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Creation of solitons in a system of weakly coupled equations
1 January
1996
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 210 (1996) 85, 86
Creation of solitons in a system of weakly coupled equations A.V. Nedzwedzky, B.I. Verkin Physicotechnicaf
Institute
O.L. Samborsky
ofLow Temperarures, Ukrainian Academy ofSciences, 310164 Kharkou, .!Jkraine
Received 24 April 1995; accepted for publication Communicated by A.R. Bishop
19 October
1995
Abstract The Cauchy problem of the system consisting of the coupled sine-Gordon equation and a linear equation of hyperbolic type is considered. It is shown that such a coupling reduces the threshold of soliton creation and allows additional soliton generation. Keywords:
Cauchy problem;
1. Consider
Sine-Gordon
the following
equation;
system
Solitons; Coupled equations
where qo(n) is an arbitrary function decreasing as x + fm. As follows from IQ. (21, under zero initial conditions (4) (see, e.g. Ref. [3])
of equations
U,, - U,, + sin U = EVE,,
(1)
v,, - s2vxx = &u,x,
(2)
U = U( x, t) and V = V( x, t) are scalar fields in a two-dimensional x-t coordinate space. The subscripts x and t denote partial derivatives with respect to these variables. E is a small coupling constant (E < s -=K1). Eqs. (1) and (2) are the weakly coupled sine-Gordon (sG) equation and a linear equation of hyperbolic type respectively. Such a system describes, for example, the interaction between a charge-density wave and longitudinal ultrasound in a Peierls dielectric [1,2]. Let the initial conditions for Eqs. (1) and (2) be where
U(x,O)=O,
U,(x,O)=O,
U,(x,O)=cp(x), (3)
V( X, 0) = 0,
V,( x, 0) = 0,
Elsevier Science B.V. SSDIO375-9601(95)00833-O
(4)
Substituting
(5) into Eq. (1) we obtain
U,,( x, t) - Or,,( x, t) + sin (I( x, t) =~[-2ux,(x,?)+ux,(x+.~t,0) + U,,( x - sr,
O)] .
Using the initial condition
(3) yields
u,, - [ 1 - ( E/s)‘]U~~ + sin U = 0.
(6)
2. We start by considering the case E = 0. The Cauchy problem of the nonlinear differential equation (6) can be solved by the inverse scattering transform (IST) method [4]. By the IST method the sG equation (6) associates with the linear scattering problem i*=o
(7)
86
A.V. Nedzwedzky,
O.L. Samborsky/
for the wave function *=
Physics Letters A 210 (1996) 85. 86
where
($1I.
S(X)=$~~X’
(p(x’).
2
In this case the operator i is i=1;71-+i{[h-(l/4h) -(
cos
a,/4h)a’
I/la,
sin U + +a,( U, - U,)}.
(8)
Here I^ is the unit matrix, a, are the Pauli matrices (a, = 1, 2, 31, and A is a real spectral parameter (0 < A < m>, i2 = - 1. The eigenfunctions $ of the linear problem (7) are defined by the boundary conditions
and x++w,
A) +
a(A) exp(-iikr)
1
where k = A - 1/4A. The zeros of the analytical function u(A) in the upper half-plane Im A > 0 correspond to solitons by the IST method. For the initial conditions (3) Eq. (7) can be transformed into the following form, - aicp&,
+!J~~= - $ik$,
- ii&,.
is defined as
$,( X) = cos s,,
(‘5)
where
A simple analysis of (15) allows one to conclude that soliton generation starts when S, & ir, the total number of solitons being N = [ $ + so/n-], where the square brackets stand for the integer part. to rewrite
( 16)
where 2 = [l - (.c/s)~]- ‘/‘.x = CXX.The initial condition accordingly changes, cp(x) -+ q(X), and we get $, = (Y& as alternative to 5,. Since E < s we obtain S, > S, and, consequently, the threshold of soliton creation reduces. If S, G irr(2n + 1) and S, CY > irr(2n + 1) for given n an additional soliton can be created, i.e. i = N + 1.
(11) (12)
Let us determine the total number of solitons created in the system. To do so, we have to find the number of zeros of a(A). The threshold condition corresponds to the zero crossing the real axis A, and in this case k(A) = 0. Then the formal solutions of the scattering problem (I 1) and (12) become qli,( X) = - i sin S( x),
(13)
$qx)=cos
(14)
S(x),
u(0) = JnX
U,, - U,, + sin U = 0,
’ (‘0)
$,X = iik+,
amplitude
3. Let now E # 0. Then, it is possible Eq. (6) in the form
b( A) exp( iikx) $(x7
The reflection
References [l] A.S. Rozhavsky, A.V. Nedzvedzky and 1.0. Kulik, Sov. J. Low Temp. Phys. 14 (1988) 89. [Z] AS. Rozhavsky, Yu.S. Kivshar and A.V. Nedzvedzky, Phys. Rev B 40 (1989) 4168. [3] A.N. Tikhonov and A.A. Samarsky, Equations of mathematical physics (Nauka, Moscow, 1972) [in Russian]. [4] V.E. Zakharov, S.V. Manakov, S.P. Novikov and L.P. Pitaevsky, Theory of sobtons (Consultants Bureau, New York, 1984).
