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 observation 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). One gets (with φ2 = 0):

∂η ∂φ ∂η ∂φ ∂η ∂2φ 3 3 = 2 2 1 1 ∂t − ∂y − ∂x ∂x − ∂x ∂x∂y ∂2φ ∂3φ + 1 η + 1 1 η2 ∂y 2 2 2 ∂y 3 1 2 2 (2 .36) ∂φ 3 1 ∂φ 1 ∂ φ2 ∂φ 1 ∂ φ1 gη 3 + = 2 + 2 η1 ∂t −2 ∂x ∂x ∂y ∂y ∂y 2 ∂2φ 1 ∂3φ 1 η 1 η2 − ∂t ∂y 2 − 2 ∂t∂ 2y 1 All terms on the l.h.s. except the ﬁrst one in the ﬁrst equation come from evaluating the boundary conditions at y = 0 instead of y = η(x). 5. Asymptotic Techniques 25

Evaluation of the l.h.s. proceeds as before, by substituting the previously found expressions for φ1, η1 and η2 (eq. 2.30, 2.35). After multiplication 2 ∗ 3 of the exponentials, one obtains terms proportional to A1A1 and A1 and their complex conjugates:

∂η ∂φ 5 ∗ − 3 3 = ωk 2 Re iA 2A ei(kx ωt ) ∂t − ∂y 8 − 1 1 n o 9 2 3 3i(kx −ωt ) + ωk Re iA 1 e 8 − (2 .37) ∂φ 3 n ∗ − o gη + 3 = ω2k Re A2A ei(kx ωt ) 3 ∂t −8 1 1 3 n − o + ω2k Re A3 e3i(kx ωt ) 8 1 n o The terms proportional to exp 3i(kx ωt ) on the l.h.s. do not pose − any special problems and give contributions in φ and η which are 3 3 uniformly small compared to previous terms in the expansion. On the contrary terms proportional to exp i(kx ωt ) have the same functional − form as first-order terms and are a potential problem. If one tries to determine their contribution to φ3 and η3 under the same form as above (eqs. (2.26–27)), the linear inhomogeneous system has a determinant equal to zero (2.29) and can only be solved if the inhomogeneous term satisfies a particular relation. In general, the presence of this resonant inhomogeneous term produces contributions in η3 and φ3 which grow linearly as compared to η1 or φ1, in the time or spatial domain. An alternative way to determine the resonant contribution is to combine equations (2.37) into a single equation. Taking the time derivative of the second one and subtracting g times the first one, one gets:

2 ∂ φ ∂φ ∗ − 3 + g 3 = ω3k Re iA 2A ei(kx ωt ) ∂t 2 ∂y 1 1 n o 3 − + ω3k Re iA 3 e3i(kx ωt ) (2 .38) 4 1 n o If one looks for a solution φ3 which is uniformly small in space which respect to φ1, then the contribution to φ3 of the resonant ﬁrst term is:

1 2 2 ∗ i(kx −ωt ) [φ3]r = 2 ω k tA 1A1 e (2 .39)

Which grows linearly in time and become larger than the ﬁrst order contribution. Therefore, the expansion breaks down in the long-time limit. 26 V. Hakim

2.2.3 Stokes’s nonlinear dispersion relation

Stokes’s idea was that the appearance of the secular term (2.39) was simply a reﬂection of a small frequency change due to nonlinear interactions. The diﬀerence between the unperturbed and perturbed frequency gives rise to a long-time scale and we can exploit Stokes’s idea by using the multiscale method. We introduce a long-time scale T = ǫ2t and write φ and η as if they depended separately on t and T

2 η(x, t ) = ǫη 1(x, t, T ) + ǫ η2(x, t, T ) + . . . 2 (2 .40) φ(x, y, t ) = ǫφ 1(x, y, t, T ) + ǫ φ2(x, y, t, T ) + . . .

