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ISSN 2074-1871 Уфимский математический журнал. Том 15. № 1 (2023). С. 122-134.
INTEGRABLE TOLKYNAY EQUATIONS AND RELATED YA JIM A-OIK AWA TYPE EQUATIONS
Zh. MYRZAKULOVA, G. NUGMANOVA, N. SERIKBAYEV, K. YESMAKHANOVA, R. MYRZAKULOV
Abstract. We consider some nonlinear models describing resonance interactions of long waves and short-waves (shortly, the LS waves models). Such LS models were derived and proposed due to various motivations, which mainly come from the different branches of modern physics, especially, from the fluid and plasma physics. In this paper, we study some of integrable LS models, namely, the Yajima-Oikawa equation, the Newell equation, the Ma equation, the Geng-Li equation and their different modifications and extensions. In particular, the gauge equivalent counterparts of these integrable LS models (equations), namely, different integrable spin systems are constructed. In fact, these gauge equivalent counterparts of these LS equations are integrable generalized Heisenberg ferromagnet type models (equations) (HFE) with self-consistent potentials (HFESCP). The associated Lax representations of these HFESCP are presented. Using these Lax representations of these HFESCP, they can be studied by the inverse scattering method. For instance, the equivalence established using the Lax representation also makes it possible to find a connection between the solutions of the corresponding integrable equations.
Keywords: Integrable equations, Heisenberg ferromagnet equation, Yajima-Oikawa equation, gauge equivalent, Lax representation.
Mathematics Subject Classification: 35C08, 35Q51
1. Introduction
Nonlinear models of long wave-short wave resonant interactions, the so-called LS equations, play an important role in modern physics as well as in modern mathematics [l]-[3]. Some of these models are integrable nonlinear partial differential equations in 1+1 and 2+1 dimensions (see, e.g. [4]-[13] and references therein). In 1+1 dimensions, such LS equations of long wave-short wave resonant interactions, formally in general, can be written as
iqt + Qxx + fi(v,vx,q,q,...) = 0, vt + 6(1 q\2)x = 0, (1.1)
where q(x, t) represents an envelope (complex-valued) of the short wave and v(x,t) denotes a real-valued amplitude of the long wave (potential), ó = const. Here f1 is some function of its arguments. System of equations (1.1) can be considered as the nonlinear Schrodinger type equations with self-consistent potentials. It combines all well-known integrable LS models, namely, the Yajima-Oikawa equation (YOE), the Newell equation (NE), the Geng-Li equation (GLE) etc. This outcome is similar to the one that proves that the Korteweg-de Vries and the modified KdV equations are just two particular cases of the Gardner equation. There exists
Zh. Myrzakulova, G. Nugmanova, N. Serikbayev, K. Yesmakhanova, R. Myrzakulov, Integrable Tolkynay equations and related Yajima-Oikawa type equations.
(r) Myrzakulova Zh., Nugmanova G., Serikbayev N., Yesmakhanova K., Myrzakulov R. 2023.
The work is financially supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan, Grant API 1971227.
Submitted August 15, 2020.
another interesting class of integrable systems, namely, integrable generalized spin systems or Heisenberg ferromagnet equations with self-consistent potentials (GHFESCP) (see, e.g. [14] [18] and references therein). Such spin systems in general can be written as
i St + f 2 (S,SX,SXX,U,UX,...) = 0, ut + utr(S [Sx,St])x = 0, (1.2)
where a 2 x 2 spin matrix S reads as
5 =(£ -'s,) ™
and satisfies the following condition
S2 = I. (1.4)
Some time we use the following form of integrable generalized spin systems
iRt + MR, Rx, Rxx, u, ux ,...) = 0, (1.5)
where, in contrast to (1.3) and (1.4), the symbol R denotes a 3 x 3 spin matrix satisfying the condition
R3 = eR, (e = ±1).
2 u
function (potential) and (u, 5) are real constants.
