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2.2 the Nonlinear Schrödinger Equation As an Envelope Equation

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2.2 the Nonlinear Schrödinger Equation As an Envelope Equation 20 V. Hakim Another interesting elementary example of application of the multiscale method is the computation of the size of the limit cycle of a Van der Pol oscillator: x¨ + ǫ x2 1 x˙ + x =0 for ǫ 1 (2 .13) − ≪ The zeroth order solution is x0 = A cos( t + φ) (2 .14) With an arbitrary amplitude A. Introducing a slow time scale like in (2.6), the cancellation of secular terms at first order gives the evolution of the amplitude dA A3 2 = A (2 .15) dT − 4 This shows that the amplitude of the limit cycle is equal to 2 and that (2.15) describes the approach to the limit cycle on a time scale that is long compared to the oscillation frequency. At next order, the change of the period of the limit cycle appears 1 ω = 1 ǫ2 (2 .16) − 16 This computation is left as an exercise to the reader. It is described in detail in S. Fauve’s lectures. 2.2 The nonlinear Schr¨odinger equation as an envelope equation for small amplitude gravity waves in deep water 2.2.1 A short introduction to water waves The problem of water waves has played a central role in the development of nonlinear physics, with Stokes’s discovery that dispersion relation in- volves the amplitude for periodic nonlinear wave trains, Russell’s first observation of a solitary wave, the derivation of model equations by Boussinesq, Rayleigh and Korteweg-de Vries . Moreover, the observa- tion of water waves is a common but fascinating experience. A good introduction to the subject can be found in Stoker (1957) and Whitham (1974). We consider the problem in the simplified setting of a two dimensional flow of an inviscid incompressible liquid. We assume that 5. Asymptotic Techniques 21 the flow is irrotational and introduce the potential φ for the fluid velocity v v = φ (2 .17) ∇ The incompressibility of the fluid implies that φ satisfies Laplace’s equation div v = 0 , 2φ = 0 (2 .18) ∇ The potential φ is determined by two boundary conditions. The first one is of dynamic nature. If surface tension is neglected, the pressure in the liquid and the pressure in the air should be equal at the surface. The pressure in the liquid is determined by Bernouilli’s equation, as can be seen by substituting (2.17) into Euler’s equation. Assuming a constant air pressure (i.e., neglecting air density) one obtains at the surface ∂φ 1 + ( φ)2 + gy = 0 (2 .19) ∂t 2 ∇ surface where y denotes the vertical coordinate and all space independent constants have been cancelled by a suitable redefinition of the potential. There is also a second boundary condition at the surface of purely kinematic nature. By the very definition of the surface, the normal velocity of the surface should be equal to the normal velocity of the fluid at the surface: N v = N φ (2 .20) · int · ∇ In the case of a layer of liquid of finite depth, one gets an other kinematic boundary condition from the bottom impermeability. For simplicity, we will only consider the case of waves in deep water and we require simply that the fluid velocity vanishes as y . → −∞ At this stage, some readers may feel that our system of equations is overdetermined since we have imposed two boundary conditions on Laplace’s equation (2.17). This is not the case, because we are considering a free-boundary problem. The surface of the liquid and therefore the shape of the region where one should solve Laplace’s equation is not fixed in advance but it is itself determined in the course of solving the equation. A good way to get an intuitive feeling may be to say that (2.18) together with (2.19) determines φ at time t as a function of the surface and the field at time t ∆t. Then equation (2.20) can be − solved as an equation of motion for the fluid surface and gives the new position of the surface at time t. 22 V. Hakim In order to analyze these equations perturbatively, it is convenient to rewrite them in a less compact but more explicit fashion. We denote by η(x, t ) the vertical position of the water surface at position x and time t. Then the surface normal N and surface velocity vint are: ∂η ∂η N , 1 , v = 0, (2 .21) −∂x int ∂t .. The kinematic boundary condition (2.20) is thus ∂η ∂φ ∂η ∂φ = (2 .22) ∂t − ∂y −∂x ∂x y=η(x) where we have explicitly indicated that this relation should be satisfied at the liquid surface. With the same notation, the other boundary condition (2.19) reads: ∂φ 1 ∂φ 2 ∂φ 2 + g η = + (2 .23) ∂t − 2 " ∂x ∂y # y=η(x) 2.2.2 Weakly nonlinear expansion of periodic waves and secular terms We study water waves as a perturbation of the simplest case, a liquid at rest with an horizontal surface. This case is described with a suitable choice of vertical coordinates by φ0(x, t ) = 0 , η 0(x, t ) = 0 (2 .24) For a slight perturbation, φ and η will be small and the nonlinear terms in the l.h.s. of equations (2.22) and (2.23) can be neglected at first order. Moreover, the boundary conditions can be evaluated at y = 0 instead of y = η(x). One should therefore find a Laplacian field φ1 satisfying the two boundary conditions: ∂η ∂φ 1 1 = 0 ∂t − ∂y (2 .25) ∂φ 1 + g η = 0 ∂t 1 The equations being linear, the motion of an arbitrary surface can be determined from the superposition of the motions of its Fourier components. Therefore, we consider η1 under the form: i(kx −ωt ) η1 = Re A1 e (2 .26) n o 5. Asymptotic Techniques 23 The form of the associated periodic Laplacian field which vanishes as y is: → −∞ i(kx −ωt ) ky φ = Re F1 e e , k > 0 (2 .27) n o Substitution of (2.26) and (2.27) in (2.25) gives the homogeneous linear system iωA 1 + kF 1 = 0 (2 .28) gA iωF = 0 1 − 1 A non trivial solution of this homogeneous linear system exists only if its determinant is zero. That is the frequency is related to the wave number by ω2 kg = 0 (2 .29) − In this case, the liquid flow is related to the wave shape by ω F = i A (2 .30) 1 − k 1 We have thus recovered the well-know linear dispersion relation of gravity waves. Now, we want to analyze the influence of the neglected nonlinear terms and therefore consider η1 and φ1 as first order terms in a systematic expansion of a nonlinear periodic wave, solution of the full equations (2.18), (2.22–23), as Stokes first did in 1847. At this stage it is worth clarifying what the actual expansion parameter is. The acceleration of the gravity g appears explicitly in the equation. If we admit, moreover, that for a nonlinear periodic wavetrain we can impose arbitrarily the wave height a and the wave length 2 π/k then the solution is determined by three dimensional parameters. Out of these, only one dimensionless parameter can be formed, the so-called “wave steepness” ak . Therefore, the weakly nonlinear expansion is really an expansion in powers of the wave steepness. It can be explicitly checked that the wave steepness is the dimensionless parameter which stands in front of the nonlinear terms in eq. (2.22), (2.23) when dimensionless variables are introduced by setting x =x/k ˜ , y =y/k ˜ , t = τ/ √kg , η = aη˜ and φ = a g/k φ˜. Without finding it necessary to do so, we can now proceed and compute the nonlinear corrections to η and φ . We formally expand p 1 1 φ and η as: 2 3 η = ǫη 1 + ǫ η2 + ǫ η3 + . 2 3 (2 .31) φ = ǫφ 1 + ǫ φ2 + ǫ φ3 + . where we have introduced a formal expansion parameter ǫ which can be set to one at the end of the computation to remind us that terms of order 24 V. Hakim n are proportional to ( ak )n, as pointed out above. Substituting (2.31) in equations (2.22, 2.23) one gets at second order ∂η ∂φ ∂η ∂φ ∂2φ 2 2 = 1 1 + 1 η ∂t − ∂y − ∂x ∂x ∂y 2 1 y=0 (2 .32) ∂φ 1 ∂φ 2 ∂φ 2 ∂2φ gη + 2 = 1 + 1 1 η 2 ∂t −2 ∂x ∂y −∂y∂t 1 " # y=0 The underlined terms in the l.h.s. of these equations come from the fact that the boundary conditions are evaluated at y = 0 instead of y = η(x). Using the expressions (2.26, 2.27) and of φ1 and η1, the l.h.s. of (2.32) can be computed and one obtains ∂η 2 ∂φ 2 2 2i(kx −ωt ) = Re iωkA 1 e ∂t − ∂y y=0 − n o (2 .33) ∂φ − gη + 2 = Re 1 ω2A2 e2i(kx ωt ) 2 ∂t 2 1 n o One can seek for η2 and φ2 under the form 2i(kx −ωt ) 2i(kx −ωt ) 2ky η2 = Re A2 e , φ 2 = Re F2 e e . (2 .34) n o n o A2 and F2 are easily determined by substitution of (2.34) into (2.33) 1 2 A2 = 2 k A 1 , F 2 = 0 (2 .35) Therefore, nothing special happens at second order. Things change at third order. The equations for φ3 and η3 are found as before by substitution of the formal expansions (2.31) into equations (2.22, 2.23).
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