Nonlinear Schrödinger equation with a white-noise potential: Phase-space approach to spread and singularity

Transcription

1 Physica D 212 (2005) Nonlinear Schröinger equation with a white-noise potential: Phase-space approach to sprea an singularity Albert C. Fannjiang Department of Mathematics, University of California, Davis, CA , Unite States Receive 9 August 2004; receive in revise form 26 September 2005; accepte6 September 2005 Available online 15 November 2005 Communicate by C.K.R.T. Jones Abstract We propose a phase-space formulation for the nonlinear Schröinger equation with a white-noise potential in orer to she light on two issues: the rate of sprea an the singularity formation in the average sense. Our main tools are the energy law an the variance ientity. The metho is completely elementary. For the problem of wave sprea, we show that the ensemble-average ispersion in the critical or efocusing case follows the cubic-in-time law while in the supercritical an subcritical focusing cases the cubic law becomes an upper an lower boun respectively. We have also foun that in the critical an supercritical focusing cases the presence of a white-noise ranom potential results in ifferent conitions for singularity-with-positive-probability from the homogeneous case but oes not prevent singularity formation. We show that in the supercritical focusing case the ensemble-average self-interaction energy an the momentum variance can excee any fixe level in a finite time with positive probability. c 2005 Elsevier B.V. All rights reserve. Keywors: Nonlinear Schröinger equation; Ranom potential; Wave sprea; Blow-up 1. Introuction High-intensity laser beams propagating in ielectric meia such as optical fibers, films or air are important problems both from funamental an practical perspectives. The physical process is nonlinear an the amplitue moulation Ψ is customarily escribe by a nonlinear Schröinger equation with an aitional inhomogeneous potential V representing the ranom impurities in the meium. Tel.: aress: cafannjiang@ucavis.eu. URL: fannjian /$ - see front matter c 2005 Elsevier B.V. All rights reserve. oi: /j.phys

2 196 A.C. Fannjiang / Physica D 212 (2005) Let z an x R be the longituinal an transverse coorinates of the wave beam. In the physical setting, the transverse imension may be 0, 1, 2. The simplest non-imensional form of the nonlinear Schröinger (NLS) equation with a white-noise potential z V then reas [11] i z Ψ + γ 2 xψz + γ 1 g Ψ 2σ Ψz + γ 1 Ψ z V = 0, σ > 0. (1) Similar equations also arise in many other contexts such as in plasma physics an the Bose Einstein conensation, see [13] an references therein. Here we have written the equation in the comoving reference frame at the group velocity an non-imensionalize the equation with the longituinal an transverse reference lengths L z, L x ; γ = L z /(kl 2 x ) is the Fresnel number; g is the nonlinear coupling coefficient with g > 0 representing the selffocusing case an g < 0 the self-efocusing case. For a nonlinear Kerr meium σ = 1 leaing to the cubic NLS equation. We shall let σ be an arbitrary positive constant. We shall consier white-noise-in-z, x-homogeneous ranom potential V with z V (z, x) z V (z, x ) = δ(z z )z e i(x x ) p Φ(p)p where Φ(p) is the power spectral ensity an the prouct V Ψ in Eq. (1) stans for the Stratonovich prouct. Here an below the bracket enotes the ensemble average w.r.t. the ranom meium. In view of the real-valueness of V we may assume Φ( p) = Φ(p), p R. Such a short-range-in-z potential arises in long-istance propagation when the longituinal length L z of the wave beam is much larger than the correlation length of the ranom impurities in the meium resulting in rapily fluctuating potential in the non-imensionalize coorinates. An it is well known that in the short-rangecorrelation scaling limit a non-white-in-z multiplicative noise gives rise to a Stratonovich, instea of Itô, integral with respect to a white-in-z noise [12]. We are particularly intereste in the problems of wave sprea an singularity formation. To this en we shall use the phase-space approach of the Wigner istribution which is particularly useful in the regime of low Fresnel number γ 1 which can be viewe as a high-frequency limit. The main ingreient of our analysis is the phasespace variance ientity by which we erive various estimates for the mean sprea, efine in (9), incluing precise behavior in the efocusing g 0 or critical case σ = 2. To our knowlege, these are significant improvements over previous results (e.g. [1,14,3]) which are mostly for the linear problem. Wave sprea is an important physical quantity in itself an in technological applications such as estimating the effect of nonlinearity on the information capacity of optical fiber communications [8,10]. Using the variance ientity we further show that the scattering by the ranom potentials results in conitions for singularity-with-positive-probability that are ifferent from those for the homogeneous case. In the present setting the physical roles of t an z are interchange: the variable t is space-like while the variable z is time-like an we will refer to it as time occasionally, especially in the iscussion of finite-time singularity. (2) 2. Wigner istribution We consier the Wigner istribution of the form W (z, x, p) = 1 ( (2π) e ip y Ψ z, x + γ y ) ( Ψ z, x γ y ) y (3) 2 2 which is always real-value an may be thought of as quasi-probability ensity on the phase space [9,6]. First its marginals are essentially the position an momentum ensities

