GENERALIZED INVARIANT MANIFOLDS FOR INTEGRABLE EQUATIONS AND THEIR APPLICATIONS Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 13. № 2 (2021). С. 141-157.

Dedicated to the memory of A.B. Shabat and R.I. Yamilov

GENERALIZED INVARIANT MANIFOLDS FOR INTEGRABLE EQUATIONS AND THEIR APPLICATIONS

I.T. HABIBULLIN, A.R. KHAKIMOVA, A.O. SMIRNOV

Abstract. In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature, the method of the differential constraints is well known as a tool for constructing particular solutions for the nonlinear partial differential equations. Its essence is in adding to a given nonlinear PDE, another much simpler, as a rule ordinary, differential equation, consistent with the given one. Then any solution of the ODE is a particular solution of the PDE as well. However the main problem is to find this consistent ODE. Our generalization is that we look for an ordinary differential equation that is consistent not with the nonlinear partial differential equation itself, but with its linearization. Such generalized invariant manifold is effectively sought. Moreover, it allows one to construct such important attributes of integrabilitv theory as Lax pairs and recursion operators for integrable nonlinear equations. In this paper, we show that they provide a way to construct particular solutions to the equation as well.

Keywords: invariant manifold, integrable system, recursion operator, Lax pair, algebro-geometric solutions, Dubrovin equations, spectral curves.

Mathematics Subject Classification: 35Q51, 35Q53, 35Q55

1. Introduction

In the article, a notion of the generalized invariant manifold for nonlinear integrable equation is discussed. Recently in our works [l]-[7] it was observed that the objects of such kind provide an effective tool for evaluating the Lax pairs and recursion operators.

The approach developed in [l]-[7] explains the essence of the Lax pair phenomenon. In fact, the Lax pair in 1 + 1 dimension is naturally (internally) derived from the nonlinear equation under consideration. First we find the linearization (Fréchet derivative) of the nonlinear equation. The linearized equation obviously includes the dynamical variables of the original equation as well, which are here considered as functional parameters. Now we find an ordinary differential equation consistent with the linearized equation, which also depends on the dynamical variables of the original equation. We call this ordinary differential equation a generalized invariant manifold. For a given equation, there are many such manifolds, including nonlinear ones. In order to evaluate the generalized invariant manifold, we use the consistency with the linearized equation that allows us to derive a system of differential (difference) equations that is highly overdetermined due to the presence of the independent parameters, which are the dynamical variables of the original nonlinear equation. In all of the examples discussed in [l]-[7] (KdV, Kaup-Kupershmidt equation, Krichever-Novikov equation, Volterra type lattices from Yamilov list, two equations of KdV type found by

I.T. Habibullin, A.R. Khakimova, A.O. Smirnov, Generalized invariant manifolds for

integrable equations and their applications.

© I.T. Habibullin, A.R. Khakimova, A.O. Smirnov. 2021.

The work of A.R. Khakimova is supported in part by Young Russian Mathematics award. The work of A.O. Smirnov is supported by the Ministry of Science and Higher Education of the Russian Federation, Grant Agreement No. FSRF-2020-0004.

Submitted March SO, 2021.

Svinolupov and Sokolov, Garifullin-Mikhailov-Yamilov non-autonomous lattice, sine-Gordon equation and several hyperbolic type equations, etc.) the corresponding overdetermined systems are effectively solved and the desired non-trivial manifolds are found. Trivial generalized invariant manifolds are constructed quite elementary by using the classical or higher symmetries, see examples in [5]. A manifold, which is consistent with the linearized equation if and only if the original nonlinear equation is satisfied, is called non-trivial. Actually, this condition means that a pair consisting of the linearized equation and the generalized invariant manifold defines a Lax pair. It is curious that usual Lax pairs do not belong to this class, but they can be derived from properly chosen nonlinear generalized invariant manifolds by suitable transformations. Note that new Lax pairs are of an independent interest. For instance, a generalized invariant manifold generated by a consistent pair of linear invariant manifolds is easily transformed into the recursion operator. It was shown in [7] at the example of the Volterra lattice that a nonlinear Lax pair can be used for constructing particular solutions of the nonlinear equation.