The set of equations (1.1) is some kind generalizations of the nonlinear Schrodinger equation (NLSE):
i Qt + qXx + 2 v lql2q = 0. (1.6)
At the same time system (1.2) and (1.5) are certain extensions of the following Heisenberg ferromagnet equation (HFE)
i St + 2[S,S^]=0. (1.7)
Both NLSE (1.6) and HFE (1.7) are integrable and admit some integrable extentions in 1+1 and 2+1 dimensions (see, e.g., [19] [38] and references therein). It is well-known that between NLSE (1.6) and HFE (1.7) are related by a gauge and geometrical equivalence [39]-[40], The aim of this paper is to find such gauge equivalence between some particular reductions of systems of equations (1.1), (1.2) and (1.5).
This paper is organized as follows. In Section 2, the integrable Tolkynay equation (TE) is presented. The Yajima-Oikawa-Mewell equation (YONE) is considered in Section 3 and its gauge equivalence with the TE is studied in Section 4. In the next two Sections 5 and 6 the gauge equivalent counterparts of the Yajima-Oikawa equation (YOE) and the Ma equation (ME) are presented. The relation between the TE and NE is established in Section 7. The same problem was studied in Section 8 for the Geng-Li equation (GLE). In Section 9, the MXXXIV equation is investigated. In Section 10, the M-V equation and its relation with the LS equations were considered. The last section is devoted to some conclusions and discussions.
2. The Tolkynay equation In this paper, in particular, we study the following Tolkynay equation (TE)
i Rt + [ R2,Rx]x = 0. (2.1)
R
23
f Rii Ri2 R
R = 1 R21 R22 R
V R31 R 2 R
and satisfies the following conditions
R3 = R, tr(R) = 0, det(R) = 0.
The TE is one of integrable generalized Heisenberg ferromagnet type equation. The Lax representation (LR) of the TE is given by
%
% = V1%,
where
U1 = iXR,
- 3')
Vi = -xXUR2 - - I - X[R2, Rx]
The YONE
One of most general integrable LS equations is the Yajima-Oikawa-Newell equation (YONE)
[8]
iqt + qxx + (iavx + a2v2 - fiv - 2alql2)q = 0, vt - 2(|g|)x = 0, (3.1)
where the parameters a, P are arbitrary real constants. These parameters may be considered as independent constants describing a long-short wave cross-interaction. This system reduces to the YOE for a = 0, P = 1 and to the Newell equation as a = a, P = 0. System YONE (3.1) is integrable. Its LE has the form [8]
where
Here
E
Bo
Bn
3
^x
Uo = iXE + Q,
10 0 0 0 0 0 0 -1
100 0 -2 0 0 0 1
—i a\q\2
U2&, = V2$,
Vo = -\2B2 + t\B1 + B0.
Q
aq
0
i(a2v - ß) aq
0
B1 = I iaq
i q 0
0 - i q
0 -iaq 0
-avq + i qx
-a2vq + ßq - iaqx 2ia\q\2
ia2\q\2 -a2v q + ßq + iaqx
\ \2 avq - iqx
- a\ \ 2
4. Gauge equivalence between TE and YONE
In this section we establish the gauge equivalence between TE (2.1) and YONE (3.1). In order to do this, we consider the gauge transformation
We obtain where
As a result, we have where
% = g-1§, g = $x=0.
% = Ui%, % = Vi%, Ui = g-1 {Ü2 - gxg-l)g, Vi = g-1(V2 - gtg-1)g. Ui = iXR, V1 = -X2g-1B2g + iXg-1B1g, R = 9-1e9 .
0
After some calculations, we obtain
QS + SQ
or
V
g-1Qgg-1Sg + g-1 Sgg-1Qg = g~ On the other hand,
Rx = g-1[S,Q]g = g-1
Hence, we get
0 0
aq 0 - qq
0 - aq 0
0 q 0 \
aq 0 -q 1
0- aq 0
-iBi
-ig lBxg = g 1QgR + Rg 1Qg
-i,
0
-aq „2
q 2 i v
0 q I g.
