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Solitons in Nonlinear Schrödinger Equations Universidade de São Paulo Biblioteca Digital da Produção Intelectual - BDPI Departamento de Física Aplicada - IF/FAP Artigos e Materiais de Revistas Científicas - IF/FAP 2013-07-26 Solitons in nonlinear Schrödinger equations. Publicação do Instituto de Física, São Paulo, n.1678, p.1-16, 2013. http://www.producao.usp.br/handle/BDPI/44645 Downloaded from: Biblioteca Digital da Produção Intelectual - BDPI, Universidade de São Paulo SOLITONS IN NONLINEAR SCHRÖDINGER EQUATIONS Instituto de Física, Universidade de São Paulo, CP 66.318 05315-970, São Paulo, SP, Brasil M.Cattani1 and J.M.F.Bassalo Publicação IF 1678 26/07/2013 SOLITONS IN NONLINEAR SCHRÖDINGER EQUATIONS M.Cattani1 and J.M.F.Bassalo2 1Instituto de Física, Universidade de São Paulo, São Paulo, SP, Brasil [email protected] 2Fundação Minerva, Belém, Pará, Brasil [email protected] Abstract. Using a mathematical approach accessible to graduate students of physics and engineering, we show how solitons are solutions of nonlinear Schrödinger equations. Are also given references about the history of solitons in general, their fundamental properties and how they have found applications in optics and fiber-optic communications. (1)Introduction. In a first approximation and we can say that a soliton is a solitary wave which preserves its shape and velocity when it moves, exactly as a particle does. A soliton is a solitary wave, i.e. a localized wave, with spectacular stability properties.1-3 The first observation of this kind of wave was done in 1834 in water channels by the engineer John Scott Russel.1,3,4 Only at 1895 a theory developed by Korteweg and de Vries1,3,4 was able to explain the fascinating behavior of the hydrodynamic soliton observed by Russel. This amazing phenomenon was forgotten until a numerical experiment carried in 1953 by Fermi, Pasta and Ulam3 using computers that appeared to contradict thermodynamics. Only ten years later this effect was explained by Zabusky and Kruskal5 taking into account solitary waves that they named solitons. The study of Zabusky and Kruskal5 is a landmark in the history of solitons. After this physicists noticed that solitons are solutions of nonlinear equations. Before, theoretical approaches were trying to avoid nonlinearities or to treat them as perturbations of linear theories. The 19th.century and the first half of the 20th.century can be viewed3 as the triumph of the linear physics (like Maxwell´s equations and quantum mechanics) based on a linear formalism emphasizing a superposition principle. This picture was dramatically changed after the discovery of solitons from the mathematical and physical points of view. The body of knowledge1 that is presently associated with the term “soliton” is enormously broad involving several significant fields with no previous contact with each other. Today, the scientific community gravitating around “soliton equations” (or integrable dynamical systems) includes, on the one hand, nonlinear-optics engineers, astrophysicists, 1 theoretical biologists, oceanographers and, on the other hand, pure and applied mathematicians in algebra, geometry and functional analysis. According to Degasperis1 the formation of the underlying concepts took place independently, in physics and in mathematics, and the discovery of solitons may be compared to the opening of the “Pandora´s box”. Strictly speaking, however, the term “soliton” indicates, in general, a peculiar solitary wave whose propagation is modeled by a nonlinear equation and whose space profile is such that the nonlinearity and the dispersion or the diffraction effects of the medium balance each other. It is a spatially localized wave with spectacular stability properties. The name soliton sounds like the name of a particle. It is a wave but moves exactly as a particle does; it is a solution of a classical field equation which simultaneously exhibits wave and quasi-particle properties.3 These are features that one would expect from quantum systems and not from a classical one. The quantum analogue goes so far that soliton tunneling has been found.6 There are different kinds of solitons which are solutions of different nonlinear equations like,3 for instance, of Kortweg-de Vries (KdV) equation, sine-Gordon equation and nonlinear Schrödinger (NLS) equation. In a recent paper, written to graduate students of physics and engineering, we have shown how to obtain the hydrodynamic KDV solitons. In a preceding paper7 we have studied the existence and stability of Gaussian solitons in 1-dim nonlinear Schrödinger equation In Section 1 we obtain solitons that are solutions of the 1-dim nonlinear Schrödinger equation (NLS) with no external potential. In Section 2 we study the 1-dim motion of a free particle with mass m which obeys a NSL equation. In Section 3 are analyzed the optical solitons2,3,8 (spatial and temporal solitons) that are predicted by 1 and 2-dim NLS equations assuming the Kerr nonlinearity for the optical medium. 