Integration of the toda-type chain with an integral type
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Современные проблемы дифференциалных уравнений и смежных раз..., Фергана-2020 1 UDK 517.946 INTEGRATION OF THE TODA-TYPE CHAIN WITH AN INTEGRAL TYPE SOURCE
B.A.Babajanov 1 , M.M. Ruzmetov 1 , A.B. Babajanov 1 1
a.murod@mail.ru Abstract. In this paper, it is shown that the solutions of the Toda-type chain with a self- consistent integral type source can be found by the inverse scattering method for the discrete Sturm-Liuville operator. Keywords. Toda-type chain, self-consistent integral type source, inverse scattering method, moving eigenvalues, one-soliton solution. The formulation of problem The Toda lattice [1, 2] is a simple model for a nonlinear one-dimensional crystal that describes the motion of a chain of particles with exponential interactions of the nearest neighbors. This equation has different practical applications. For example, the Toda lattice model of DNA in the field of biology [3]. Moreover, one important property of the Toda lattice type equations is the existence of so called soliton solutions. There is a close relation between the existence of soliton solutions and the integrability of equations: the known research results show that all the integrable systems have soliton solutions [4]. Soliton solutions of the Toda lattice are obtained in the works [2] and [5]. Soliton equations with self-consistent sources have received much attention in the recent research literature[6]-[12]. Physically, the sources appear in solitary waves with non-constant velocity and lead to a variety of dynamics of physical models. They have important applications in plasma physics, hydrodynamics, solid-state physics, etc. [13]-[18]. For example, the KdV equation, which is included an integral type self-consistent source, was considered in [16]. By this type equation the interaction of long and short capillary-gravity waves can be described [18]. Other important soliton equations with self-consistent source are the nonlinear Schrodinger equation which describes the nonlinear interaction of an ion acoustic wave in the two component homo– geneous plasma with the electrostatic high frequency wave [19]. The purpose of this paper consists on develop the scattering method for the Toda type lattice equation with self-consistent integral type source. An effective method of integration of the Toda-type chain with an integral type self-consistent source is presented. We consider the system of equations with a special self-consistent source
da n dt = a n (a 2 n+1
− a 2 n−1 ) + a n (b 2 n+1
− b 2 n ) + a n H |µ|=1 1 µ (f n+1
(µ, t)g n+1
(µ, t) −f n (µ, t)g n (µ, t))dµ, db n dt = 2a 2 n (b n+1
+ b n ) − 2a 2 n−1
(b n + b n−1 ) + a
n H |µ|=1 1 µ (f n (µ, t)g
n+1 (µ, t)+
+f n+1
(µ, t)g n (µ, t))dµ − a n−1 H |µ|=1 1 µ (f n (µ, t)g
n−1 (µ, t) + f n−1 (µ, t)g
n (µ, t))dµ, a n−1
f n−1
+ b n f n + a
n f n+1 = µ+µ
−1 2 f n , a n−1 g n−1 + b n g n + a
n g n+1 = µ+µ
−1 2 g n , n ∈ Z., (1) 1
Современные проблемы дифференциалных уравнений и смежных раз..., Фергана-2020 2 and the initial condition a n
0 n , b n (0) = b
0 n , n ∈ Z, (2)
where {a 0 n } ∞ −∞ , {b 0 n } ∞ −∞ satisfy the following properties: (i) a
0 n > 0, Imb 0 n = 0, n ∈ Z, (ii)
P ∞ n=−∞ | n | a 0 n − 1 2 + b 0 n < ∞, (iii) The operator (L(0)y) n
n−1 (0)y
n−1 + b
n (0)y
n + a
n (0)y
n+1 (3)
has exactly N eigenvalues λ k (0) = z k (0) + z −1 k (0) 2 , k = 1, 2, ..., N, which are out of the interval [−1; 1]. In system (1), the functional sequences of the functions {a n (t)}
∞ −∞ , {b n (t)}
∞ −∞ , {f n (µ, t)}
∞ −∞ and {g n (µ, t)}
∞ −∞ are unknown vector-functions. Moreover for all t ≥ 0 and |µ| = 1 the following asymptotic properties are fulfilled: g n (µ, t) ∼ n→ − ∞ p(µ, t)µ n + q(µ, t)µ −n , f n (µ, t)
∼ n→ − ∞
r(µ, t)µ n + s(µ, t)µ −n . Here p(µ, t), q(µ, t), r(µ, t) and s(µ, t) are given continuous functions that satisfy Holder’s condition with some degree ν ∈ (0, 1] on |µ| = 1 for all nonnegative t. The main aim of this work is to obtain the expressions of the solutions {a n (t)}