-2i(a2v - ß) -aq 0
[ R, Rx] = g-1
and therefore,
0
aq
q 4i v 0 q
9 = 9 lQ9 +
i
0 0
0
00
4 i(a2v - ß) aq 0
g-1Qg = [R, Rx] - 3ig
( a2v -ß) 0 0
i
0 0
0
00
Taking into consideration the formula
/0 0 v
E I 0 0 0 | +
\ (a2v - p) 0 0
we obtain
( a2 - ß) 0 0 00
0 0 0 I E = 0,
( a2 - ß) 0 0
g-1Big = i(g-1Qgg-1Eg + g-1Egg-1Qg) = ¿([R, RX]R + R[R, Rx]) = i[R2, Rx].
Thus, we have shown that
2
g-1B29 = -- 1 + iR2, g-1B19 = i[R2,Rx}.
3
Finally, we have the following Lax pair for TE (2,1)
2
Ux = x\R, V = -i\2(R2 - - I) - X[R2, Rx\.
3
We provide some useful formulas:
tr(R2) = -4alql2 + 8v(a2v - p), det(R^) = 2ip|g|2.
As integrable equation, the YONE admits the infinity number of integrals of motion. For example, here we present the following integral of motion for the YONE (3,1):
P = J J dx,
where a, b are some constants and
J = 4a[a(5/3 - 1)|g|2 + 2v{a2v - /)] = atr(S2) + bdet(Sx), J
Jt = 16a[za(qxq - qqx) - /|q|2 + 2a2v|q|2]x,
10 aa
iß .
and hence,
Pt
J dx^j
J dx I =0
with the boundary conditions
limg(x,t) ^ 0, \imv(x,t) ^ 0 as x ^
5. Gauge equivalence between the YOE and the MM-IIE
The first example of integrable long-short waves interactions models is the following Yajima-Oikawa equation (YOE) [1]
i Qt + ^ Qxx -uq = 0,
ut + Ux + lq\X = 0
Its Lax representation reads as
U3 = Aq + 2i AS + (2A) A_1
Fa = -U + 2% A2Al + ABi + Bq + ¿(4A)"
where
1 0 0 S= | 0 0 0 0 0 1
Bi
B i
0 0 0
0 -$* 0 Aq = 1 0 0 0 0 $* 0
0 2 $x + $*
BQ = | 1 $ 0
|$|2 0 |$|2 \
$x 0 $x I
-|$|2 0 -|$|2 )
/ -ni 0 - n
A_1 =| $ 0 $
\ n 0 n
0 —i $* — $* u 2 x
0
-1 $ 2
0
-1 $ 2 $
0 |$|2 0 $x 0 -|$|2
$ = qe2 x).
0 -$* 00
0 $*
|$|2
$x
-|$|2
We consider a gauge transformation
0 = g(x, t, Aq) = 0(x, t, A) | a=Aq C GL(£, C)
as A = Aq, where
9x = U3(x,t,Ao)g, gt = V3(x,t,Ao)g,
Then
U4 = g_1u3g - g-1 gx, V4 = g_1vsg - g-1 gt.
We obtain:
U4 = 2i(A - Aq)9-1Aig -
A - A0 _1 a
1A\9 A_19,
(5.1)
V4 = 2*(A2 - A20)g_1A\g - U4 - ^^9_1B_19 + (A - Aq)B1 .
We introduce two new matrices R and a
R = g_1Sg, a = g_1A_1g, R3 = R.
Then
U4 = 2x(A - Aq)[R + (4^AqA)_V],
V4 = -U + (A - Aq){2í(A + Aq)R2 + R(R2)x + A_1 [í(4Ao)_1[a, R2] + 2[a, R2]R]}.
This Lax pair gives the following Makhankov-Myrzakulov-II equation (MM-IIE) [24]
R + Ri + (2rR - 2\BR'i)x + («A0)-1k = 0,
( + + (^k, R]x - i\0[(J,R2])x + i\0h = 0,
where
(6.1)
h = [a(Rx - 2 i\oI),R2] + ([R2,a]R)x, tr( U4) = tr( V4) = tr( U3) = tr( V3) = tr( R) = tr(<r) = 0.
