On a General Class of Solutions of a Nonholonomic Extension of Optical Pulse Equation

Transcription

1 EJTP 8, No. 25 (2011) Electronic Journal of Theoretical Physics On a General Class of Solutions of a Nonholonomic Extension of Optical Pulse Equation Pinaki Patra, Arindam Chakraborty 1, and A. Roy Chowdhury 2 1 Department of Physics, Swami Vivekananda Institute of Science and Technology, Govindapur, Kolkata , India 2 Department of Physics, Jadavpur University, Kolkata , India Received 23 November 2010, Accepted 10 February 2011, Published 25 May 2011 Abstract: A Nonholonomic extension of an equation obeyed by short pulse in non-linear optics is obtained.a general class of solutions of such an equation is obtained with the help of Riemann- Hilbert technique. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Integrable System; Nonholonomic; Lax Pair; Reimann Hilbert PACS (2010): k; Pq; Ac; a 1. Introduction Recent literature of integrable systems has been enriched by quite a few publications discussing a new class of systems which are nonholonomic in the sense that they are accompanied by differential constraints [1] those cannot be explicitly solved. This actually enlarges the integrable class itself. People have studied some interesting cases associated with AKNS [2] systems Kaup-Newell equation and so on. Here in this communication we have obtained a nonholonomic generalization of a very special type of equation which governs the propagation of a short optical pulse in nonlinear optics [3]. In the second part we have shown how a general class of solutions of such an equation can be generated through the Riemann Hilbert method [6]. (Corresponding author) asesh r@yahoo.com

2 274 Electronic Journal of Theoretical Physics 8, No. 25 (2011) Formulation To start with, we consider the Lax pair L = φ x 1 λ φ x ; N = A B C A whence consistency leads to Next we set ψ x = Lψ (1) ψ t = Mψ (2) φ xt + A x C λb =0 (3) B x +2A +2φ x B =0 (4) C x +2λA 2φ x C =0 (5) A = a 1 λ 1 + a 2 + a 1 λ + a 0 λ 2 (6) B = b 1 λ 1 + b 2 + b 1 λ + b 0 λ 2 (7) C = c 1 λ 1 + c 2 + c 1 λ + c 0 λ 2 (8) The unknown coefficients are determined in a recursive manner and are given as a 0 = b 0 =0 c 0 = η 2 a 1 = η 2 φ x (9) b 1 = η 2 c 1 = 1 2 η ( ) 2 φxx + φ 2 x a 2 = 1 4 η ( ) 2 φxxx 2φ 3 x b 2 = 1 2 η ( ) 2 φxx φ 2 x c 1 = η 1 exp (2φ) (10) where η 1 and η 2 are constants. Along with ; and the nonlinear equation ; a 1x = η 1 exp (2φ) (11) b 1xx +2φ xx b 1 +2φ x b 1x +2η 1 exp (2φ) = 0 (12) c 2xx 2φ xx c 2 2φ x c 2x +2η 1 exp (2φ) = 0 (13) φ xt + a 2x c 2 b 1 = 0 (14)

3 Electronic Journal of Theoretical Physics 8, No. 25 (2011) It is interesting to note that c 2, b 1 are to be determined only as solutions of eq(8) and eq(9), which really leads to nonlocal nonholonomic constraints. To proceed further we consider the special case η 1 = 0, which leads to a 1 = constant ; along with c 2 = η 3 exp (2φ) andb 1 = η 4 exp ( 2φ) so that the nonlinear equations turns out to be ; φ xt = φ xxxx 6φ 2 xφ xx +2η sinh (2φ). (15) by proper choice of η i. Equation (11) is nothing but the nonlinear equation obeyed by a short optical pulse propagating in an optical fibre. So, if we assume η 1 0, then eq(10) can be consider as a nonholonomic generalization of eq (11) General Solutions To analyze the solutions of the new equation [14] we take recourse to the Riemann- Hilbert technique [4]. The usual approach is to assume that the analytic wave function φ(λ), with the constraint φ 1 (λ) φ 2 (λ) =G (λ) (16) where φ 1, φ 2 are the boundary values of φ on the interior and exterior of a close Jordan curve Γ in the complex λ plane. In case of Riemann - Hilbert problem with simple poles we assume G (λ) =1. Letφ 0 be the starting seed solution of eq [14] with ψ 0,the corresponding Lax eigenfunction [5]. We can set ψ = χψ 0 (17) If the lax operator corresponding to φ 0 be denoted as L 0,thenonegets L=χ x χ 1 + χl 0 χ 1 (18) To formulate the Riemann-Hilbert problem we assume that χ contains simple pole in λ - plane, ( χ = 1+ S ) (19) λ λ 1 along with χ 1 = ( 1+ R ) λ λ 1 (20) From the condition χχ 1 =1onegets S= R =(λ 1 μ 1 ) P (21) where P is the projection operator (P 2 = P ). Also from Eq.[18] we get L=L 0 (λ 1 μ 1 )[P, σ ] (22)