Here, it is to be understood that one should make the substitution ∂ ∂ ∂ + ǫ2 (2 .41) ∂t −→ ∂t ∂T in every time differentiation. At first and second order, nothing changes except that all appearing constants depend now on the long time. The interesting result appears when we collect the third order terms. Supplementary terms appear from differentiation with respect to the slow variable:

∂η ∂φ ∂A − 3 3 = Re 1 ei(kx ωt ) ∂t − ∂y old − ∂T h i (2 .42) ∂φ 3 ∂F 1 i(kx −ωt ) g η 3 + = Re e ∂y old − ∂T h i As before, the two boundary conditions can be combined into a single one for φ3 (see eq. (2.38))

2 ∂ φ ∂φ ∗ − 3 + g 3 = ω3 k Re iA 2A ei(kx ωt ) ∂t 2 ∂y 1 1 2n o ω ∂A − (2 .43) + 2 Re 1 ei(kx ωt ) k ∂T + non-secular terms where we have used the relation (2.30) between F1 and A1 and the linear dispersion relation. To avoid linearly growing terms in φ3, we can now set the coeﬃcients of exp i(kx ωt ) and exp i(kx ωt ) separately − − − to zero. This is a constraint that must be satisﬁed at all values of T ∂A ∗ 1 = i 1 k2ωA 2A (2 .44) ∂T − 2 1 1 5. Asymptotic Techniques 27

As anticipated A1 evolves on a time scale which is longer than the basic time scale 2 π/ω by the inverse square wave steepness. This is Stokes’s celebrated result for the nonlinear dispersion relation

1 2 2 ω . . = kg 1 + k A + . . . (2 .45) N L 2 | 1| p 2.2.4 Evolution of slowly modulated wavetrains

In the previous subsection, we considered a perfectly periodic wavetrain. It is also very interesting to consider the evolution of wave packets and the interaction between dispersion and nonlinearity. A wave packet centered around wave number k0 can be represented as:

− w(x, t ) = A(x, t ) ei(k0x ω0t) (2 .46)

We want to determine the time evolution of A(x, t ) when A(x, 0) is a slowly varying function of x. For convenience, we have explicitly extracted the linear pulsation ω0 = √k0g. We begin with a simple approach and ﬁrst consider the linear evolution of A(x, t ). We expand A(x, 0) in Fourier series and write the initial condition as

i(k0+k)x w(x, 0) = Ak e (2 .47) k X where k is small compared to k0. The form of the wave packet at all times is given by the linear dispersion relation

i[( k0+k)x−ω(k0+k)t w(x, t ) = Ak e . (2 .48) k X

Since k is small compared to k0, we can expand the dispersion relation in the neighborhood of k0

′ 1 2 ′′ ω(k0 + k) = ω0 + kω (k0) + 2 k ω (k0) + . . . (2 .49)

Substituting this relation into (2.48) and comparing with (2.46), one obtains the linear evolution of A(x, t )

′ k2 ′′ i kx − ω (k0)k+ 2 ω (k0)+ ... t A(x, t ) = Ak e . (2 .50) k X 28 V. Hakim

We thus find that A(x, t ) satisfies the equation 2 ∂A ′ ∂A i ′′ ∂ A + ω (k0) = + ω (k0) + . . . (2 .51) ∂t ∂x 2 ∂x 2 For a slowly varying amplitude ( A(x) = f(ǫx )), this equation reduces at first order to its first two terms, i.e., one obtains the well-known result ′ that the wave-packet moves at the group velocity ω (k0). At next order, in a reference frame which moves at the group velocity, one sees the dispersion of the wave packet 2 ∂A i ′′ ∂ A = ω (k0) (2 .52) ∂t 2 ∂x 2 The combined effects of dispersion and nonlinearity are simply described by adding the r.h.s. of (2.44) and (2.52). One obtains the so-called “nonlinear Schrödinger” equation: 2 ∂A i ′′ ∂ A i 2 2 2 = ω (k0) k ω A A . (2 .53) ∂t 2 ∂x 2 − 2 0 0| | This is a well-known equation which appears in a variety of contexts and which possesses many interesting properties, as we will shortly discuss in the next section. Before doing that we want to show briefly how (2.53) is more formally derived by the multiscale method. Again, the strategy is to separate explicitly the slow variations from the fast ones. We therefore introduce “long” space variables X and Y , and two slow time variables T1 and T2 since dispersion occurs on a longer time-scale than motion of the wave packet, as seen above:

2 X = ǫx, Y = ǫy, T 1 = ǫt , and T2 = ǫ t . (2 .54) For simplicity, we focus on the velocity potential φ and therefore combine the two boundary conditions (2.22), (2.23) into a single one for φ, as explained above: ∂2φ ∂φ ∂η ∂φ ∂2φ ∂η ∂φ ∂2φ + g = g ∂t 2 ∂y ∂x ∂x − ∂t ∂y ∂t − ∂x ∂x ∂t (2 .55) ∂φ ∂2φ ∂φ ∂2φ ∂η ∂φ ∂2φ ∂η − ∂y ∂y ∂t − ∂x ∂x ∂y ∂t − ∂y ∂y 2 ∂t y=η(x)

where η should be understood as being expressed in function of φ using boundary condition (2.23)

1 ∂φ 1 ∂φ 2 ∂φ 2 η = + + (2 .56) −g ∂t 2 ∂x ∂y " !# η(x) 5. Asymptotic Techniques 29

In Laplace’s equation and equations (2.55), (2.56) we then make the substitutions

∂ ∂ ∂ ∂ ∂ ∂ + ǫ , + ǫ , ∂x −→ ∂x ∂X ∂y −→ ∂y ∂Y (2 .57) ∂ ∂ ∂ ∂ + ǫ + ǫ2 . ∂t −→ ∂t ∂T 1 ∂T 2

At ﬁrst order, one obtains again eq. (2.30) but now A1 and F1 depend on the slow variables ω F1(X, 0, T 1, T 2) = i A1(X, T 1, T 2) . (2 .58) − k0

At second order, supplementary terms appear in Laplace’s equation

∂2φ ∂2φ 2φ = 2 1 2 1 ∇ 2 − ∂X ∂x − ∂Y ∂x (2 .59) ∂F ∂F − = 2k Re i 1 + 1 ei(k0x ω0t) ek0y − 0 ∂X ∂Y

Secular terms do not appear in φ2 if

∂F ∂F 1 = i 1 , (2 .60) ∂Y − ∂X that is if F is a function of X iY . New terms also appear in the − boundary condition for φ2

2 2 ∂ φ2 ∂φ 2 ∂ φ1 ∂φ 1 2 + g = 2 g ∂t ∂y old − ∂t ∂T 1 − ∂Y h i ∂F 1 ∂F 1 i(k0x−ω0t) = Re 2iω 0 + g e old − − ∂T ∂Y 1 h i (2 .61) The requirement that secular terms do not appear reduces to set the new contribution to zero. This gives back that the wave packet propagates at the group velocity

∂A g ∂A 1 + 1 = 0 (2 .62) ∂T 2ω ∂X 1 0 Y =0

where we have utilized relation (2.60) between X and Y derivatives of

F1 and replaced F1 Y =0 by A1 (eq. 2.58).

30 V. Hakim

At third order, no secular terms appear in Laplace’s equation since F1 satisﬁes eq. (2.60) and φ2 = 0,

2 2 2 ∂ ∂ φ3 = + φ1 = 0 (2 .63) ∇ − ∂X 2 ∂Y 2

New contributions appear in the boundary condition for φ3

2 2 2 ∂ φ3 ∂φ 3 ∂ φ1 ∂ φ1 2 + g = 2 2 + non-secular terms (2 .64) ∂t ∂y old − ∂t ∂T 2 − ∂T 1 h i We have only explicitly written the new secular contributions. Secular contributions are of two types: – They involve a single exponential and diﬀerentiation with respect to the slow variables. The only such terms are the second and third term on the l.h.s. of (2.64). – They involve the product of three exponential and cannot involve diﬀerentiation with respect to the slow variables. These term have already been computed (l.h.s. of eq. (2.38). They are denoted by [ ] old in the above equation. Setting the secular contribution to zero gives 2 3 2 ∗ ∂F 1 ∂ F1 iω kA 1A1 + 2 iω 2 2 = 0 (2 .65) ∂T 2 − ∂T 1