∞ −∞ , {b n (t)}
∞ −∞ , {f n (µ, t)} ∞ −∞ and {g n (µ, t)}
∞ −∞ of the problem (1)-(3) in the framework of inverse scattering method for the operator L(t). Under condition (ii) the equation (3) has the Jost solutions with the asymptotic properties: ϕ n
n + o(1) as n → ∞, |z| = 1, ψ n
−n + o(1) as n → −∞, |z| = 1. (4) It is very well known that such solutions exist and, moreover, they are identified by the asymptotic expressions (4) unique and analytically expended into the circle |z| < 1. For |z| = 1 the functions {ϕ n (z), ϕ
n (z −1 )} and {ψ n (z), ψ n (z −1 )} are the pairs of the linearly independent solutions of (3), therefore ψ n
n (z −1 ) + β(z)ϕ n (z), ϕ n (z) = α(z)ψ n (z −1 ) − β(z −1 )ψ n (z),
(5) with
α(z) = 2 z − z −1 W {ψ
n (z), ϕ
n (z)} ,
(6) and
W {ψ n (z), ϕ n (z)} ≡ a
n (ψ n (z) ϕ n+1
(z) − ψ n+1
(z) ϕ n (z)). The reflection coefficient is given by the formula R(z) = − β(z
−1 ) α(z) . The function α(z) is analytically expended into the circle |z| < 1 and inside it has a finitely many zeros z 1 , z 2 , ..., z
N . The pointsλ k =
k +z −1 k 2 , k = 1, 2, ..., N correspond to eigenvalues of the operatorL. From (6) we arrive at the following expression 2
Современные проблемы дифференциалных уравнений и смежных раз..., Фергана-2020 3 ϕ k n = B
k ψ k n , k = 1, 2, ..., N (7) where ψ
k n ≡ ψ n (z k ). The set of the quantities {R(z), z 1 , z
2 , ..., z
N , B
1 , B
2 , ..., B
N } is called the scattering data for equation (3). The values of {a n (t)} ∞ −∞ and {b n (t)}
∞ −∞ can be found from the scattering data. Theorem. If the functions a n (t), b n (t),f
n (µ, t),g
n (µ, t), n ∈ Z are solutions of the problem (1)-(2), then the scattering data of the operator (L(t)y)
n ≡ a
n−1 (t)y
n−1 + b
n (t)y
n + a
n (t)y
n+1 , are given by relations dz k dt = 0, k = 1, 2, ..., N, (8)
∂R(z, t) ∂t = z 2 − z −2 2 + 1 z 2 − 1
v.p. I |µ|=1 D(µ, t)dµ ! R(z, t)+ +2π i (Q(z, t) + Q(z −1 , t))R(z, t) + 4π iP (z −1 ), (9) dB k (t) dt = (
z 2 k −z −2 k 2 − 1 z 2 k −1 H |µ|=1 (µ+z k )(µz k −1)
µ(µ−z k ) (b(µ, t)c(µ, t) + q(µ, t)r(µ, t))dµ − 1 z 2 k −1 H |µ|=1 (µ−z k )(µz k +1)
µ(µ−z −1 k ) (a(µ)d(µ) + p(µ)s(µ))dµ)B k (t)k = 1, 2, ..., N, (10) where
D(µ, t) = (q(µ, t)r(µ, t) + p(µ, t)s(µ, t)) (µ + z)(µz − 1) µ(µ − z) + (µ − z)(µz + 1) µ(µ − z −1 ) , a(µ, t) = p(µ, t)β(µ −1 , t) + q(µ, t)α(µ, t), b(µ, t) = p(µ, t)α(µ −1 , t) + q(µ, t)β(µ, t), c(µ, t) = r(µ, t)β(µ −1 , t) + s(µ, t)α(µ, t), d(µ, t) = r(µ, t)α(µ −1 , t) + s(µ, t)β(µ, t) and α(µ, t) = N Y
µ − z j µz j − 1
exp ( 1 4πi Z |ζ|=1 ln
1 − |R(ζ, t)| 2
µ + ζ µ − ζ dζ ζ ) , β(µ −1 , t) = −R(µ, t)α(µ, t). The relations (8), (9) and (10) specify completely the evolution of the scattering data of the operator L(t); this allows inverse scattering method to find solution of the system (1) under initial conditions (2). References [1] Toda M. Waves in nonlinear lattice. Progr Theoret Phys Suppl 1970; 45: 74-200. [2] Flashka H. On the Toda lattice. II Progr Theoret Phys 1974; 51: 703-716. [3] Muto V., Scott A.C., Christiansen P.L. Thermally generated solitons in a Toda lattice model of DNA. Physics Letters A 136, 1989; 33-36. 3
Современные проблемы дифференциалных уравнений и смежных раз..., Фергана-2020 4 [4] Lou S.Y., Tang X.Y. Method of Nonlinear Mathematical Physics. Beijing: Science Press 2006; 116-120. [5] Manakov S. V. Complete integrability and stochastization of discrete dynamical systems. Zh. Eksper Teoret Fiz 1974; 67: 543-555. [6] Cabada A., Urazboev G.U. Integration of the Toda lattice with an integral-type source. Inverse Problems 2010; 26: 085004 (12pp). [7] Liu X., Zeng Y. On the Toda lattice equation with self-consistent sources. J. Phys A: Math Gen 2005; 38: 8951-65. [8] Babajanov B.A., Fechkan M., Urazbaev G.U. On the periodic Toda Lattice with self- consistent source. Communications in Nonlinear Science and Numerical Simulation 2015; 22: 379-388.
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