4 276 Electronic Journal of Theoretical Physics 8, No. 25 (2011) The Lax operator has some interesting symmetry property σ 1 L T σ 1 = L( φ, λ) (23) σ 2 L T σ 2 = L (φ, λ) (24) with σ 1 σ 2 being Pauli matrices and σ = σ 1 iσ 2. An easy seed solution is φ =0, whence we require the Lax eigenfunction as solution of ; ψ x = 0 1 ψ (25) λ 0 and ψ t = η 1 xλ 1 η 1 λ 1 η 1 x 2 + η 2 λ 2 η 1 x 2 λ 1 η 2 λ ψ (26) η 1 xλ 1 writing out in component form we get one partial differential equation ψ 2t + ( ηx 2 + ηλ 1 λ 2) ψ 2x λ = ηxλ 1 ψ 2 (27) Now it is easy to show that an equation of the form ; has a solution of the form f(x)ω x + g(y)ω y =[δ 1 (x)+δ 2 (y)]ω (28) [ ] ( δ1 (x) ω = exp f (x) dx + δ2 (y) g (y) dy Θ 1 f (x) dx ) 1 g (y) dy (29) where Θ is an arbitrary function of its argument. Finally using the explicit of function δ 1 and δ 2 from Eqs.[26], we arrive at ψ 1 = ψ 2 = μ(θ) λ ηλ 1 x 2 (30) ηλ 1 xμ(θ) λ ηλ 1 x + μ x (θ) (31) 2 λ ηλ 1 x 2 where θ = t + 1 2aηλ 1 constructed as, ln ( ) x+a x a, along with a 2 = λ3. η P= p><q <p q> The Projection operator is now (32)

5 Electronic Journal of Theoretical Physics 8, No. 25 (2011) where p>and q>are given by; p>= a 1ψ 1 (λ 1 ) = u 1 (33) a 2 ψ 2 (λ 1 ) u 2 q>= b 1ψ 1 (μ 1 ) = v 1 (34) b 2 ψ 2 (μ 1 ) where Eq.(22) leads to u 1 v 2 φ x =(λ 1 μ 1 ) (35) u 1 v 1 + u 2 v 2 The form of the solution is clearly different from the usual soliton like profile sustained by original optical pulse equation. The absence of any wavefront (x vt) suggests that it is a nonpropagating solution. v 2 Acknowledgement One of the author (P.P) is grateful to CSIR(Govt. Fellowship which made the work possible. of India) for a Junior Research References [1] A. Karasu, A.Karasu, A. Sakovich, S. Sakovich and R. Turhan - J. Math. Phys. 51, (2010) [2] Ruguang Zhoua, J. Math. Phys. 50, , 2009 [3] Leblond H, Melnikov I, V Mihalache D - Phys. Rev. Lett. A 78(2008) [4] A. Chakraborty and A. Roy Chowdhury - EJTP. No.24(2010)1 8 [5] Boris A. Kupershmidt, Phys. Lett. A 372, 2634 (2008) [6] R. G. Zhou, J. Math. Phys. 36, 4220 (1995)

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