6. The MM-IE and its relation with the Ma equation Let us consider the following Makhankov-Myrzakulov-I equation (MM-IE) [24]
iRt + 2[ R,RXX] - 4(RXR)X = 0,
R
R3 = R.
The MM-IE is integrable. The corresponding LR has the form
% = U5 %,
% = V5^,
where
U5 = %\R, V5 = 2%\2R2 + 2\(R2)x. Note that the MM-IE is gauge equivalent to the following Ma equation (ME) [24]
iqt - 2qxx + 2uq = 0, ut + |<?|2 = 0.
After some simple scale transformations, the ME takes becomes YOE (5,1), In contrast to the YOE, for the ME the LR takes a simpler form [3]
= U6$,
(6.2)
where Here
U = C0 + i\E,
Co
D,
o =
' I 0 1 n \ E* I 2
0 0 J
0 —i Ex - 2 |E |
E* 0
0 E 0
Vfi
E
Di
= V>$, D0 + XD1 + 2i\2D2.
10 0 0 0 0
0 0 -1
0 E 0 0 0 - E* 0 0 0
Do = E2
As we mentioned, the MM-IE and the ME are gauge equivalent one to other. In order to prove this, let us consider the gauge transformation ^ = u-1$, where u = $(x, t, A) |a=o- Then we have
ux = Ue0u, ut = V60u,
where
U6o = Ui |A=0, V60 = V1 |A=0.
As a result, we get
U5 = u 1U6u - u 1ux, V5 = u 1V6u - u 1 u
-1
-1,
2
or
U6 = iXu Eu, V6 = Xu D1u + 2iX u D2u. After some calculation we obtain
u-lDiu = 2(R2)x, u-1D2u = R2,
where
R = u-1Eu.
The matrix function R satisfies the following conditions
tr(R) = 0, det(R) = ^E|2, det(Rx) = det(u-1[Cb Co]u) = ^E|2.
It is not difficult to confirm that the matrix function R satisfies MM-IE (6,1), This is proves that MM-IE (6,1) and ME (6,2) are gauge equivalent one to the other,
7. The TE and its relation with the Newell equation The TE reads as
i Rt + [ R2,Rx]x = 0, where the spin matrix R is of the form
13 23 33
and satisfies the following conditions
R3 = R, tr(R) = 0, det(R) = 0.
Here a = ±1 R12 = R- = R1 — iR2 is a complex function and R11 functions. Let us calculate some useful expressions
(7.1)
l Rii Ri2 R
R = I R2i R22 R
i \ R3i R32 R
Rii Ri2 i / R co R-
-oRi2 0 a Ri2 I =I —aR+ 0 aR+
— — Ri2 —Rii ) 1 \ — — R- — R
R3, Ri, R2,v are real
alRnl2
R2 = I -(Rii + tv)aRi2 —2<r | Ri212 -(Rii + tv)aRi2
Rh — vlRnl2 + v2 (Rii — iv)Ri2 >)
2
elRnl
Rii
R3 = R2R =R = (R2i — 2a|Ri2|2 + v2) 1 —a Ru
( Rii — iv) Ri2 R2i — alRi2l2 + v2
Ri2
0 aRi2 Ri2 —Rii
( r2n — 2alri2l2 + v2)r.
We hence obtain the following condition
R\l — 2alRi212 + v2 = R2 — 2alR+l2 + v2 = R¡ — 2a(R¡ + R2) + v2 = 1.
The TE is one of integrable generalized Heisenberg ferromagnet type equation. The LR of the TE is given by
^ = U7^, = Vj^,
where
Uj = tXR, Vj = — X2 IR2 — -I ) — X[R2, Rx].