3 A.C. Fannjiang / Physica D 212 (2005) W (z, x, p)p = Ψ 2 (z, x) ρ(z, x) 1 (2π) W (z, x, p)x = 1 γ ˆΨ(z, p) 2 which are nonnegative; inee any marginal on a -imensional subspace is nonnegative everywhere. Also, the energy flux is given by 1 2i (Ψ Ψ Ψ Ψ) = pw (z, x, p)p. (4) R In the whole phase space, however, the Wigner istribution is not everywhere nonnegative in general an hence cannot be a true probability ensity. As γ 0, the Wigner istribution has a nonnegative-measure-value weak limit, calle the Wigner measure [9]. Therefore the Wigner istribution is particularly useful for analyzing high frequency behaviors of waves. 3. The evolution equations From the NLS equation (1) it is straightforwar to erive the close-form (Wigner Moyal) equation for the Wigner istribution in the Itô sense [4], [5] z W + p x W z + U γ W z Q γ W z + z V γ W = 0. (5) Here the self-ajoint operator Q γ is the Stratonovich correction term Q γ W (z, x, p) = 1 Φ(q)γ 2 [ 2W (z, x, p) + W (z, x, p γ q) + W (z, x, p + γ q)]q 2 an U γ an V γ are the nonlinear an linear Moyal operators, respectively U γ W (z, x, p) = i e iq x γ 1 [W (z, x, p + γ q/2) W (z, x, p γ q/2)]û(z, q)q, V γ W (z, x, p) = i e iq x γ 1 [W (z, x, p + γ q/2) W (z, x, p γ q/2)] V (z, q) with U = gρ σ. Here we have somewhat abuse the notation by writing Û as the Fourier-transform-in-x of the function U(z, ) an ˆV (z, q) as the spectral measure of the white-noise potential V (z, ). The spectral measure ˆV (z, p) is relate to the power spectral ensity Φ of V as z ˆV (z, p) z ˆV (z, q) = δ(z z )δ(p q)φ(p)zpq. An important property of these integral operators is Q γ W p = U γ W p = V γ W p = 0 (6) which will be useful in eriving the energy law an the variance ientity. A major avantage of formulating the wave process on the phase space using the Wigner istribution is that the (high-frequency) low-fresnel number limit γ can be easily obtaine. Formally we see that as γ 0

4 198 A.C. Fannjiang / Physica D 212 (2005) U γ W (z, x, p) U 0 W (z, x, p) x U(z, x) p W (z, x, p) V γ W (z, x, p) V 0 W (z, x, p) x V (z, x) p W (z, x, p) Q γ W (z, x, p) Q 0 W (z, x, p) = p D p W (z, x, p) with the iffusion coefficient D = Φ(p)p pp. (7) We shall refer to Eq. (5) as the nonlinear Wigner Moyal Itô (NWMI) equation when γ > 0 an as the nonlinear Liouville Itô (NLI) equation when γ = 0. Another avantage of working with Eq. (5) is that one can use it to evolve the mixe-state initial conition. The mixe-state Wigner istribution is a convex combination of the pure-state Wigner istributions (3) escribe as follows. Let {Ψ α } be a family of L 2 functions parametrize by α which is weighte by a probability measure P(α). Denote the pure-state Wigner istribution (3) by W [Ψ α ]. A mixe-state Wigner istribution is given by W [Ψ α ]P(α). (8) The limiting set, as γ 0, of the mixe state Wigner istributions constitutes the nonnegative Wigner measures [6,9]. The NWMI or NLI equation preserves the mixe-state form (8) Basic properties The NWMI an NLI equations conserve the total mass, i.e. z N = 0, N = W (z, x, p)xp an the L 2 norm W 2 (z, x, p)xp = 0. z One can absorb the effect of the total mass N into g by the obvious rescaling of W in Eq. (5). Henceforth we assume that N = 1. Let S x an S p be the spreas (or variances) of position an momentum, respectively S x (z) = x 2 W (z, x, p)xp (9) S p (z) = p 2 W (z, x, p)xp. A natural space of initial ata an solutions is the subspace W L 2 (R 2 ) with a finite Dirichlet form W Q γ W xp < an finite, positive (pre-ensemble-average) variances S x (0, ), S p (0, ). In aition, we shall also assume that the initial ata have a finite Hamiltonian H (, ) with (10)

