A MODULATION METHOD FOR SELF-FOCUSING IN THE PERTURBED CRITICAL NONLINEAR SCHR ODINGER EQUATION. GADI FIBICH AND GEORGE PAPANICOLAOU y

Size: px

Start display at page:

Download "A MODULATION METHOD FOR SELF-FOCUSING IN THE PERTURBED CRITICAL NONLINEAR SCHR ODINGER EQUATION. GADI FIBICH AND GEORGE PAPANICOLAOU y"

Rosa Waters
5 years ago
Views:

1 A MODUATION METHOD FOR SEF-FOCUSING IN THE PERTURBED CRITICA NONINEAR SCHR ODINGER EQUATION GADI FIBICH AND GEORGE PAPANICOAOU y Abstract. In this etter we introduce a systematic perturbation method for analyzing the eect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schrodinger equation (PNS) to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simplied, depending on whether PNS is power conserving or not. An important and somewhat surprising result is that various small defocusing perturbations lead to a universal form for the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations. () PACS 4.65.J keywords: self-focusing, nonlinear Schrodinger equation, modulation theory. Introduction. The perturbed critical nonlinear Schrodinger equation (PNS) i z +? + j j + F ( ; z ; r? ;...)=; (;x;y)= (x; @y ; < ; arises in various physical models in nonlinear optics, plasma physics and uid dynamics (e.g. table ). When = eq. () reduces to the critical nonlinear Schrodinger equation (CNS) () i z +? + j j =: We recall that for the nonlinear Schrodinger equation with a general nonlinearity and transverse i z + + j j =; D we distinguish between three dierent cases: ) When D <, the subcritical case, diraction always dominates and focusing singularities do not form. ) In the supercritical case D >, there is a large class of smooth initial amplitudes for which a focusing singularity forms in nite distance z. Since in supercritical self-focusing the nonlinearity dominates over diraction, addition of small perturbations to the equation has a small eect. 3) In the critical case D = (as in the case of eq. ), solutions can also become singular in a nite z. However, in this borderline case between subcritical and supercritical self-focusing, singularity formation is characterized by a near-balance between the focusing nonlinearity and diraction. As a result, critical self-focusing is extremely sensitive to small perturbations, which can have a large eect and can even lead to the arrest of collapse. Self-focusing is a genuinely nonlinear phenomenon and standard linearization methods cannot be used to analyze singularity formation in eqs. () and (). In addition, methods such as the inverse scattering transform (IST), which is so successful in the D cubic subcritical case, cannot be applied to eq. (), because () is not integrable. Self-focusing in () or () is, moreover, a local phenomenon which cannot be accurately captured by global estimates. For these reasons, despite considerable progress the present theory of critical self-focusing in the presence of small perturbations is still far from complete. In this etter we present a general method for analyzing the eect of a any deterministic or random perturbation on critical self-focusing. In this method PNS () is reduced to a simpler system of modulation equations which do not depend on the transverse variables. The reduced system is much easier to analyze and simulate, and provides insights that are hard to get directly from PNS. Department of Mathematics, UCA, os-angeles CA , bich@math.ucla.edu y Department of Mathematics, Stanford University, Stanford CA 9435, papanicolaou@stanford.edu The amplitude may depend on additional variables, such ast in the case of time-dispersion.

2 . Review of critical self-focusing. CNS () has two important conserved quantities: The power N := j j dxdy and the Hamiltonian H( ) := jr? j dxdy, A sucient condition for singularity formation in () is while a necessary condition is CNS has waveguide solutions of the form R(r) + r H( ) < j j 4 dxdy N( ) N c = :86 : ; r? = = exp(it)r(r) ; r =(x + y ) @ R, R + R 3 =; R () = ; lim R(r) r! The solution of eq. (3) with the lowest power (`ground state'), sometimes called the Townes soliton, has an important role in self-focusing theory. This positive, monotonically decreasing solution has exactly the critical power for self-focusing (4) and its Hamiltonian is equal to zero (5) R rdr = N c H(R) =: Analysis of blowup in CNS is based on the assumption (which is supported by numerical and analytical evidence) that near the singularity the solution is roughly a modulated Townes soliton: (6) (7) and (8) R ; R := R() exp(is) ; = ; S = + z d dz = : More precisely, near the singularity, s, the inner part of the solution, whose power is slightly above critical, collapses towards the singularity in a quasi-self-similar fashion, s V (;) exp(is) ; r 4 ; : A possible denition is s = for c, with c constant.

