — 6C3)u) + Ci, (4.20)

2( 2 — 3)

and

utttt + 2(c2 — 6C2 — 6C3 — 10u)utt — 10 (ut)2 + 40u3 — 12(c? — 6C2 — 6C3)u2

+ (ci — 12c2(C2 + C3) + 8 (4c2 + 10C2C3 + 4c3 — CiC2 + CiC3)) u + C2 = 0, (4.21)

where C1 and C2 are integration constants. We can rewrite equation (4.21) in the form

I2 + 2 A h + (A2 — 4 B2 + 8 CiB )u + C2 = 0,

where A = cf — 6(c3 + c2), B = c3 — c2,

I 2 = utttt — 20uutt — 10 (ut)2 + 40u3, h = utt — 6u2.

(4.22)

(4.23)

It follows from equations (4.22) and (4.23) that the function 2u(t) is a two-gap potential of the Schrodinger operator [37, 38]

^tt — 2wtp = E^. (4.24)

It is well known that linear independent solutions to equation (4.24) with two-gap potential 2u(t) can be written as

dt

' (4.25)

where

and

fa,2 = £i,2^expj±u(E) j

^ = E2 + ( 7i — u)E + 72 — 7iu — - ii

4

A

1

7i

72 = — (A2 — 4B2 + 8CiB .

i ^2 A d2

2" '2 16

Substituting (4.25) in (4.24) and simplifying, we obtain an equation for spectral curve of two-gap potential u(t):

u2 = E5 + 271 4 + (7l2 + 212)E3 ^27172 — 8C2 J E2 + C3E + C4, (4.26)

where

^yl 1 o 1 / r\2 5 4 3 2 1 ^ 2 1

6s =-ututtt - —utt + 7(71 - 5u)u + 7« - 71« + 72« + 0C2U + 72 - ö 8 16 4 4 8 8

C =~1u2ttt + 7i — 3u)utum — ^uu« + -1 (C2 + 1672u — 247iu2 + 40u3 + 2u2t) utt 64 8 8 32

1 7

+ ^ (27? — 272 — 107iu + 15u2) u? — 3u5 + -^u4 — (7? + 272)u3

+ "1(167i72 — 3C2)u2 + 17iC2u — 172C2. 16 8 8

By (4.20) we have

K 1 t f 1 A \ 1

ui =2(u + W) = — 4BJi + U — 4bJ u +2Ci,

K 1 T ( 1 A \ 1

u2 =2 — w) = 4B Ji + U + 4Bju — 2Ci.

Therefore, functions pj and q are solutions to the equations

5t2pi — 2upi = — 1 (A + 2B) pi,

(4.27)

ô2^ - 2up2 = -- (A - 2B) p2.

1

4

It follows from (4.24), (4.25), (4.27), and (4.28) that

(4.28)

jPi = £iiV^exp <j u(E) J ^ jp2 = £21 v^exp <| u(E) Î ^

, pi = ei2^expj -u(E) î ^

E=E1 I ^

, p2 = £22 exp j - u(E) î y

e=e2 i j ^

E=Ei

E=E2

where

£ii£i2 = 1/B, £2^22 = —1/B, Ei = —(A + 2B )/4, E2 = —(A — 2B)/4.

It is easy to see that the functions pj and q are bounded as Ej satisfies the conditions Re( v (Ej )) = 0.

Two-soliton potential of operator (4.24) has the form (b > a > 0)

(a2 — b2)(b2 — a2 + a2 cosh(26i) + b2 cosh(2ai)

u(t) ^-. (4.29)

2(6cosh(6i) cosh(ai) — asinh(6i) sinh(ai))2

Substituting (4.29) into (4.22), we get

A = —2(a2 + b2 ), Ci = — — 62)2 — B 2, C2 = 0. v ; ' i 2B 2

The spectral curve (4.26) of potential (4.29) is determined by the equation

J = E (E — a2)2(E — b2)2.