1) Solitons of 1-dim NLS equation. Let us consider the 1-dim nonlinear differential equation given by3 i∂ψ/∂t + P(∂2ψ/∂x2) + Q |ψ|2ψ = 0 (1.1), where t is the time, x is the coordinate along the x-axes, P and Q are coefficients that depend on the particular problem which is being analyzed. This equation appears very similar to the Schrödinger equation (SE) if we write it as i∂ψ/∂t = [-P(∂2/∂x2) - Q |ψ|2]ψ = 0 (1.2), and is formally analogous to the SE if P > 0. If P < 0 we take the complex conjugate of (1.2) obtaining an equation for ψ* in which the coefficient of 2 (∂2ψ*/∂x2) is positive. So, without any restriction we can assume P > 0 in (1.1) or (1.2). Note that the complex conjugate transformation change the signs of P and Q, that is, P → -P and Q→ -Q so that it does not affect the sign of the product PQ. This invariance, as will be seen, is of fundamental importance to determine the nature of the solutions of (1.1). The potential function of the SE is here equal to the nonlinear term -Q|ψ|2. As will be shown, when Q > 0 the ψ solution is localized, with a bell shape. Thus, the NLS equation is such that ψ generates its own potential well which, as will be seen, is a necessary condition for the existence of a solution named spatially localized solution. In this case the soliton is named bright soliton. This is a “self-trapping” phenomenon which is essential for the physics of systems obeying a NLS equation. Let us look for a solution of (1.1) of the form Ψ(x,t) = ϕ(x,t) exp[iΘ(x,t)] (1.3), where the amplitude ϕ and the phase factor Θ are real functions. If we assume that Θ varies between 0 and 2π we can restrict the search of ϕ only to positive values. Thus, putting (1.3) in (1.1) we get, separating real and imaginary parts 2 3 -ϕΘt + Pϕxx - Pϕ Θx + Qϕ = 0 (1.4) ϕt + PϕΘxx +2PϕxΘx = 0 (1.5). Let us look for a particular wave solution of (1.4) and (1.5) such that ϕ(x,t) = ϕ(x-vet) and Θ(x,t) = Θ(x-vpt) (1.6), where the envelope and the phase propagate, respectively, with velocity ve and vp that can assume different values. Thus, from (1.4) and (1.5) we have 2 3 vpϕΘx + Pϕxx - Pϕ Θx + Qϕ = 0 (1.7) -veϕx + PϕΘxx +2PϕxΘx = 0 (1.8). Multiplying (1.8) by ϕ and integrating we obtain 2 2 -veϕ /2 + Pϕ Θx = C (1.9), where C is a constant. In order to obtain spatially localized solutions of the NLS equation it is necessary to assume that for |x| → ∞ we have ϕ → 0 and Θ → 0. Consequently, from (1.9) with ϕ ≠ 0 we see that C = 0 and 3 Θx = ve/2P (1.10). Integrating (1.10) results Θ = (ve/2P)(x – vpt) + C´ (1.11), where C´ is an integration constant that we impose to be equal to zero by an appropriate choice of the time origin. Putting (1.11) into (1.7) we obtain 2 3 (ve vp/2)ϕ + Pϕx – (ve /4P) ϕ + Q ϕ = 0 (1.12), Multiplying (1.12) by Pϕx we get an expression that can be readily integrated resulting 2 2 (P /2) ϕx + Veff(ϕ) = 0 (1.13), where Veff(ϕ) is a “pseudo-potential” defined by 4 2 2 Veff(ϕ) = (PQ/4) ϕ – (ve - 2vevp) ϕ /8 (1.14), where the constant of integration has again taken equal to zero in order to 2 have a spatially localized solution. Since ϕ is real ϕx ≥ 0. In this way from (1.13) we verify that the “motion of a particle” must be in a ϕ region where Veff(ϕ) ≤ 0. Consequently, after a simple analysis, we see that there are two different functions Veff(ϕ) x ϕ that are shown in Fig.1 (a) and (b) as a function of PQ: (a) PQ > 0 and (b) PQ < 0.3 3 Figure 1.Shapes of Veff(ϕ) x ϕ for (a) PQ > 0 and (b) PQ < 0. The motion of a localized soliton evolves between the points 1 and 2. This soliton, when P > 0 and Q > 0 is known as bright soliton. We see that the “particle motion” governed by (1.13) must occurs only when PQ > 0 and between the points 1 and 2 shown in Fig.1(a). The 4 2 1/2 points 1 and 2 are ϕ1 = 0 and ϕ2 = ϕo = {(ve - 2vevp) /2PQ} , respectively. 2 In this case the amplitude ϕo is finite. We also verify that ve - 2vevp ≥ 0 which does not impose a sign for ve and vp, but shows that ve = vp is not allowed.
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