3 V! R as z! c and c is the location of the singularity. Based on this modulation ansatz, it was shown that near the singularity self-focusing can be described by the reduced system [,,3] (9) () z =, () ; zz =, 3 () c exp(,= ) ; c = 45: : In order to motivate (9,), we note that the modulation variable is the radial width as well as =amplitude of the focusing part s, and that is proportional to the excess power above critical of s [4]. Therefore, at the point of blowup ( c )=( c )=. The() term arises from radiation eects (power losses of s ) during self-focusing. Since near the singularity (z) ; () is exponentially small and self-focusing is essentially adiabatic... Adiabatic approach. Originally, the reduced system (9,) was analyzed by solving (9) to leading order near c and then using (). This leads to the loglog law for the rate of critical blowup [,,3]. However, it turns out that the loglog law does not become valid even after amplication of the peak amplitude by a factor of a billion or more, which is long after the nonlinear Schrodinger equation ceases to be physically relevant. Fortunately, this can be `xed' by solving (9,) using an adiabatic approach. Since changes in (i.e. the power of s ) are exponentially small compared with the focusing rate, we rst solve () with constant, and only then add non-adiabatic eects (9) as the next-order correction. Application of this approach leads to an adiabatic law for critical self-focusing which is valid almost from the onset of self-focusing [5]. 3. Modulation theory for self-focusing in the perturbed CNS. The adiabatic law, which provides an accurate description of critical self-focusing in domain of physical interest, is obtained in two stages: ) Derivation of the reduced modulation equations (9,) which do not depend on the transverse variables and ) Solving these equations with the radiation term () neglected to leading order (adiabatic approach). In this section we extend this approach to self-focusing in PNS: ) The modulation ansatz (6) is used in proposition 3. to reduce eq. () to the system () and ) The reduced system is analyzed with the adiabatic approach (propositions 3.{3.3). More details, as well as proof of results are published else 3 [6]. For modulation theory to be valid, the following three conditions must hold:. The focusing part of the solution is close to the asymptotic prole (6).. The power of the focusing part is close to critical j s (z; x; y)j dxdy, N c ; or equivalently, j(z)j : 3 The paper is available at the web sites and 3

4 3. The perturbation is small: jf jj? j and jf jj j 3 : In general, at the onset of self-focusing only condition 3 holds. As the solution approaches c (the blowup point in the absence of the perturbation), conditions { are also satised and modulation theory becomes valid. The main result of modulation theory is the following: Proposition 3.. If conditions {3 hold and if F is an even function in x and y, self-focusing in PNS () is given to leading order by the reduced system () z (z)+ () = M (f ) z, M f ; zz (z) =, 3 : The auxiliary functions f and f are given by () f(z) = (z) Re F ( R ) exp(,is)[r() +r? R()] dxdy (3) f(z) = Im R F ( R) dxdy M = 4 r 3 R(r) dr = :55 : We note that assuming that we can carry out the transverse integration, f and f are known functions of the modulation variables,, and their derivatives. 3.. Conservative and non-conservative perturbations. Considerable simplication is achieved if we distinguish between conservative perturbations i.e. those for which the power remains conserved d j (z;x;y;)j dxdy dz in (), and non-conservative perturbations: Proposition 3.. et conditions {3 hold.. If F is a conservative perturbation i.e. Im then f, and to leading order () reduces to (4) F ( ) dxdy ;, 3 zz = + M f ; = (), M f () ; is independent of z.. If F is a non-conservative perturbation i.e. Im F ( ) dxdy 6 then to leading order () reduces to (5) z =, M f ; zz =, 3 : Note that in both cases, non-adiabatic eects disappear from the leading order behavior of (). 4