Let us briefly describe the content of the article. In the second section we recall the definition of the invariant manifold and generalized invariant manifold for the differential equations in partial derivatives. We explain how to look for the generalized invariant manifold and why it can be effectively found. We conjecture that each integrable equation admits a consistent pair of linear invariant manifolds and give examples supporting such conjecture. We assert, based on our previous work, that consistent pairs of linear invariant manifolds can be used to construct both recursion operators and Lax pairs. We illustrate the algorithm by the examples of NLS system and mKdV equation in Sections 3-5, The consistent pair of the linear generalized invariant manifolds usually can be reduced to nonlinear one of smaller order. In this form, the invariant manifold provides an efficient way to derive the Dubrovin equations, from which finite-gap solutions are obtained; on method of finite-gap integration see [8] [12], The description of the spectral curve, the derivation and study of the Dubrovin equations for the NLS equation are presented in Sections 3,1-3,3, The corresponding solutions of the generalized invariant manifolds and their relation with the Novikov equation are considered in Section 3,4, Examples of one-phase and two-phase solutions of the NLS equation are given in Section 3,5, Derivation of the Dubrovin equations for mKdV equation is presented in Section 4,

2. Invariant manifolds and their generalization

The concept of an invariant manifold is well known in the theory of partial differential equations. It forms the basis of the method of differential constraints, widely used to construct particular solutions of nonlinear equations. We recall briefly the main points of the method of the invariant manifolds using the example of equations of evolutionary type

d ^ u

ut = f (x,t,u,ux,uxx ,...,uk), Uj = —. (2,1)

An ordinary differential equation of the order r

ur = g(x,t,u,ux,uxx,... ,ur-i) (2,2)

is called an invariant manifold for the equation (2,1) if it is consistent with (2,1), or, in other words, if the following condition is obeyed:

Drxf - Dtg\(2 l) (2 2) = 0. (2.3)

Here Dx and Dt are operators of the total derivative with respect to x and to t.

It is clear that if a solution u(x,t) of equation (2,1) satisfies equation (2,2) for some moment t = t0, then it remains a solution of (2,2) at all values of time t. This is the invarianee of equation (2,2),

Obviously relation (2,3) defines a PDE for the desired function g. Sometimes this equation can be solved explicitly, although in the general case the problem of finding the function g is rather complicated.

The situation changes essentially if we look for an ordinary differential equation that is consistent not with the nonlinear equation (2,1) itself, but with its linearization

TT 9^TT + df TT + df TT + + df TT (9A\

Ut = u + -— Ux + --Uxx +-----+ -— Uk. (2,4)

OU OUx OUxx OUk

Let give rigorous definitions. We consider an ordinary differential equation of the form

Um — FU, Ux, Uxx,..., Um—i; u, ux, uxx,..., U'n), (2-5)

where U — U(x, t) is a sought function, while an arbitrary solution u — u(x,t) of the original equation (2,1) is interpreted in (2,5) as a functional parameter. In fact, the variables x, t, U, Ux, Uxx, ..., Um-\, u, uXl uXXj ..., un in (2,5) are regarded as independent.

Definition 2.1. Equation (2,5) determines a generalized invariant 'manifold if the relation

D™Ut - DtUm\^A)!(2.4)!(2.5) = 0 (2-6)

is satisfied identically for all values of the variables [uj}, x, t, U, Ux, ..., Um-\.

Here the variables ut, Ut as well as their derivatives with respect to x are expressed due to equations (2,1) and (2,4), the variables Um,Um+\,... are replaced by means of (2,5), To emphasize that the solution u(x, t) is arbitrary, we consider the variables u, uXl uxx,... as independent ones. By virtue of this assumption, the problem of finding the function F (x, t, U~, Ux, Uxx,..., — i; u, , ^xx,..., ^n ) numerous examples, can be effectively solved.

Linear generalized invariant manifolds, that is, those of the form

LU — 0,

where L is a linear differential operator

N

L — Oi(u, ux,uxx,.. .)Dl

i=0

are of a special interest.

Definition 2.2. Let equations L\U — 0 and L2U — 0 define linear generalized invariant manifolds for the equation (2,1). We call these two manifolds consistent if for all E C the linear combination

(ALi + vL2) U — 0

is a generalized invariant manifold for (2,1).

The following conjecture is supported by numerous examples, see [1]—[T].

Conjecture 2.1. Equation (2,1) is integrable if and only if it admits a pair of the consistent linear generalized invariant manifolds such that the quotient

R — L— iL2

is a pseudodifferential operator; in fact, it is the recursion operator for (2,1). Examples can be found below in Section 3,2 and at the end of Section 5,

3. Invariant manifolds for NLS equation

In this section we find an invariant manifold of the first order (the simplest nontrivial!) for the nonlinear Sehrodinger equation. It is determined by the system

i ut = uxx + 2u2v,

• (3-1)

ivt = — vxx — 2v u

under appropriate additional condition. Let us first find the linearized equation for the system by rule (2,4):

i Ut = Uxx + AuvU + 2u2V.

2 (3.2)

iVt = —Vxx — 2v U — 4uvV. V

According to Definition 2.1, the generalized invariant manifold is a system of the ordinary differential equations consistent with (3.2) for arbitrary solution u = u(x, t), v = v(x, t) of (3.1). We look for it in the form

Ux = f(U, V, u, v), Vx = g(U,V, u, v).