Calculating v2(Ej ), we have

u2(Ei) = 32(a2 + b2 — B) ((a2 — b2)2 — B2) ,

AE2) = 32(a2 + b2 + B) ((a2 — b2)2 — B2) . The conditions v2(Ej) ^ 0, j = 1, 2, imply B = ±(b2 — a2).

IfB = b2 - a2, then Ex = b2, E2 = a2

C2 = -2(c2 + 8b2 - 4a2), C3 = -2(c2 + 8a2 - 462),

and

ib^/b2 - a2 cosh(ai)

g g1 (6cosh( bt) cosh(at) - asinh(6i) sinh(at)), ^ ^1

zaV^2 - a2 sinh(6i)

(6cosh( bt) cosh(at) - asinh(6i) sinh(ai)), The corresponding solution to equations (3.8) has the form

pi(i i) = ¿pi (t -a Zl)e101 (Ml \ qi(t, z i) = -p *(t, z i), P2(t, *i) = ip2(t - cizi)eld2(t'zi\ q2(t, zi) = -p*(t, zi),

(4.30)

where

, s b^/W-a2 cosh(aT + ta)

pi( 1 )

P2(T )

and

(bcosh(bT + tb) cosh(aT + ta) - a sinh(6T + tb) sinh(aT + ta)) '

a\jb2 - a2 sinh(6T + tb) (bcosh(bT + tb) cosh(aT + ta) - a sinh(6T + tb) sinh(aT + ta)) '

9i(t, zi) = + ( 62 - 4 c?) zi -a J (p2(T ) + p2(T )) —t, 02(t, Zi) = + ( a2 - 4 c?) Zi -a J (p2(T ) + p2(T )) —t.

Here (ta, tb) is an initial two-dimensional phase. The magnitudes of solutions (4.13) are shown on Figure 3.

5. Orthogonal transformation

Since the matrix J has two equal diagonal entries, an orthogonal transformation of the vectors of solutions to equations (3.3) again gives solutions to these equations. To prove this statement, we consider the equation

i$t + U $ = 0, (5.l)

where

U = Uo + rJ, Uo = -\J + Q,

'l 0iN

$ = T$ T = ,

$ T$ T >0 TJ-

It follows from equations (2.l) and (5.l) that QT = TQ. Therefore, the identities

q = T q, p = (Ti)-1 p (5.2)

hold true. Thus, if the matrix T satisfies the condition

TSTf = 5, (5.3)

then the vectors p and q (p and q) are related as

q = S p*, q = S p*.

The identities

Gk = TGk, Hfc = Hfc, Fk = TFkT 1, Tk = Tk

The magnitude p1(t — c1z1)

The magnitude p2(t — c1z1 )

Figure 3. The magnitudes of solutions (4.30) for a = 3, b = 5, c? = 1, ta = 2, tb = 3.

follow from recurrence relations (2.4). Therefore, if the vectors p and q are solutions to evolutionary equations (3.3), then the vectors p and q are also solutions to the same equations. Thus, using formula (5.2) with matrix

Ti

( cos p sin p\ у— sinp cos pj

and solution (4.13), it is possible to construct new elliptic solutions to equation (3.8):

p1 = (cos p dn(i — c1z1; k) + k sin p cn(t — c1z1; k)) e%d, cf1 = —P1, 21

72

rp2 =--(sin p dn(i — c1z1; к) — к cos p cn(i — c1z1 ; к)) егв, ç2 = —P2.

(5.4)

We note that the vectors p and q satisfy the stationary equation (3.1) with a non-diagonal matrix

_ rpft rp — 1

n — 1 CnT .

For solution (5.4) the matrix C1 reads as

C = 1 (3(c3 + c2) — (c3 — c2) cos(2p) (C3 — C2) sin(2p) N

1 2\ (C3 — C2)sin(2p) 3(C3 + C2) + (C3 — c2)cos(2p)j .