3
It is not difficult to confirm that the gauge equivalent of TE (7.1) is a Newell equation (NE) which reads as [2]
i qt + qxx + (iux + u2 — 2alql2)q = 0, ut — 2a(|g|2)x = 0,
where a = ±1, Note that in addition to a long wave-short wave coupling, the short wave has the same self-interaction as the NLSE (1,6), The NE is integrable. Its Lax representation is [21, [111
$ A m
$ = Vs$,
where
U8 =
i X q i v a q 0 aq iv q -iX
-\iX2 - ialql2 - Xq + iqx - vq
Vs = I -a(Xq + qx + vq) \iX2 + 2ialql2 a(Xq - iqx -vq)
i^qf
Xq + iqx - vq -1iX2 - ialql2
It is not difficult to verify that these matrices satisfy the following conditions
U+(A) = -AU (A) A, V+(A) = -AV (A) A,
U+(X) = -BV (-X)B, U+(X) = -BV (-X)B,
where
100
A = I 0 -a 0 0 0 1
0 0 1 B = I 0 1 0 100
8. Geng-Li equation
Again we consider TE
i Rt + [ R2,Rx]x = 0.
(8.1)
In contrast to the previous cases, now we assume that the spin matrix R satisfies the conditions
R3 = -R, tr(R) = 0, det(R) = 0. (8.2)
Then the LR for TE (8.1) becomes
= Ug^, % = , (8.3)
where
Ug = AR, Vg = -I A2R2 + xA[R2,Rx]. R
R = g-1Jg,
where
/0 -10 J = I 1 0 0 000
We consider the gauge transformation
$ =
Hence, it follows from equations (8.3) that the new matrix function $ satisfies the equations
$ = Ug$, $t = Vg$,
where
i iv -A 0 \ / iA2 - 2iIql2 0 -iAq
Us = I A 0 -q I , V8 = I 0 tA2 - t^2 iqx + vq y 0 q 0 y y -iAq iqx - vq ilql2
The compatibility condition
Ust - Vsx + [Us,Vs] = 0 gives the Geng-Li equation (GLE) [4]
iqt + qxx + 2lql2q + i(vq)x = 0, vt + 2(lql2)x = 0. (8.4)
This proves that TE (8,1) with conditions (8,2) and LE (8,3) is gauge equivalent to GLE (8,4),
9. The M-XXXIV equation
One of integrable HFE with self-consistent potentials (HFESCP) is the following Myrzakulov-XXXIV (M-XXXIV) equation
1 2'
where S = ( Si, S2, S3) is the unit spin vector that is S2 = Sf + S2f + S| = 1 and u is a real function (potential). The M-XXXIV equation is integrable. The corresponding LE has the form
1
^ = 1 [S + = 2 [S + (2b + 1)mxx + 2W (9-1)
where W = Wi + W2 and
St + S A S^ -uS = 0, ut + -(SX)x = 0,
Wi = (2b + 1)E + (2b + 1)SSX + (2b + 1)FS,
W2 = FI + -Sx + ES + aSSv, S± = Si ± iSf, 2
E = -—Ux, F =- (UX - 2u>] , S ÍS3 S
) ■ s =(Ü .
2 a ^ 2\a V' \S+ — S3/
In fact, the compatibility condition = gives the following system of equations
iSt + 1 [ S, S^] — iwS% = 0, 1
4
where
wv - — tr(SS, Sv]) = 0,
£ = x +— y, r> = —x, w = . a
Hence after the simple transformation r] = t, w ^ u, £ ^ x, we obtain
iSt + 2[S, Sxx] — iuSx = 0, ut — ^MS[Sx, St]) = 0. (9.2)
The following equations hold:
[Sx,Si] = — (SX)xS, S [Sx,Si] = — (SX)x I, SX = , S [Sx,Si] = — (SX)x I,
and
t r( S [ Sx,St]) = — 2z(sx)x. Hence, M-XXXIV equation (9.2) can be written as
iSt + 2[S, Sxx] — iuSx = 0, Ut + 2(sx)x = 0. (9.3)
Let us find the equation which is gauge equivalent to M-XXXIV equation (9.3). To this end, we consider the following tranformation
$ = g-1^, (9.4)
where ^ is the matrix solution of linear problem (9.1), $ and g are a temporally unknown matrix functions. Substituting (9.4) into (9.1), after some calculations we get
a$y = Bi$x + Bo$, $t = iC2 $xx + Ci$x + Cq$, (9.5)
with
(0 0) , B = 0) ,
(b +1 , Ci = f° 0), c = fcii C12)
0 1 0 0 21 22
2 1 0 bj ' 1 y I r °
c12 = i(2b + 1) qx + i a qy, c21 = — 2 % brx — i ary.