The eomptabilitv condition for the equations (3.2) and (3.3) gives an overdetermined system

invariant manifold given by a system of the form (for the details see Appendix below)

Ux = XU — 2uVC — UV,

,__(3.4)

K = —AV — 2v Vc —UV,

where A and C are arbitrary constants. Due to the obtained equations, linearized equation (3.2) converts into a system of the ordinary differential equations:

i Ut = (2uv + A2 )U — 2(ux + AuWC — UV,

,__(3.5)

i Vt = —(2uv + A2 )V + 2(vx — Av )VC — UV.

The following statement can be easily proved by straightforward computations.

Theorem 3.1. A pair of systems (3.4) and (3.5) is consistent if and only if the functions u

Therefore, the pair of equations (3.4) and (3.5) defines a Lax pair for the NLS equation. Unlike the usual Lax pair found by V.E. Zakharov and A.B. Shabat, this pair is nonlinear and contains two arbitrary constants, but with the help of a simple technique it is reduced to the usual one [13]. Indeed, by setting C = 0, U = p2, V = we reduce equations (3.4) and (3.5) to the form

= ^Ap — iuip,

and, respectively,

= —ivp — ^Aip,

Wt = (uv + 7)A2^j p — i(ux + Au)ip,

^t = i(Vx — Av— ^uv + 2a2^J ^.

3.1. Invariant manifolds and spectral curves. Let us show that the found nonlinear Lax pair is of an independent interest since it provides opportunities for building particular solutions to the NLS equation. We change the variables in the nonlinear Lax pair as U = u$, V = wV and this casts the pair into the form

— $ + $x — A$ = — 2VC—$Vwv,

v (3.6)

—V + Vx + AV = —2^JC — $Vuv

and

i—$ + i$t = (2uv + A2)$ — 2( — + A) VC — $Vuv, u \u J (3 7)

i-V + iVt = —(2uv + A2)V + 2 (— — A) VC — $Vuv.

C A C A

coefficients:

- 2N+2 i

C = - n (A — Afc) = ±"2(A). (3.8)

k=l

We see solutions to the nonlinear Lax equations in the form

N N

$ = n( A — ), V = — n(A — f3k). (3.9)

k=l k=l

We note that identity (3.8) defines the equation for the spectral hvperelliptie curve of the N-gap solution of the NLS equation, see [10] [12],

We substitute representations (3,8) and (3,9) into system (3,6) and compare the coefficients AN

N 2 N+2

£ = — + 2 A* ■

k=1 k=1 , ,

N 2 N+2

| = — -Z A,

fc=l fc=l

We substitute polynomials (3,9) into system (3,6), take A = 7j in the first equation and A = f in the second. Then we get the well known Dubrovin formulae [8]

7' =_^_, f =__»rn_, (3.11)

7 Uk=3(73 —7k), f Uk=3(f3 — fk), 1 J

where 7j = ^¡¿i f'j = By applying the same manipulations to (3,7) we obtain

. , (— Efc= 7k + ^ Efc=+2 Ak) v(73) % 7 • = --

, ^(7i — 7fc) x ' (3.12)

.. _ (— Efc= fk + 2 ES^) Kf) lfffj = Uk=3(f3 — fk) ,

where 7j = fij = In order to get the focusing NLS equation

iut = uxx + 2 \u\2u,

to system (3,1) we add a constraint of the form v = m, where the bar over a letter means the complex conjugation. Then solution ($, to the nonlinear Lax pair (3,6), (3,7) can be chosen in such a way

$ (—A) = (—1)N+1^(X). Function C(X) and parameters Xj, fij, jj satisfy the involution

C(X) = C (—X) , Xj = —Xj, ft = — Aj.

Evolution of jj in x and t is determined by a pair of the systems of ordinary differential equations

jj n ) ), (3-13)

(—Y,k=j ik + i Efc=+2 Xk) u(jj)

Hj = A-~~rr 2-^—-. 3.14)

j rik=j(lj — lk)

Thus, we arrive at systems of ODE describing the well-known algebro-geometric solutions for the NLS equations, see [10].

Theorem 3.2. A pair of systems (3,13) and (3,14) is consistent.

Proof. Let us show that a pair of systems (3,13), (3,14) is consistent. To do this, we differentiate

x

.d_ , ') — (.j) _ u(jj) iv(jj) | Uk=(ij — ik)

dt ' dx Uk=j(7j - Ik) - Ik)2

-Y( 7' )_'M_

% (%) nk=(7, -7k)

12N+\ ^ £ K7-) (3-15)

- I -

1 2N+2 \

S 7k + 2 S Xk) k=j k=l /

Ilk=i(7j - 7k)

( ^ 1 vM

-5> + 2 EM

V k=j k=l J

+ | -> 7k + 1 ^ > * n*' (7' -7k)

2 k=i j Ylk=, (1, - 7k):

We find the derivatives

ljtu (7 ), ix)' lJt n(7'- 7k), ix n(7 - 7k)

k= k=

separately. We have:

2 N+2

dt y ,3> 2 f-^ 7, - Ak

k

= 7j - Ak

2 N+2 2 2 N+2

1 f V^ 1 V^ \ y2\ii) v^ 1

2 r U'kk+ 2 hAkJ Uk= (7,-7k) £

and

d ( ) 7^) 2N+2 1 1 1/2 (7j) 2N+2 1

dx 2 7j -Ak 2Uk=1(7j - 7k) 7j - Ak

k=i u 1 lk=j v IJ IV k=l

In the same way we find:

ne» - - *) E '^r

a k=J k=' s=' 7' 7s

1 2N+2 \ 1

Ys^k + g E AM ^^) E

k=' k=1 / s=' " ,s

+ n 7 - -T* ) E (E e-vrrL(-)

J (7' - ^s)Uk=s (7s - Ik)

2 N+2

Ak n (73 - 7k) £ "(7s)

2 " ' (7 - ^ rik=,(7 - 7/k)

and

d n(7' -7k)= n(7' - 7k) 7

dxHy" 7k J~H( 73 - 7k ) 7--7

k= k= =

='/(7i} 5 — n <7— *' £ .

We substitute the obtained identities into (3,15) and after simple transformations we arrive at

4 (7') - U1,)- )

dty'J dx Uk='(7' -7k)

"(ls)

s= k=

( ' - s) k= ( s - k)

, V^ v(7s) , y^ K7s)

(3.16)

Uk=s(7s — 7k) (7j — 7,)Uk=s(7, — 7k)

We observe that the first two terms in the brackets can be simplified as

— y^ /V^ \_^i^s)_+ y^ u(7s)

7=1 \I=s J (7i — 7s) Uk=s(7s — 7k) nfc=s(7s — 7k)

= S i73 — 7s — Vs7)

Then, taking into consideration the obtained relation, we see that identity (3.16) becomes

i—r (7') — -f- (i7) = 0. dt v h) dx '3>

This completes the proof. □

3.2. Consistent pair of linear invariant manifolds. Here we present an example supporting Conjecture 2.1. From nonlinear invariant manifold (3.4) we derive a consistent pair of linear invariant manifolds for the system (3.1). We differentiate both equations in (3.4) with

x

UXV + VXU

Uxx = XUX — 2uX\/C — UV + u X

Vxx = —XVx — 2v xVC — UV +

vc — uv '

UxV + VxU

vC — uv

and then exclude irrationalities in the obtained equations due to relations:

UxV + VxU

V c — uv

= —2(u V + v U),

VC—UV = XU—U = -XV-VX..

— 2 u 2

As a result, we obtain a linear relation

L2W = XLlW, (3.17)

where W = (U, V)T and the operators are as follows

L = (Dx — ^ 0 \

Ll V 0 M —Dx + v-f)

and

T fD2x — vi:Dx + 2m; 2u2

L2 = ^ 2v2 Dx — n*Dx + 2uv

We confirm straightforwardly that the linear constraint defined by equation (3.17) is consistent with linearized equation (3.2), i.e., it defines a linear generalized invariant manifold with a X

L-lL2 coincides with the recursion operator for NLS system (3.1)

(Dx + 2uD~lv 2uD-lu \ R =\ -2vD-lv —Dx — 2vD-lu) '

3.3. Integrals of systems. The overdetermined system of equations (3.13), (3.14) admits integrals of the form

N m(xt , dj

E/ ^ =0, k = 0,1,...,^—3' (3-18)

j=lJlj (0,0)

^ J-2 J7 —[Jl! (0,0) u(l)

N n*(xt) n7

A n(xt) d7 1 2N+2

E/ JN-l J7L)=x + ? EXk (3.20)

(0.0) v(7) 2 t!

that are derived directly from the systems by using some elementary manipulations and subsequent integration.

3.4. Novikov equation. We express coefficients of the polynomial P = ($, ^)T in terms of ( u, )

(3.17) in a convenient form

RP = XP, (3.21)

where

R=(0 0Mu 0)

We introduce notations for the coefficients of the polynomial P

P = (_\)AN + )AN-i + ... + ^) . (3,2)

By virtue of the above expansion, equation (3,21) gives rise to

1 N 1 N 1 N

«I i^r + ^-jr-1 + ... + v^

1 \ \N +1 * (r A \N , , ( rN

An +1 + 'M An + ... + 1 A.

1 + ........