After replacing F1 by A1 using eq. (2.59) and replacing T1- derivatives by X-derivatives (eq. (2.63)), one recovers the nonlinear Schr¨odinger equation:

2 2 ∂A 1 ω0 ∂ A1 ω0k0 2 ∗ = i 2 2 i A1A1 (2 .66) ∂T 2 − 8k0 ∂X − 2

2.2.5 Miscellaneous remarks

In the previous subsection, we have described Stokes’ result for weakly nonlinear periodic wavetrains and a generalization for slowly modulated wavetrains. Some remarks may be needed to put them in a proper perspective (see Stoker, 1957; Benjamin, 1974; Whitham, 1974). Stokes’s investigation started a mathematical controversy about the actual existence of nonlinear periodic gravity waves with a continuum ratio of amplitude over wavelength. This culminated in Levi-Civita’s (1925) rigorous constructive proof of existence of periodic waves of small 5. Asymptotic Techniques 31 amplitude. The restriction to small amplitude was then removed in the fifties when it was shown that for every value of the wave maximum slope between 0 and 1/3 (i.e., for sup ∂η/∂X ]0 , 1 [ ) a steady progressive | | ∈ 3 wave train exists. #1 After these difficult mathematical proofs, Benjamin and Feir’s (1967) discovery comes as a bad surprise. They showed that the strictly periodic wavetrains are unstable. This is very simply derived, at least for weak amplitude, by using the nonlinear Schrödinger equation. Writing it under the form: 2 ∂A ∂ A ∗ = iγ iβA 2A (2 .67) ∂t ∂x 2 − it is easily checked that the solution:

A = a exp( iβa 2t) (2 .68) − is unstable to x dependent disturbances when βγ < 0 (see Fauve’s Chapter). The later development of the instability is to break the wavetrain into a number of pulses which are described by solitary waves of the nonlinear Schrödinger equation. These solitary waves have been called solitons because they show many remarkable properties associated with the complete integrability of the NLS equation. This means that the integration of this one-dimensional nonlinear equation can be reduced to the solution of linear equations. This beautiful mathematical development is well explained in a number of recent books (for example Ablowitz and Clarkson, 1991; Novikov et al. ,1984) to which we refer the interested reader. The nonlinear Schrödinger equation appears in different physical contexts, like plasma physics and nonlinear optics, since it simply describes the interaction of dispersion and weak nonlinearity, as shown by the simple but general derivation above. Due to its use in nonlinear optics, the case βγ < 0 is sometimes called “focusing” in the literature (vs. defocusing for βγ > 0). In higher dimension, the NLS equation produce singularities in finite time in the focusing case. This is shown by an ingenious argument of

#1 The value 1/3 can be understood from another Stokes’ result which states that when a sharpcrest is attained in a steady state proﬁle the angle ◦ there must 120 . This is simply derived by a local analysis of the equation in the neighborhood of the wavecrest (see Whitham, 1974) 32 V. Hakim

Zakharov (1972) for spherically symmetric solution. Normalizing the equation by taking γ = 1, β = 1, in two dimensions two conserved − quantities are:

C = r2 A 2 dr . (2 .71) | | Z

One ﬁnds: ∞ d2C = 6 I 4 A 4 < 6I (2 .72) dt 2 2 − | | 2 Z0 Therefore for a negative I2, C goes to zero at a ﬁnite time t0. Since A cannot vanish everywhere by conservation of I1, it is clear that a singularity should be produced for t t . The formation of these ≤ 0 singularities has been the subject of recent analyzes (for example Le Mesurier et al. , 1988; Zakharov et al. , 1989; Merle, 1992).