Here Cjj satisfy the following system of equations
ciix — aciiy = i qrx + r C12 — q C21, a c22y = —ir qx + r C12 — q C21.
The compatibility condition of equations (9,5) gives the following (2+l)-dimensional nonlinear Sehrôdinger equation:
i qt + gçç + v q = 0, i rt — r^ — v r = 0, vv + 2(r q)% = 0,
or, after î] —y t, we have
i qt + qxx + vq = 0, i rt — Txx — V r = 0, Vt + 2(rq)x = 0.
It coincide with ME (6,2), Thus, we have presented a new LE for the ME or for the YOE, Consequently, we have found a new form of the gauge equivalent counterpart of the ME and/or the YOE, namely, the M-XXXIV equation,
10. The M-V equation
Our next example of integrable generalized HFE is the so-called Mvrzakulov-V (M-V) equation, The M-V equation reads as
iRt + 2[R, Ry]x + 3(R2RyR)x = 0
or
iRt + -[R, Ry]x + 3[R2, (R2)y]x = 0, (10.1)
2[R,Ry ]x + 2 R
R3 = R, tr(R) = 0, det(R) = 0. M-V equation (10,1) is a (2+l)-dimensional integrable equation. Its LE reads as
% = Ui%, % = Vi%,
where
U1 = -AR, Vi = -aA'R + A ([R,Ryi + aifl2, (R2)y]).
In order to find its gauge equivalent equation, we consider the transformation
R = $-1E$,
where £ = diag(1, 0, -1) Let us we assume that $ satisfies the following equations $x = - A£ + Q, $t = (112A2 + inA + /i0 )$y + V $,
with a given matrix Q; here V is an unknown matrix and = consts. The compatibility condition $xt = $tx gives the following two equations:
Qt -Vx + [U, V] - (12A2 + I1A + |)Qy = 0 (10.2)
and
At - (12A2 + 11A + |)Ay = 0.
Equation (10,2) is the desired nonlinear Sehrodinger type equation coupled with the equation for the potential v(x, t). At the same time, equation (10,2) indicates that in this case, we have a nonisospeetral problem, where A = A(y, t).
11. Conclusions
Nonlinear models describing interactions of long and short (LS) waves are given by the Yajima-Oikawa type equations. These long wave-short wave interaction models were derived and proposed with various motivations, which mainly come from fluid and plasma physics. It is well known that in these long wave-short wave equations is that a long wave always arises as generated by short waves. In this paper, we study some of integrable LS models, namely, the Yajima-Oikawa equation, the Newell equation, the Ma equation, the Geng-Li equation and etc. Any integrable equations admitting the Lax representations, generally speaking, are gauge equivalent to some integrable generalized HFE, In this context, it is interesting to find the gauge equivalent counterparts of the above mentioned integrable LS models. In this paper, the gauge equivalent counterparts of integrable LS models (equations) are found. In fact, these gauge equivalents of the LS equations are integrable generalized Heisenberg ferromagnet equations with self-consistent potentials (HFESCP), The associated Lax representations of these HFESCP are given.
Acknowledgements
This work was supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan, Grant AIM 1971227.
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Zhaidarv Mvrzakulova, Gulgassvl Nugmanova, Nurzhan Serikbavev, Kuralay Yesmakhanova, Hal bay Mvrzakulov Eurasian National University, Satbavev str. 2,
010008, Astana, Kazakhstan
Ratbay Mvrzakulov Eurasian International Centre for Theoretical Physics, 38th str., 27/1,
010009, Astana, Kazakhstan E-mail: rmyrzakulovSgmail. com