-1) \sij \Sn

Comparing the coefficients at XN+i in (3,1), we find

'0\ ( 1

CO=(-) ■

Rl or

We recall that the operator R involves an integration. It is easy to confirm that equation (3,23)

is satisfied for an appropriate choice of the constants in the integration,

AN

that implies

/VA (7 {ux + ciu)

V S J UK -ClA

By continuing this process we find for k > 1:

7 (9k + ciQk-l + ■■■ + Cku)

(r>A = ( U (9k + C1 Qk-1 + ... + Cku) \ V s k) = V1 ( hk + C1hk-1 + ... + Ck (-v)) J

where q, i = 1, k are arbitrary constants, the vector (gj, hj) coincides with the generator of the homogeneous symmetry of the order k

uTj = 9j, vTj = hj

of NLS system (3,1), Finally, comparing the coefficients at X0, we find

tn+1 = 0, s n+I = 0

that actually coincides with the Novikov equation

9n + ciqn-i + ... + cnu = 0, hN + cihN-i + ... + cn (-v) = 0.

3.5. Examples. Below we present two examples illustrating the use of the Dubrovin equations (3.13), (3.14) by taking N = 1 Mid N = 2.

Example 1. In the particular case when N = 1 and

u(l) = (i - Xi)(i - X2) with Xi = r] + i£, X2 = -r] + we get a system of consistent equations for determining the unknown 7 = j(x, t):

i = (7 - Xi)(7 - X2), H= 2Î(7 - Xi)(7 - X2").

It can be solved easily:

7 = ??tanh(2£ i]t + r]x + s 0) + i

In order to find u = u(x, t), we first solve the equation

^ = -7 + 2i e u

Integration of this equation yields

g i£xA(t)

u

cosh(2^rjt + r]x + s o)

We substitute the obtained ansatz into the NLS equation iut = uxx + 2 |u|2u and find

A(t) = rie^2. Then finally we get the well-known soliton solution

u( x, )

i(Zx+(v2-Z2 )t+Lpo)

cosh(2i; l]t + T]X + s 0)

Example 2. Let us take N = 2 and assume that the hvperelliptie curve is as follows

u2(X) = (X2 - 4)3.

In order to find the functions 71 (x, t) and j2(x, t), we use the integrals (3,18)-(3,20), which in this case take the form:

2 ,7j (x,t) dl ^

^ i -- = ^

j=1 J (o,o)

m Q dj 1

4

V 7—t~\ =x + -ty Xk. (0,0) 7^ 2 t! k

These integrals are evaluated in a closed form and generate a system of algebraic equations for A^x, t), j2(x, t), which is easily solved and gives rise to

(xt) = _2X(- - lT) + (--V(lT -X - -)(iT + X - -)R h(x, V 2 (X2 + T2 + -)(X2 + T2 + 4%T - 3) , 3 2,

T = 4 X = 2 x

R = \J(T2 + 2%T + X2 + 3)(X - iT + -)(X + iT - -).

u( x, )

u(x, t)= (- - v42(\ - PI \ ) Z-2U.

X2 + T2 +

The obtained solution obviously coincides with the well known two-phase Peregrine soliton [14],

4. Invariant manifolds for mKdV equation: complex-valued case

A complex-valued version of the modified KdV equations

uT + uxxx + 6|u|2 ux = 0 (4,1)

has important physical applications, see [15],

The equation is obtained by imposing an involution of the form v = u, where the bar over the letter means complex conjugation, in the system of equations

uT + uxxx + 6uvux = 0,

VT + Vxxx + 6uvvx = 0. i^-ty

Linearization of (4,2) leads us to the system

UT + Uxxx + 6uvUx + 6v uxU + 6uuxV = 0, VT + Vxxx + 6uvVx + 6v uxU + 6uuxV = 0.

It should be stressed that generalized invariant manifold (3,4) found in the previous section for the NLS equation, is also a generalized invariant manifold for mKdV equation (4,2), This is not surprising since the systems (3,1) and (4,2) mutually commute. It can be easily confirmed that system

Ux = XU - 2uVC—UV,

x ,_- (4.4)

Vx = -XV - 2vVo -UV

is consistent with (4,3) for each solution (u, v) of (4,2), Due to (4,4), linearized equation (4,3) is reduced to the form

Ut = 2 Uxx + 2u2v + Xux + X2u) V-UV + C - (2uVx - 2vux - 2Xuv - X3) U,

__(4,5)

VT = 2 (vxx + 2uv2 - Xvx + X2v) V-UV + C - (2uvx - 2vux - 2Xuv - X3) V.