1996
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 210 (1996) 85, 86
Creation of solitons in a system of weakly coupled equations A.V. Nedzwedzky, B.I. Verkin Physicotechnicaf
Institute
O.L. Samborsky
ofLow Temperarures, Ukrainian Academy ofSciences, 310164 Kharkou, .!Jkraine
Received 24 April 1995; accepted for publication Communicated by A.R. Bishop
19 October
1995
Abstract The Cauchy problem of the system consisting of the coupled sine-Gordon equation and a linear equation of hyperbolic type is considered. It is shown that such a coupling reduces the threshold of soliton creation and allows additional soliton generation. Keywords:
Cauchy problem;
1. Consider
Sine-Gordon
the following
equation;
system
Solitons; Coupled equations
where qo(n) is an arbitrary function decreasing as x + fm. As follows from IQ. (21, under zero initial conditions (4) (see, e.g. Ref. [3])
of equations
U,, - U,, + sin U = EVE,,
(1)
v,, - s2vxx = &u,x,
(2)
U = U( x, t) and V = V( x, t) are scalar fields in a two-dimensional x-t coordinate space. The subscripts x and t denote partial derivatives with respect to these variables. E is a small coupling constant (E < s -=K1). Eqs. (1) and (2) are the weakly coupled sine-Gordon (sG) equation and a linear equation of hyperbolic type respectively. Such a system describes, for example, the interaction between a charge-density wave and longitudinal ultrasound in a Peierls dielectric [1,2]. Let the initial conditions for Eqs. (1) and (2) be where
U(x,O)=O,
U,(x,O)=O,
U,(x,O)=cp(x), (3)
V( X, 0) = 0,
V,( x, 0) = 0,
Elsevier Science B.V. SSDIO375-9601(95)00833-O
(4)
Substituting
(5) into Eq. (1) we obtain
U,,( x, t) - Or,,( x, t) + sin (I( x, t) =~[-2ux,(x,?)+ux,(x+.~t,0) + U,,( x - sr,
O)] .
Using the initial condition
(3) yields
u,, - [ 1 - ( E/s)‘]U~~ + sin U = 0.
(6)
2. We start by considering the case E = 0. The Cauchy problem of the nonlinear differential equation (6) can be solved by the inverse scattering transform (IST) method [4]. By the IST method the sG equation (6) associates with the linear scattering problem i*=o
(7)
86
A.V. Nedzwedzky,
O.L. Samborsky/
for the wave function *=
Physics Letters A 210 (1996) 85. 86
where
($1I.
S(X)=$~~X’
(p(x’).
2
In this case the operator i is i=1;71-+i{[h-(l/4h) -(
cos
a,/4h)a’
I/la,
sin U + +a,( U, - U,)}.
(8)
Here I^ is the unit matrix, a, are the Pauli matrices (a, = 1, 2, 31, and A is a real spectral parameter (0 < A < m>, i2 = - 1. The eigenfunctions $ of the linear problem (7) are defined by the boundary conditions
and x++w,
A) +
a(A) exp(-iikr)
1
where k = A - 1/4A. The zeros of the analytical function u(A) in the upper half-plane Im A > 0 correspond to solitons by the IST method. For the initial conditions (3) Eq. (7) can be transformed into the following form, - aicp&,
+!J~~= - $ik$,
- ii&,.
is defined as
$,( X) = cos s,,
(‘5)
where
A simple analysis of (15) allows one to conclude that soliton generation starts when S, & ir, the total number of solitons being N = [ $ + so/n-], where the square brackets stand for the integer part. to rewrite
( 16)
where 2 = [l - (.c/s)~]- ‘/‘.x = CXX.The initial condition accordingly changes, cp(x) -+ q(X), and we get $, = (Y& as alternative to 5,. Since E < s we obtain S, > S, and, consequently, the threshold of soliton creation reduces. If S, G irr(2n + 1) and S, CY > irr(2n + 1) for given n an additional soliton can be created, i.e. i = N + 1.
(11) (12)
Let us determine the total number of solitons created in the system. To do so, we have to find the number of zeros of a(A). The threshold condition corresponds to the zero crossing the real axis A, and in this case k(A) = 0. Then the formal solutions of the scattering problem (I 1) and (12) become qli,( X) = - i sin S( x),
(13)
$qx)=cos
(14)
S(x),
u(0) = JnX
U,, - U,, + sin U = 0,
’ (‘0)
$,X = iik+,
amplitude
3. Let now E # 0. Then, it is possible Eq. (6) in the form
b( A) exp( iikx) $(x7
The reflection
References [l] A.S. Rozhavsky, A.V. Nedzvedzky and 1.0. Kulik, Sov. J. Low Temp. Phys. 14 (1988) 89. [Z] AS. Rozhavsky, Yu.S. Kivshar and A.V. Nedzvedzky, Phys. Rev B 40 (1989) 4168. [3] A.N. Tikhonov and A.A. Samarsky, Equations of mathematical physics (Nauka, Moscow, 1972) [in Russian]. [4] V.E. Zakharov, S.V. Manakov, S.P. Novikov and L.P. Pitaevsky, Theory of sobtons (Consultants Bureau, New York, 1984).