Periodic problem for the sine-hilbert equation

Volume 120, number 4

PHYSICS LETTERS A

16 February 1987

PERIODIC PROBLEM FOR THE SINE-HILBERT EQUATION Y. MATSUNO Department ofPhysics, Faculty ofLiberalArts, Yamaguchi University, Yamaguchi 753, Japan Received 16 October 1986; accepted for publication 11 December 1986

A method for constructing periodic solutions of the sine-Hubert (sH) equation is developed. It is shown that the sH equation can be reducible to a linear differential equation. The explicit periodic solutions are then derived by solving the initial value problem for this linear equation.

In this paper we consider the following sine-Hilbert (sH) equation HO~=—sinO, 6=O(x,t),

(1)

with

(5) Eq. (5) is furthermore modified in the form f+Lf_,~—W2i)(f_

HO(x,t)=-~-P It

I

0’t)~,,

(2)

where the operator H is the Hilbert transform and the subscript t denotes partial differentiation with respect to t. The sH equation first appeared in a paper [1] which treated a class of matrix nonlinear integro—differential equations. Subsequently, it was shown to be associated with a Riemann—Hilbert scatteringproblem [2,3]. These works are, however, not concerned with the periodic problem. The purpose of the present paper is to develop a systematic method for constructing periodic solutions of the sH equation. First, introduce the following dependent variable transformation O=iln(f÷/f_),

f± =f±(x,t)

(3)

f+)]

=—f_[f÷,~—(1/2i)(f_—f÷)] (6) Therefore, eq. (5) is satisfied provided the following .

system of linear partial differential equations for f÷ andf_ holds:

f÷t (1 /2i) (f_ —f+ ) + i1f÷ ,

(7)

f_ ,

,

(8)

=

=

(1/2i) (f_

—f÷) i).f_ —

where 2 is a real parameter depending generally on t. In this paper we shall focus our attention on real periodic solutions and seekf÷andf in the form N 1 f_f=[T~sin[I3(x_x1)], x~=x~(t), (9)

f~=fK. Here, x~(j=1, 2,

(10) ...,

N) are complex functions of t

where f+ (f_) is a complex analytic function with zeros lying in the lower (upper) half complex x plane. Then, it readily follows by using the property of the H operator that

satisfying the conditions Im x~>0, fi is a positive parameter and * denotes complex conjugate. In this situation, eqs. (7) and (8) are reduced to a linear equation forf

HO~=—[ln(f÷f_)]~.

f~=(l/2i)(f_f*)_iAf.

(4)

Substitution of (3) and (4) into (1) yields the bilinear equation 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(11)

Furthermore, assume, for simplicity, that 2 is a positive constant with a constraint 2—~0when fl—~0. 187

Volume 120, number 4

PHYSICS LETTERS A

The initial value problem for the linear equation (11) is given explicitly as f(x,

1) =f(x,

0)(cos yt— ~i(l +22)

16 February 1987

~,

c\( t) = (2ifl) ~ exp(

ifl ~ x1

)

(19)

Sin ~t)

cv(t)=(~2iflY~exp(ifl ~ \

sin yt +~if*(x,0)—,

(12)

and consequently

(13)

c~(t)c_,~( t) = (2/i) 2X (21) At the same time, we obtain for N= 2M (M is a positive integer)

with 2A)2—1]~2 .

(20)

j=I

y=~[(l+ It should be noted here that the motion of the vanables x 1, x2, X~is governed by the following system of nonlinear ordinary differential equations ...,

rn—~O,1 Al—I and for N=2M+1

1J5~1fl-’sin[fl(x~—x~’)] 2i fTj_IUn) fi-’ sin[fl(x~—x,)] 1

(22)

,

,n=0, I ,...,M. (23) The time dependence of x~is found by solving the c~,(t)=c~7,?(t)=0,

(n = 1, 2

N)

(14)

,

where the dot denotes differentiation with respect to t. The system of equations (14) follows from (9) and (11). In order to derive the explicit time dependence of x,~,we expand fas

algebraic equation of order N, c2,~~(t)z’ = 0

(24)

.

where (25)

2iJJx

f(x, t)=

c~(t)e”~

~

,

(15) The initial data c,1(0) (n=0, ±1 ±N)must be chosen such that the conditions Im x~(0)>0(n= 1, 2 N) hold since if these conditions are satisfied, they hold for later time. This statement can be confirmed by the relation which stems from the equations of motion (14):

and substitute (15) into (12). One then finds cn(t)cn(0)(cosYt_~i(l+2A) ~±~2’-t) sin yt y

(16)

The initial data c~(0)and c~~(0) are related to the initial values off(x, t) andf*(x, t) through (15) as f(x,0)=

~

c~(0)e’~,

(17)

N

N

~

c(0)e~’~.