It is easy to confirm that (4,4), (4,5) define a nonlinear Lax pair for system (4,2),

Since systems (4,4) and (4,5) are very similar to those studied in the previous section (see (3,4), (3,5)), we investigate them in the same way. First we change the variables as U = u$, V = and get

—$ + $x -X$ = -2VC - $Vuv,

- X

U

—V + + XV = -2VC -

(4.6)

and

v

—$ + $T = 2( — + 2uv + X— + X2) VC - - (2uvx - 2vux - 2Xuv - X3) u V u u / v '

—V + ^t = 2( — + 2uv -X — + X2) VC - &Vuv - (2uvx - 2vux - 2Xuv - X3) V. We use the same spectral curve:

-, 2 N+2

1 "IT ,, , s 1 2,

(4.7)

C = 4 II (X - X*) = -1/2(X) (4.8)

k=l

and seek solutions to the nonlinear Lax equations in the same form:

N N

& = n(X - 7k), v = - n(X -f3k). (4.9)

k=l k=l

We substitute representations (4,8) and (4,9) into system (4,6) and by comparing the coefficients at XN, we derive trace formulae (3,10), The next step is to substitute polynomials (4,9) into system (4,6), Then we set X = 7j in the first equation and X = [j in the second and get the system of ordinary differential equations defining the dynamics of the roots on x: [8]

7' =__, [' =__^_, (4.10)

lj Uk=j(ij -ik), [j nk=j(Pj -3k), 1 j

where ij = %% =

Let us derive equations describing the time evolution of the functions j(x, t), [(x, t). We substitute explicit representations of the functions V and C into (4.7) and set X = 7j in the

first equation and À = fa in the second:

' uxx 0 \ux \2\ u(ij )

i = + 2uv + À^ + À2) V u u J

u u y Uk=jiij -ikY

fa = - ( ^ + 2uv -À ^ + À2)

* v JYlk=3(fr -Pk)

fa = ¡j- To get a closed system ft — and — due to (3.10). For the term uv we deduce two equations:

V ' V U v ' ^

where we used notations ij = ^r, fa = To get a closed system for ij, fa, we exclude

2 N+2 N 1 /2N+2 \ 2

2Y,i'k + EÀkEik-2Y,ikis-2E(ik)2 + 4( EÀA -E

k=j k=l k=j k=s k=l \ k=l / k=s

2 N+2 N 2 N+2 2

-2 E Pk + E Àk - 2 £ Pk fa - 2 £ (fa ) + 4[ £ Àk - £ .

k=j k=l k=j k=s k=l \ k=l ) k=s

4uv = -2^fak + ^ Àk^fa - 2^fafa - 2^(fa)2 + Àk I -^ÀkÀs

k=j k=l k=j k=s k=l \ k=l ) k=s

by comparing coefficients at the power

ÀN+l

in (4.7). As a result, system (4.11) converts into a system of equations describing the time evolution of the roots

1 2 N +2 3 ¡2 N +2 \ 2 1 \ ^( )

v = I-2 EÀkEik + E ikis + 3\ EM -2EÀkÀsU -ik)

k= - k

2

Pj = - I^>,Àkyjk - v fafa - ^ ( y'Àk] 'ÀkÀs

(1 2N+2 3 /2N+2 \ 2 1 \

-2 E ÀkEik + E ikis + 3 E M -2EÀkÀs)

k=l k=j k=s=j \ k=l / k=s J

(-, 2N+2 (2N+2 \ 2 \

2 E Àk YjPk - £ №. - 3 £ àJ +1 £Àk.Às)

k=l k=j k=s=j \ k=l / k=s J

It can be proved by a direct computation that systems (4.10) and (4.12) mutually commute.

In what follows, we impose reductions of two types on system (4.1). Complex reduction v = u in (4.1) is related to the involution

$ (-X) = (--)N+1^(X)

of the eigenfunetions, that generates conditions for the zeros = -fa and Xk = -Xj of the polynomials C such that

N +1

C(X)= n (X2 - 2XReX3 + X|2) . 3=1

In the case of the reduction v = u we obtain

$(-X) = (--)N+1^( X)

and = - fa, Xk = -Xj, such that

N +1

\2 \ 2 ^

k

c (à) = ii (à2 - àk). =l

It is obvious that both reductions are consistent with dynamics (4.10), (4.12). By using the found solution {Aj }NN=1 of the Dubrovin equations, one can find solution u of equation (4.1) due to formula (3.10). Solutions of such kind were earlier studied in [16].

The overdetermined system of equations (4,10), (4,12) admits integrals of the form

V = 0, k = 0,1,...,N - 4, (4.13)

f^J7j (0,0) Hl)

^ pi (X'T) rlry

™ (X'T) n-2 dl ( 1

T 1N-24L: = \ ~Y Xk) T, (4.15)

(0,0) 7 ^) V2 kk )

N

™(x>t) N_ldl 13(2N+\X i

R ^ - ^=x +1H 1>J- 21^) - ^

we find that

Pi (x,t,r) 7

y = 0, k =0, 1,...,N - 4,

j^Jj, (0,0,0) Hi)

nj (x,t,t) dry

V 1N-3— = T, ^ I ..... 1 u(j) ',

i=\Jli (0,0,0)

f n ^) N-2 dr /1

y.^ ^=t+\2 IkV

^ n(x,**) N, d-y (i2N+2, \ 13 f2N+\ V

E/ ^ ^ = x + UEMt+13 EaH -^xkxs 000) \2 uv £i J 2 U

(0,0,0) v(l) V tl J \8\ ^ ' 2

We note that formulae (3,18)-(3,20) and (4,13)-(4,16) can be easily generalized to solutions which simultaneously satisfy several equations from the AKNS hierarchy. In this case we obtain the following representation for the integrals