5 H = 1 2 S p g σ + 1 A.C. Fannjiang / Physica D 212 (2005) ρ σ +1 x, ρ(z, x) = W (z, x, p)p. (11) The first term in H is the kinetic energy an the secon term is the (self-interaction) potential energy. Note that in the presence of a ranom potential, the value of the Hamiltonian (11) is not conserve uner the evolution (5). Inee, the ensemble-average value of the Hamiltonian increases in time, cf. (27) below. 4. Energy an variance ientity Let us analyze the evolution of the mean Hamiltonian H an the average sprea S x. First note the result of U γ acting on the quaratic polynomials: U γ x = 0 (12) U γ p = i e iq x qû(q)q = x U (13) U γ x 2 = 0 (14) U γ x p = i e iq x x qû(q)q = x x U (15) U γ p 2 = i e iq x 2p qû(q)q = 2p x U. (16) A remarkable fact is that these results are inepenent of γ 0. Using (12) (16) an (6) we obtain by integrating Eq. (5) that z S p = x U 2pW xp + Q γ p 2 W px g ρ σ +1 x = x U pw px z σ + 1 from which the evolution equation for the mean value of the Hamiltonian follows z H = 1 Q γ p 2 W px. 2 We shall refer to (17) as the energy law. (17) 4.1. Variance ientity The variance ientity has been long use to erive the wave collapse conition for the NLS in the homogeneous case [13]. Below we reformulate it for the phase space evolution equation (5). We have another set of remarkably simple properties of Q γ inepenent of γ 0: Q γ x = 0 (18) Q γ p = 0 (19) Q γ x 2 = 0 (20) Q γ x p = 0 (21) Q γ p 2 = Φ(q) q 2 q 2R. (22)

6 200 A.C. Fannjiang / Physica D 212 (2005) For the iffusion operator Q 0 we have R = 2 tr(d) where the iffusion matrix D is given by (7). We shall use the above ientities to perform integrating by parts in the erivation of the variance ientity. Combining the results from the previous section an (18) (22) we obtain the rate of change of S x z S x = 2 S xp where S xp is the cross-moment S xp = x pw xp. Differentiating S xp an taking expectation we obtain z S xp = S p gσ ρ σ +1 x. σ + 1 Hence the secon erivative of S x becomes z 2 S 2(2 σ )g x = 4 H + ρ σ +1 x. (24) σ + 1 Alternatively using the efinition of H we can rewrite the variance ientity (24) as z 2 S x = 2σ H + (2 σ ) S p (25) which in the critical case σ = 2 becomes z 2 S x = 2σ H. Both (24) an (25) will be useful for estimating wave sprea. (23) (26) 5. Dispersion rate Although the meium is lossless, reflecte in the fact that the total mass N = 1 is conserve, the value of the Hamiltonian, however, is not conserve by the evolution since the ranom scattering is not elastic ue to the time-varying nature of the ranom potential. Inee, by (17) an (22), the average Hamiltonian is an increasing function of time H = R, z H (z) = H(0) + Rz (27) ue to the iffusion-like sprea in the momentum p. In the critical case σ = 2, we obtain from Eqs. (26) an (27) the exact result z 2 S x = 4 H = 4H(0) + 4Rz before any singularity formation causes a possible breakown of the variance ientity. Integrating (28) twice we obtain the exact sprea rate as state below. Proposition 1. If σ = 2 or g = 0, then S x (z) = S x (0) + 2S xp (0)z + 2H(0)z 2 + 2R 3 z3. (29) (28)