(18)

n~ —N

It should be observed from the expressions (9) and (15) that 188

z=e

Imx,~(t)=fl_Ltanh1[tanh[flImx,~(0)]

>
(~

G(~)d1’)~ ,

0

with G(t) being a real function of I given by

(26)

(

Volume 120, number 4

1

PHYSICS LETTERS A

sin[fl(x~—x~)J sin[fl(x~—x~)]

N

f~=(l/2i)(f_f*),

fi

(27)

sin[fl(x~_xJ)J) sin[fl(x~—x~)]

~, ~n)

(36)

f(x,t)=f(x,0)

(J#n)

—

16 February 1987

.

+ (l/2i) [f(x, (x~—x~) 0) _f*(x, 0)Jt, 1 fl~,

Xfl=~

Thus, we have completed the procedure for constructing periodic solutions of the sH equation. We shall now write the results for N= 1 explicitly: c0(t)=0,

(n 1, 2,

1)

...,

N)

(38)

,

respectively. Ifwe expandf(x, (28)

(37)

(x~—x

fiN

1)

in powers ofx as

N

f(x, 1) = ~ (—1 ys~.x~’~J, s~=s3(t),

2e’’~4~

,

(29)

c,(t)=~ (l+22+2y) 2c,(t),

(39)

the time1=0 evolutions of s 1 are found from (36) and

(30)

(39) as ~=(l/2i)(sj—s~)

(31)

Eqs. (40) are readily integrated as s~ =a~t+b~ +ia~ (1=1,2 N),

(j=l,2,...,N).

(40)

c_,(t)=l/4fl x,(t)

=~

t+b+~ ln(l +22+2y),

where b is a real constant and y is given by (13). If we put y=fla,

(32)

with a being a positive constant, (31) becomes to be a more transparent form as x,(t)=at+b+i

sinh’ (2/ia) 2/I

where a 1 and b~are real con~tants.The results (39)—(4l) coincide with those obtained in a previous paper [4]where the system of equations (38) has been solved instead of the method developed here. Finally, we conclude this paper by noting that the periodic solution (34) reduces, in the limit offl—40,

(33)

to

The expression of 0 is derived from (3), (9), (10) and (31) in the form

0

9

2 tan’

(

2a2)”2+l 2/ia

(l+4fl

xcot[fl(x_at_b)])~

(34)

which represents a periodic function with a period it/fl.

Next, we consider the limit fl—~0.The expressions (9), (11), (12) and (14) are reduced, in this limit, to N

f(x,t)=fl (x—x~),

(41)

—2 tan

a

\x—at—b

(42)

and the x derivative of (42) yields a pulse with a brentzian profile 2a (43) (x—at—b)2+a2~ The functional form of (43) is the same as that for the one-soliton solution of the Benjamin—Ono equation [5,61. However, the propagation velocity of the pulse is inversely proportional to the amplitude. Therefore, a tall pulse propagates more slowly than a small pulse unlike the behavior of the usual soliton which propagates with a velocity proportional to its amplitude.

(35)

1=l

189

Volume 120, number 4

PHYSICS LETTERS A

The author thanks Professor M. Nishioka for his encouragement. References

[21 A. Degasperis, P.M. Santini and M.J. Ablowitz, J. Math. Phys. 26 (1985) 2469. [3] P.M. Santini, Mi. Ablowitz and A.S. Fokas, On the initial value problem for a class of nonlinear integral equations including the sine-Hilbert equation, INS #47 (July 1985) [4] Y. Matsuno, Phys. Ecu. A 119 (1986) 229.

[51T.B. Benjamin, J. Fluid Mech. 29 (1967) [I] A. Degasperisand P.M. Santini, Phys. Lett. A 98 (1983) 240.

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16 February 1987

559. [61 H. Ono, J. Phys. Soc. Japan 39 (1975) 1082.