N pi (io,ii,i2,...) fa

E/ iN-fc=tk + E, k ^ i,

j^Jn (00,.) ix

where t\ = x, t2 = t, t3 = t and tj, for j > 3 correspond to higher symmetries. Here Ck,j are constant coefficients,

5. Invariant manifolds for mKdV equation: real case

In this section we seek for the generalized invariant manifold for the mKdV equation which can be obtained from (4,2) by imposing the real involution u = v:

Ut = uxxx + 6u2ux (5-1)

and then show how to derive from it the Lax pair, recursion operator and a consistent pair of the linear invariant manifolds. First, we linearize equation (5,1):

U = Uxxx + 6u2Ux + 12uuxU. (5,2)

Since equation (5,1) is reduced to the equation with cubic nonlinearity

Wt = wxxx + 2w3

by substitution u = wx the linearizations of these two equations are related by a similar replacement. Indeed, if we put U = Wx, then we arrive at the equation

Wt = Wxxx + 6u2Wx (5.3)

that is much simpler than (5.2). Hence, it is more convenient to work with this one. Below we will seek an ODE of the form

= F(Wx, W, u),

u( x, )

the computations, we present only the answer

Wxx = 2u^-W2TXW2TC + XW, (5.4)

X C turns into the form

Wt = (X + 2u2) Wx + 2uxy/-Wi+XW2TC. (5.5)

It worth mentioning that the found nonlinear equations provide a Lax pair for (5.1), namely, the following theorem holds.

Theorem 5.1. A pair of equations (5.4) and (5.5) are consistent if and only if the function

u

Remark 5.1. We note that generalized invariant manifolds (4.4) and (5.4) are related by the following change of the variables. We let X = in (5.4) and introduce U, V in such a way

U = WX -£W, V = WX +

Then we get

Ux = £U - 2uVC - UV, Vx = -£V - 2v yJC -UV.

C = 0

Lax pair of equation (5.1). We change the variables in the following way

W = 2ip^, Wx = VX (p2 + ^2) then in the new variables equation (5.4) becomes a system of linear equations

px = -i up + 2 VXip,

= - Vx<p + iu'0

and similarly, (5.5) turns into

< = — (uxx + 2u3 + uA) < + — -\/\ (2iux + 2u2 + A) 0, 0t = ——/A (2iux — 2u2 — A) << + i (uxx + 2u3 + uA) '0.

In order to bring it to a standard form, we make a replacement

•=C —•

where $ = ( (,^)T and $ = ((p,rip)T. Then we get

(x = - iwtp,

= -iu(p -

and

(t = (4Ç3 + 2u2£) (p - i (4<2 + 2uxC + Uxx + 2u2) ■p, p = -i (4<2 - 2ux£ + uxx + 2u2) (p - (4£3 + 2u2f) tp,

X = 4 2

We get rid of irrationality in identity (5,4) by squaring and then rewrite the result as

^ - + X2W + 4W2 - 4XW2 - 4C = 0.

u2 u2 u2

x

to be linear

Wxxx- —Wxx + 4u2Wx = X(Wx- —W

x x x - x x x x

u u

Actually, it defines a linear invariant manifold, consistent with the equation (5,3), Let us rewrite it in the form

L2W = ÀLlW, (5.6)

where

Ll = Dx - —, L2 = D3 - —Dxx + 4u2Dx. u u

Since U = Wx from (5.6) we get a relation

L2D-1U = XLID~1U

U

recursion operator for (5.1):

R = DxL-1LxD-1 = D2x + 4u2 + 4uxD-1u. (5.7)

6. Appendix

Let us show how equations (3.4) are constructed. The aforementioned consistency condition for equations (3.2) and (3.3) is written as

8 „ „ o^x .8

— (Uxx + 4uvU + 2u2V) - i— (f(U, V, u, v))

= 0,

(3.1),(3.2),(3.3)

8 8 — (-Vxx - 2v 2U - 4uvV) - i - (g(U, V, u, v))

(6.1)

= 0.

(3.1),(3.2),(3.3)

u x U V

variables ut, vt, Ut, Vt, D^U, D^V, where k > — in identities (6.1), will be excluded by virtue of equations (3.1),(3.2) and (3.3). Then we get an overdetermined system of equations:

F (U, V,u,v,ux, vx, vxx) = 0, G(U, V,u,v,ux, vx,uxx) = 0.

In the obtained relations, we equate the coefficients at independent variables ux x ux x x and step by step we determine the form of the sought functions f(U, V, u, v) and g(U, V, u, v).

u ux x

x x ux x

f( U,V,u, v) = fi(U,V )u + f2(U,V),

g( U,V,u, v) = gi(U, V )v + g2(U,V).