7 A.C. Fannjiang / Physica D 212 (2005) The analogous result (S x z 3 ) for the linear Schröinger equation ( = 1, g = 0) with a ranom potential has been prove previously [1,3]. Next we consier the supercritical case an the efocusing case. We have from (24) that an hence z 2 S (4 2σ )g x = 4 H + σ + 1 z 2 S x 4 H = 4H(0) + 4Rz, for g(2 σ ) < 0 z 2 S x 4 H = 4H(0) + 4Rz, for g(2 σ ) 0. On the other han, from (25) we obtain for any g z 2 S x 2σ H, for 2 σ 0 z 2 S x 2σ H, for 2 σ 0. Integrating the above inequalities twice, we obtain the following. Proposition 2. The following estimates hol ρ σ +1 x (30) S x (z) S x (0) + 2S xp (0)z + 2H(0)z Rz3, g(2 σ ) 0 (31) S x (z) S x (0) + 2S xp (0)z + 2H(0)z Rz3, g(2 σ ) 0 (32) an S x (z) S x (0) + 2S xp (0)z + σ H(0)z 2 + σ 3 Rz3, 2 σ (33) S x (z) S x (0) + 2S xp (0)z + σ H(0)z 2 + σ 3 Rz3, 2 σ. (34) Therefore Corollary 3. Assume g < 0 (hence H 0). Then an S x (z) S x (0) + 2S xp (0)z + (σ 2)H(0)z 2 + σ 2 Rz 3 3 S x (z) S x (0) + 2S xp (0)z + (σ 2)H(0)z 2 + σ 2 Rz 3. (35) 3 In other wors the variance S x is cubic-in-time in the efocusing case. For g > 0 in the subcritical case, however, the cubic law is a lower boun while in the supercritical case the cubic law is an upper boun.

8 202 A.C. Fannjiang / Physica D 212 (2005) Finite-time singularity Finite-time singularity for the critical or supercritical (σ 2) self-focusing NLS equation in the homogeneous case is a well known effect [13]. In this case the singularity is the blow-up type S p, ρ σ +1. Here we call breakown of the variance ientity (24) an (25) or the energy law (17) as finite-time singularity an seek the sufficient conitions for singularity with positive probability. We show that in the supercritical case with aitional assumptions the finite-time singularity is of the blow-up type in the sense that S p, ρ σ +1 ten to infinity. As such the blow-up phenomenon iscusse here is not necessarily a sure event but rather an event of a positive probability that S p an ρ σ +1 can excee any fixe level in a finite time. For g 0, σ 2 one can boun S x as in the inequality (33) S x (z) S x (0) + 2S xp (0)z + σ H(0)z 2 + σ R 3 z3 F(z) (36) an as motivate by the homogeneous case we look for the conitions when F(z) vanishes at a finite z 0. A sufficient conition for F(z) to vanish at a finite positive z can be erive from that F(z) takes a non-positive value F(z 0 ) 0 at its local minimum point z 0 > 0. The local minimum point z 0 is given by H(0) + H(0) 2 2RS xp (0)/(σ ) z 0 =. (37) R Therefore we are le to the following conitions for singularity. Proposition 4. Assume σ 2, g > 0. The solutions of the NWMI or NLI equation evelop singularities with positive probability at a finite time z z 0 given by (37) uner the conition F(z 0 ) 0 an either one of the following conitions S xp (0) < 0 (38) S xp (0) > 0, 2RSxp (0) H(0) <. (39) σ H by allowing the self- Remark 1. Clearly, the conition F(z 0 ) 0 requires H(0) to be sufficiently below interaction energy g ρ σ +1 (0)xp σ + 1 to be sufficiently negative. Remark 2. Using, instea, inequality (31) one can obtain an alternative expression H(0) + H(0) 2 RS xp (0) z 0 = R an the corresponing conitions for singularity formation, namely either S xp (0) < 0 or S xp (0) > 0, H(0) < RS xp (0). It may be the case that for a given initial conition the solutions evelop singularity at ifferent times epening on the realization of ranom potential [2]. To analyze such an effect we nee the probabilistic versions of variance ientity an energy law which are much more involve. (40)