Therefore, the constants c,- for transformed solutions are equal:

C2 = C3 + C2) — C3 — C2) cos(2p), C3 = 1(C3 + C2) + 2(C3 — C2) cos(2p), C4 = C5 = 2(C3 — C2) sin(2p). At the same time, since the monodromy matrices of the functions ^ and ^ are similar

M = TMT-1,

an equation of the same spectral curve corresponds to these solutions. Therefore, the Baker-Akhiezer function ^ can be constructed from a spectral curve only up to a linear transformation f.

6. Discussions and conclusions

In many works devoted to studying finite-gap solutions of the Manakov system (see, for example, [22], [39], [24], [40], [23], [41]), in contrast to our work, the following aspects were not taken into consideration. First, as we showed in [26], if the functions pj are linearly dependent, then the eigenfunctions of Lax operator (2.1) are determined on two separated spectral curves. Secondly, to the best of the authors' knowledge, other researchers have not previously taken into consideration orthogonal transformations of solutions preserving spectral curves.

And finally, as we have seen in examples discussed in Section 4 , the number of phases of the solution is less than the genus of the corresponding spectral curve. Indeed, it follows from equations (3.1) and (3.3) that the following relations hold:

n—1 / n—1 \

9Zn p = - °kdz«-k p - cndtp + i Wn + Ck^'n—k + C-n^'n p + i cn p,

k=1 V k=1 J

n- 1 n- 1

dzn q = - °kdzn-kq - Cndtp -i Irn + °krn-k + cnrn \ q - iCnq k=1 V k=1 J

Therefore, the solutions pj and qj, up to exponential multipliers, are n-phase functions (functions with n arguments):

Pj (t, Zl, . . . , Zn) Pj(^ CnZn, Z1 Cn—1 Zn, . . . , ^n-1 C1 Zn^C' J ( ),

Qj (t, Z■l, . . . , z n) qj(t Cn Zn, Z1 Cn—1 Zn, . . . , Zn—1 C1 Zn^C ' J ( ).

An equation for a spectral curve r = {(¡, A)} reads as

K(n, A) = det(il - M) = i3 + A(A)i + B(A) = 0, (6.1)

where

1 2 2n+2

A(A) = - 1A2n+2 - ^A2n+1 + ^ A3A2n+2—j,

3 3 j=2 2 2r 3n+3

B( A) = ^A3n+3 + A3n+2 + ^ BjA3n+3—j.

2 7 9

=2

If n < 3, then the discriminant of (6.1), as a polynomial of ¡, is

4n+4

A( A) = (Cn+1 - Cn+2)2A4n+4 + £ DjA4n+4—j. (6.2)

=1

Probably, identity (6.2) is also true for other values of n. It follows from equation (6.2) that when condition cn+1 = cn+2 is fulfilled, the curve r has (4n + 4) branching points. Using the Riemann-Hurwitz formula

M

q =--N + 1,

y 2 T '

where M is a number of branching points, N is a number of sheets of a covering, we get that the genus of the spectral curve r is equal

4 n + 4 g = —2--3 + 1 = 2 n.

Thus, to construct finite-gap solutions to the Manakov system or to the vector Kundu-Eckhaus equation, it is necessary to use trigonal curves, the genus of which is twice the number of phases of solutions. That is, in the finite-gap solutions of the Manakov system and the vector Kundu-Eckhaus equation, only half of the phases involves the variables , 1 , . . . , n. The second half of the phases depends on the parameters of the solutions.

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Aleksandr Olegovich Smirnov,

St.-Petersburg State University of Aerospace Instrumentation, Bolshaya Morskaya str., 67A, 1900000, St.-Petersburg, Russia E-mail: alsmir@guap.ru

Aleksandra Alekseevna Caplieva,

St.-Petersburg State University of Aerospace Instrumentation, Bolshaya Morskaya str., 67A, 1900000, St.-Petersburg, Russia E-mail: alex.caplieva@gmail.com