Note that the dependence of functions f(U, V, u, v) and g(U, V, u, v) on the variables u and v is defined, so we can equate the coefficients at these variables,

ux x u

mine the form of functions fi(U, V), f2(U, V) and gi(U, V), g2(U, V):

fi(U, V) = y/-4UV + a, f2(U, V) = c2U + C3, gi{U, V) = \J-4UV + C4, 92(U, V) = c5V + eg.

By considering the remaining equations, we obtain a relationship between the constant parameters q, i = 1,6:

C4 = CU C5 = - C2, C3 = Cg =

Thus, the functions f(U,V,u, v) and g(U,V,u, v) read as

f(U, V, u, v) = AU - 2uVC - UV, g(U, V, u, v) = -AV - 2vy/C - UV,

where A = c2, C = jC\.

Conclusion

In the article we have discussed the notion of the generalized invariant manifold for the nonlinear partial differential equations. We have found such manifolds for the nonlinear Scrodinger equation and the modified Korteweg-de Vries equation. We illustrated that this object provides an effective tool for constructing the recursion operator and the Lax pair. We have shown that the well-known Dubrovin equation, which is an important ingredient in the finite-gap integration method, can be easily derived from a suitably selected generalized invariant manifold,

REFERENCES

1. I.T. Habibullin, A.R. Khakimova, M.N. Poptsova. On a method for constructing the Lax pairs for nonlinear integrable equations //J- Phvs. A: Math. Theor. 49:3, 35 pp. (2016).

2. I.T. Habibullin, A.R. Khakimova. Invariant manifolds and Lax pairs for integrable nonlinear chains 11 Theor. Math. Phvs. 191:3, 793-810 (2017).

3. I.T. Habibullin, A.R. Khakimova. On the recursion operators for integrable equations //J- Phvs. A: Math. Theor. 51:42, 22 pp. (2018).

4. I.T. Habibullin, A.R. Khakimova. A direct algorithm for constructing recursion operators and Lax pairs for integrable models // Theor. Math. Phvs. 196:2, 1200-1216 (2018).

5. A.R. Khakimova. On desription of generalized invariant manifolds for nonlinear equations // Ufa Math. J., 10:3, 106-116 (2018).

6. E.V. Pavlova, I.T. Habibullin, A.R. Khakimova. On One Integrable Discrete System //J. Math. Sci. 241:4, 409-422 (2019).

7. I.T. Habibullin, A.R. Khakimova. Invariant manifolds and separation of the variables for integrable chains 11 J. Phvs. A: Math. Theor. 53:39, 25 pp. (2020).

8. B.A. Dubrovin, V.B. Matveev, S.P. Novikov. Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties // Russian Math. Surveys 31:1, 59-146 (1976).

9. I.M. Krichever. Integration of nonlinear equations by methods of algebraic geometry // Funct. Anal. Appl. 11:1, 12-26 (1977).

10. V. Kotlvarov, A. Its. Periodic problem for the nonlinear Schrddinger equation // Preprint: arXiv:1401.4445 (2014).

11. A.O. Smirnov. Elliptic solutions of the nonlinear Schrddinger equation and the modified Korteweg-de Vries equation // Russian Acad. Sci. Sb. Math. 82:2, 461-470 (1995).

12. A.O. Smirnov. Two-gap elliptic solutions to integrable nonlinear equations // Math. Notes 58:1, 735-743 (1995).

13. V.E. Zakharov, A.B. Shabat. Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media // Sov. Phvs. JETP 34:1, 62 pp. (1972).

14. D.H. Peregrine. Water waves, nonlinear Schrddinger equations and their solutions //J- Austral. Math. Soc., Ser. B 25:1, 16-43 (1983).

15. F. Demontis. Exact solutions of the modified Korteweg-de Vries equation // Theor. Math. Phvs. 168:1, 886-897 (2011).

16. V.B. Matveev, A.O. Smirnov. Solutions of the Ablowitz-Kaup-NeweH-Segur hierarchy equations of the "rogue wave" type: A unified approach // Theor. Math. Phvs. 186:2, 156-182 (2016).

Ismagil Talgatovich Habibullin, Institute of Mathematics, Ufa Federal Research Center, EAS, Chernyshevsky str,, 112, 450008, Ufa, Russia

E-mail: habibullinismagilSgmail. com

Aigul Rinatovna Khakimova,

Institute of Mathematics,

Ufa Federal Research Center, RAS,

Chernyshevsky str., 112,

450008, Ufa, Russia

E-mail: aigul. khakimovaSmail. ru

Aleksandr Olegovich Smirnov,

Saint-Petersburg State University of Aerospace Instrumentation, BoPshaya Morskava str., 67, 190000, St. Petersburg, Russia E-mail: alsmir@guap.ru