9 A.C. Fannjiang / Physica D 212 (2005) Blow-up We will follow the argument of Glassey [7] to show more explicitly the blow-up mechanism in the case with the supercritical, self-focusing nonlinearity an give a sharper boun on z uner certain circumstances. Proposition 5. Suppose σ > 2, g > 0. Then uner the conitions (38) an H(z ) = H(0) + Rz 0, with z = S x /z an S p blow up before or at a finite time z. 2S x (0) S xp (0)(2 σ ), (41) Proof. Since blow-up is a local phenomenon, S x is a poor inicator of its occurrence. A more relevant object to consier is S p. From (25) it follows that z 2 S x = 2 z S xp (2 σ ) S p < 0, z z. (42) Since S x (0) /z = 2 S xp (0) < 0 by (42) an (38), S x /z is a negative, ecreasing function up to the time z. Hence we have 0 S xp 2 S x S p S x (0) S p, z [0, z ] (43) which implies S p S xp 2 S x (0). Let A(z) = S x (z)/z 0, z z. We have from (42), (44) an (23) the ifferential inequality (44) z A C A2, C = σ 2 4S x (0) > 0 (45) which yiels the estimate A(z) A(0) 1 zc A(0), z < 1 C A(0) = z an thus the blow-up of A(z) before or at z. This along with (44) then implies the blow-up of S p at a finite time. The preceing argument emonstrates clearly the blow-up mechanism, namely the quaratic growth property (45). Remark 3. It shoul be note that, since H(z 0 ) > 0, the conition (41) implies that z < z 0, as given by either (37) or (40). Therefore, uner the conition (41), z provies a sharper upper boun on the time of singularity than z 0. Remark 4. Since H(z) is boune over compact sets of z, Proposition 5 implies the blow-up of the selfinteraction energy ρ σ +1 in a finite time. 7. Conclusion We have presente an elementary approach for analyzing the nonlinear Schröinger equation with a white-noise potential. We have focuse on the ensemble-average quantities such as the variance ientity an energy law in orer to she light on two problems: the rate of sprea an the singularity formation.

10 204 A.C. Fannjiang / Physica D 212 (2005) We have shown that the ensemble-average sprea in the critical or efocusing case follows the cubic-in-time law while in the supercritical an subcritical focusing cases the cubic law becomes an upper an lower boun respectively. In a separate publication we will use these estimates to analyze the limitation on channel capacity in optical fibers ue to self-phase an cross-phase moulations [8]. We have also foun singularity conitions in the critical an supercritical focusing cases. An we show the finite-time singularity in the supercritical case is of the blow-up type. The singularity/blow-up iscusse in the present paper is not necessarily a sure event but that of a positive probability. Acknowlegements Research partially supporte by the Centennial Fellowship from the American Mathematical Society, the UC Davis Chancellor s Fellowship an US National Science Founation grant DMS References [1] L. Bunimovich, H.R. Jauslin, J.L. Lebowitz, A. Pellegrinotti, P. Nielaba, Diffusive energy growth in classical an quantum riven oscillators, J. Statist. Phys. 62 (1991) [2] A. Debussche, L. Di Menza, Numerical simulation of focusing stochastic nonlinear Schröinger equations, Phys. D 162 (2002) [3] E.B. Erogan, R. Killip, W. Schlag, Energy growth in Schröinger s equation with Markovian riving, Comm. Math. Phys. 240 (2003) [4] A. Fannjiang, White-noise an geometrical optics limits of Wigner Moyal equation for wave beams in turbulent meia, Comm. Math. Phys. 254 (2) (2005) [5] A. Fannjiang, Self-averaging in scaling limits for ranom parabolic waves, Arch. Ration. Mech. Anal. 175 (3) (2005) [6] P. Gérar, P.A. Markowich, N.J. Mauser, F. Poupau, Homogenization limits an Wigner transforms, Comm. Pure Appl. Math. I (1997) [7] R.T. Glassey, On the blow-up of solutions to the Cauchy problem for nonlinear Schröinger equations, J. Math. Phys. 18 (1977) [8] L. Kazovsky, S. Beneetto, A. Willner, Optical Fiber Communication Systems, Artech House, [9] P.L. Lions, T. Paul, Sur les mesures e Wigner, Rev. Mat. Iberoamericana 9 (1993) [10] P.P. Mitra, J.B. Stark, Nonlinear limits to the information capacity of optical fibre communications, Nature 411 (2001) [11] A.C. Newell, J.V. Moloney, Nonlinear Optics, Aison-Wesley, Rewoo City, CA, [12] G. Papanicolaou, W. Kohler, Comm. Pure Appl. Math. 27 (1974) [13] C. Sulem, P.-L. Sulem, The Nonlinear Schröinger Equation: Self-Focusing an Wave Collapse, Springer, New York, [14] S. Tcheremchantsev, Transport properties of Markovian Anerson moel, Comm. Math. Phys. 196 (1) (1998)