2.3 Amplitude equation from a more general viewpoint

In the previous subsection, we have derived the NLS equation as an envelope equation for gravity waves. It is a special case of an amplitude equation for a conservative system. In the more common case where dissipation exists, the usual amplitude equation is the so-called “complex Ginzburg–Landau” equation (Newell, 1974):

∂A ∂2A = µA + α A 2A + β (2 .73) ∂t | | ∂x 2 where µ, α and β are complex coefficients which are not purely imaginary. Their physical meaning is as follows. A tends to increase or decrease depending on the balance between dissipation and injected energy. For Re µ < 0 the trivial solution A = 0 is stable. On the contrary { } for Re µ > 0 a small A tends to grow. The real part of β reflects { } the dependence of the instability growth rate on wavelength while the imaginary part comes from dispersion as above. The imaginary part of α describes the variation of the basic frequency with the amplitude of A, 5. Asymptotic Techniques 33 while its real part reflects the dependence of dissipation on the amplitude of A. If Re α < 0, dissipation increases when A grows and nonlineari- { } | | ties tend to saturate the instability. This is referred to as a “normal” or “supercritical” instability. Near the instability threshold (0 < µ 1), ≪ the restabilised value of A is small and one can perform controlled | | weakly nonlinear expansions. A typical example is given by Rayleigh- Bénard convection. In the opposite case, Re α > 0 nonlinearities de- { } creases the dissipation for small A amplitude and A saturates for val- | | ues of order unity which are not controlled by the distance to threshold. Weakly nonlinear expansions are therefore much less informative since high order terms are as important as low order ones. This is commonly referred to as a “subcritical” instability or as an “inverted” bifurcation in the literature. Examples include plane Poiseuille flow (Stewartson and Stuart, 1971) and directional solidification in metals (Wollkind and Segel, 1970; Müller-Krumbhaar and Kurz, 1990). Amplitude equations seem to have been used in three rather distinct manners in the literature. Our derivation of the non-linear Schrödinger equation is an illustration of the first one. One start with well defined equations and derives amplitude equation using weakly non linear expansion. This is how envelope equations have been introduced for Rayleigh-Bénard convection (Segel, 1969; Newell and Whitehead, 1969) and it has since been useful in the study of many different instabilities. It is worth pointing out that the rigorous mathematical derivation of amplitude equations is a subject of current interest (see, for example, Kirchgässner, 1988; Collet and Eckmann, 1990; Craig et al. , 1992). A second direction that has been followed, is to classify the possible modes of instabilities and the different types of amplitude equations. Usually, this is done by using the various possible symmetries of the system. Of course, the particular coefficients characterizing a definite physical situation cannot be obtained but similar phenomena appearing in different systems can be described in a synthetic way. This is well illustrated by the recent work of Coullet and Iooss (1990) on possible instabilities of one dimensional cellular patterns and a nice application to traveling waves is described by Coullet et al. (1989). This strategy is described in detail in S. Fauve’s lectures. Finally, amplitude equations are used in a much more radical way in the literature, in a spirit close to the original Landau proposal and to the use of Landau–Ginzburg free energies in the description of phase 34 V. Hakim transitions. The idea is that simple amplitude equations describe the dynamics of the system order parameter. For example A = 0 would characterize a laminar flow and A = 1 a turbulent one. One imagines that it would in principle be possible to obtain a coarsened description involving A only. One then argues that simple equation like (2.74) are rich enough to describe the main phenomena. This is analogous to the description of transition to chaos using one-dimensional maps although the case is much weaker at present. Example of this strategy are provided by Pomeau (1986) or Coullet and Lega (1988) and an application to spiral turbulence is described in Andereck et al. (1989). This viewpoint gives much interest to the analysis of simple structures in Landau–Ginzburg equations, like fronts and localized states which are the subject of the forthcoming section.

3 Fronts and localized states

In this chapter, we use the simplified setting of Landau–Ginzburg like equations to investigate the spatial coexistence of different states. We begin by considering a system which possess two linearly stable homogeneous states. We discuss their relative stability by considering a situation where the system is in the different stable states at different spatial locations. Then, an interface or front separates the two states. We show that this front generally moves so that one state, which is by definition the more stable, invades the other one. We then discuss the interaction between two fronts and find that in some cases stable localized inclusion of the less stable phase can exist in the more stable phase. After treating this bistable situation which usually occurs in the vicinity of a subcritical bifurcation, we turn to the case of a normal bifurcation. Specifically, we consider the invasion of an unstable state by a stable state. This turns out to be a subtle problem the basic features of which we discuss along simple lines.