5 3.. Universality of the eect of conservative perturbations. For various conservative perturbations 4 f turns out to have the universal form f, C (6) ; C = constant : The following proposition covers this canonical case. Proposition 3.3. When self-focusing is given by (4) and f is given by (6) then y := satises the universal oscillator equation (7) (8) (9) et us dene y M = y m = C M = () + (y z ) =,4H M p + CH=M +,H=M. If the perturbation is defocusing i.e. () y (y M, y)(y, y m ) ; = M +O,H p + CH=M + = C 4M C M () ; H H() + C 4 4 () : m := y = m ; M := y = M : C > ; H H +O then it will arrest blow-up in (4), i.e. (or y) will remain positive for all z. (a) If in addition to (), > and H <, then < m < M and will go through periodic oscillations between m and M (Figure A). The period of the oscillations is r MyM =,H E, y m () ; y M E(m) = R = (, m sin ) = d is the complete elliptic integral of the second kind. (b) If in addition to (), > and H >, then i. If z () <, self-focusing is arrested when = m >, after which will increase monotonically to innity (Figure B). ii. If z () >, will increase monotonically to innity.. If the perturbation is focusing i.e. C < and if in addition > and one of the following two conditions holds () H > and z () < or () H <, then the solution of (4) will blow up in a nite distance (Figure C), i.e. 9 such that < < and ( )=: ; 4 marked in table with y 5

6 M m z A m z B z C Fig.. The leading order eect of the generic conservative perturbation (6) A: Defocusing perturbation and H < (proposition a) B: Defocusing perturbation, H > and z() < (proposition 3.3-bi) C: Focusing perturbation and z() < (proposition 3.3 -). In all cases > (i.e. power above critical) Non-adiabatic eects. The results of the previous section show that the exponentially small radiation term () disappears from the leading order behavior of perturbed CNS. In the non-conservative case the eect of () iseven smaller than the (f) z term which is also ignored. However, in the conservative case when > and H <, a defocusing perturbation can lead to periodic oscillations (as in proposition 3.3-a). In this case, the non-adiabatic radiation eect () provides the only mechanism for decay of the oscillations. It can be shown that if is moderately small the total power loss during one oscillation is small and the focusing-defocusing oscillations are slowly decreasing, but that for suciently small the quasi-periodic picture breaks down and focusing is completely arrested after a few oscillations [7]. Further analysis of non-adiabatic eects in (7) can be found in [4] Modulation theory for multiple perturbations. In some cases, one is interested in the combined eect of several small perturbations e.g. randomness and quintic nonlinearity or time-dispersion and nonparaxiality (see table ). Modulation theory can easily handle these cases, since the modulation equations are linear in F. Therefore, one simply adds the contribution of each perturbation to the modulation equations. 4. Applications. The modulation approach was used by Malkin to study the eect of a small defocusing fth power nonlinearity [4]. In [8], Fibich, Malkin and Papanicolaou analyzed the eect of small normal time-dispersion, using for the rst time a systematic approach that is generalized in this etter. A similar approach was also used by Fibich to analyze the eect of beam nonparaxiality[7] and the unperturbed CNS [5] and by Fibich and Papanicolaou to analyze the combined eect of time-dispersion and nonparaxiality [9]. Additional applications, listed in table, are derived in [6]. Direct numerical conrmation of the validity of the modulation equations and the adiabatic approach was carried out in the case of the unperturbed NS [5] and in the case of small normal time dispersion [8]. In many other cases, there is qualitative agreement between the predictions of the modulation equations and the results of numerical simulations of the corresponding PNS. For example, the universal behavior of decaying focusing-defocusing oscillations was observed numerically for ber arrays [], saturating nonlinearities [3,] and nonparaxiality []. 5. Conclusion. In this etter we have introduced a modulation theory for analyzing the leading order eects of small perturbations on critical self-focusing. This theory is able to capture the delicate balance between nonlinearity and diraction in critical self-focusing, because it is based on perturbations of the solution around modulated Townes solitons ( R ). We note that the validity of other studies of PNS in which the derivation of reduced equations is based on modulated Gaussians is questionable, because modulated Gaussians cannot capture the delicate balance in critical self-focusing [e.g. Gaussians cannot simultaneously satisfy (4) and (5)]. Moreover, the derivation of reduced equations with our R - based modulation theory is just as easy as with modulated Gaussians. In fact, all that is needed is to carry out the transverse integration in evaluating f and f. We have already remarked that modulation theory becomes valid near the blowup point c.for some perturbations (e.g. nonparaxiality, saturating nonlinearities) one can show that the modulation equations 6

7 remain valid for all z [7,6]. However, in other cases (e.g. small normal time-dispersion) it is not clear for how long modulation theory remains valid, and further analysis may be needed for the advanced stages of self-focusing. 6. Acknowledgment. The work of Gadi Fibich was supported in part by the NSF grant DMS The work of George Papanicolaou was supported in part by NSF grant DMS and AFOSR grant F

8 REFERENCES [] G.M. Fraiman. Asymptotic stability of manifold of self-similar solutions in self-focusing. Sov. Phys. JETP, 6():8{ 33, Feb 985. [] M.J. andman, G.C. Papanicolaou, C. Sulem, and P.. Sulem. Rate of blowup for solutions of the nonlinear Schrodinger equation at critical dimension. Phys. Rev. A, 38(8):3837{3843, October 988. [3] B.J. emesurier, G.C. Papanicolaou, C. Sulem, and P.. Sulem. ocal structure of the self-focusing singularity of the nonlinear Schrodinger equation. Physica D, 3:{6, 988. [4] V.M. Malkin. On the analytical theory for stationary self-focusing of radiation. Physica D, 64:5{66, 993. [5] G. Fibich. An adiabatic law for self-focusing of optical beams. Opt. ett., :735 { 737, 996. [6] G. Fibich and G.C. Papanicolaou. Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension. Computational and Applied Mathematics report 97-, UCA, 997. [7] G. Fibich. Small beam nonparaxiality arrests self-focusing of optical beams. Phys. Rev. ett., 76:4356{4359, 996. [8] G. Fibich, V.M. Malkin, and G.C. Papanicolaou. Beam self-focusing in the presence of small normal time dispersion. Phys. Rev. A, 5(5):48{48, 995. [9] G. Fibich and G.C. Papanicolaou. Self-focusing in the presence of small time dispersion and nonparaxiality. Opt. ett, page in press, 97. [] A.B. Aceves, G.G. uther, C. De Angelis, A.M. Rubenchik, and S. K. Turitsyn. Optical pulse compression using ber arrays. Optical Fiber Technology, :44{46, 995. [] F. Vidal and T.W. Johnston. Electromagnetic beam breakup - multiple laments, single beam equilibria, and radiation. Phys. Rev. ett., 77:8{85, 996. [] M.D. Feit and J.A. Fleck. Beam nonparaxiality, lament formation, and beam breakup in the self-focusing of optical beams. J. Opt. Soc. Am. B, 5(3):633{64,

9 Perturbed Application Reduced conser- Derived CNS equation vative in i z +? + j j + xxxx = ber arrays (7) with C =(3=)[I6, 3N c ] yes y [6] i z +? + j j, j j 4 = quintic (7) with C =(4=3)I6 yes y [4] nonlinearity i z +? + saturating (7) with C =(4=3)I6 yes y [6] (, exp(,j j )) = nonlinearity i z +? + j j +j j = saturating (7) with C =(4=3)I6 yes y [6] nonlinearity i z +? + j j, x =; Davey- xx + yy =,(j j ) x Stewartson (9), () yes [6] i z +? + j j + zz = nonparaxiality (7) with C =4N c no [7] i z +? + j j h random, 3 zz = +4 4 h(z) yes [6] +(x + y )h(z) = i z +? + j j + ber arrays, 3 zz = + yes [6] x h(z) + xxxx = + h random 4 4 h(z), 3 [I6, 3N c ] 4M i z +? + j j + quintic, 3 zz = + yes [6] (x + y )h(z), j j 4 = nonlinearity 4 4 h(z), I6 3M +h random i z +? + j j, tt = time- z = N c M tt, no [8] dispersion (8), () i z +? + N =; Debye, 3 zz = + C D t M N t + N = j j relaxation C D = R (r? R ) r 3 dr = 6:43 i z +? + j j + zz + time- z =, N c M, h z i n c g (j j i n ) t, zt dispersion+ (6c g c c, ) N c M t,3 tt = nonparaxiality + 3N c M tt, (8), () yes [6] no [9] Table Perturbations of critical NS and their corresponding modulation equations. Here I 6 = R R 6 rdr. y - f given by (6)

. Introduction. The nonlinear Schrodinger equation in critical dimension i z +? + j j (.) =;? ; (z (x; y) is the simplest

. Introduction. The nonlinear Schrodinger equation in critical dimension i z +? + j j (.) =;? ; (z (x; y) is the simplest SELF-FOCUSING IN THE PERTURBED AND UNPERTURBED NONLINEAR SCHR ODINGER EQUATION IN CRITICAL DIMENSION GADI FIBICH AND GEORGE PAPANICOLAOU y Abstract. The formation of singularities of self-focusing solutions