ON A NONLINEAR SCHRÖDINGER EQUATION MODELLING ULTRA-SHORT LASER PULSES WITH A LARGE NONCOMPACT GLOBAL ATTRACTOR. Rolci Cipolatti.

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1 DISCETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS Volume 17, Number 1, January 7 pp ON A NONLINEA SCHÖDINGE EQUATION MODELLING ULTA-SHOT LASE PULSES WITH A LAGE NONCOMPACT GLOBAL ATTACTO olci Cipolatti Departamento de Métodos Matemáticos Instituto de Matemática Universidade Federal do io de Janeiro C.P. 6853, io de Janeiro, Brasil Otared Kavian Départament de Mathématiques Université de Versailles Saint-Quentin 45 avenue des États Unis 7835 Versailles cedex, France (Communicated by Jerry Bona Abstract. We study a Schrödinger equation with a nonlocal nonlinearity, which has been considered as a model for ultra-short laser pulses. An interesting feature of this equation is that the underlying dynamical system possesses a bounded non compact global attractor, actually a ball in L (. Existence and instability of standing waves are also proved. 1. Introduction. The interest in the physical literature of studying the mutual influence between powerful optical pulses and broadband active media can be justified by the progress in developing ultra-short laser techniques, as in the dynamics of femtosecond laser generators, amplifiers of super-short pulses, nonlinear active optical fibers, among other examples (see L. Vázquez & al. [7] and references therein. The mathematical models in these applications describing the long-term evolution of wave packets in dispersive active media exhibit simultaneously both conservative and dissipative nonlinearities. The Maxwell-Bloch equations describing the interaction of an intensive electromagnetic radiation with a nonlinear dispersive medium lead to nonlinear Schrödinger equations. In the case of the negative group velocity dispersion and in an inertial reference frame, the equation in the dimensionless form can be written as (see L. Vázquez & al. [7] iu t u xx u u = iu iu x u(t, ξ dξ, (1 where u denotes the slowly varying complex amplitude of the wave pulse. Equation (1 reminds the so-called cubic Schrödinger equation iv t v xx v v =, ( Mathematics Subject Classification. Primary: 35Q35, 35Q6, 37K45. Key words and phrases. Schroödinger equation, blow-up, global existence, global attractor. 11

2 1. CIPOLATTI AND O. KAVIAN which has been extensively studied (see for instance V.E. Zakharov & A.B. Shabat [9], or the book by Th. Cazenave [1]. However, as we shall establish in this paper, solutions of equation (1 present rather different properties than that of the cubic Schrödinger equation. For instance while it is well known that finite energy solutions of ( are global in time on, solutions of the equation (1, as we shall see below, blow up in a finite negative time when the charge, that is the L norm, of the initial data is greater than. It is known that equation ( has a family of standing wave solutions of the form v(t, x = e iωt ϕ(x for any ω > and an appropriate positive smooth function ϕ, while equation (1 has no such solution but a solution of the form u(t, x = e 3it/4 e ix ϕ(x. The zero solution v is orbitally stable for (, that is if the initial data v( is small in H 1 norm then v(t stays near the origin for all time; we shall see that this is not the case for equation (1 that is, if u(, no matter how small is its norm in any space, then u(t L converges to as t +. In this paper we consider the well-posedness of the Cauchy problem for the equation (1 in the energy space H 1 ( and in L (. We show in Section that for all u H 1 ( there exists a unique solution in H 1 ( satisfying u( = u, which is global for positive times and may blow up in a finite negative time when u >. We show also that the blow up time is precisely T (u := log[ u / u ]. In Section 3 we prove that there exists a unique global solution in L ( and we show that the closed ball of radius in L ( is the minimal global attractor for the flow of equation (1. In Section 4 we determine an explicit standing wave solution and we discuss its stability.. Existence of global solutions in the energy space. Our goal in this section is to prove that the Cauchy problem for (1 is globally well posed in the energy space H 1 (, where by a global solution we mean a solution defined in the time interval [, + [. As usual, we begin by proving the local existence via fixed points arguments. Let us write the Cauchy problem for (1 in the form { ut + i xx u = g(u, (3 u( = u, where the nonlinear operator g(u is defined by g(u = i u u + u uf(u, F(u(x = x u(ξ dξ. Since i xx is the generator of a group of isometries {T(t} t in H 1 (, we can seek solutions in H 1 ( of the integral equation (or the so-called mild version of (3 u(t = T(tu + t (4 T(t sg(u(sds. (5 Lemma 1. Let g be the nonlinear operator defined by (4. Then g : H 1 ( H 1 ( is continuous and more precisely there exists C > such that for all u, v H 1 ( g(u g(v H 1 C ( 1 + u H 1 + v H 1 u v H 1. Proof. This is an immediate consequence of the Sobolev imbedding H 1 ( L ( and the fact that H 1 ( is a Banach algebra.

3 ON A NONLINEA SCHÖDINGE EQUATION 13 Since Lemma 1 ensures that g : H 1 ( H 1 ( is locally Lipschitz continuous, it follows from the classical Segal theorem (see for instance A. Pazy [4] that: Theorem 1. For any given u H 1 (, there exist T (u, T (u > and a unique function u C ( ( T (u, T (u ; H 1 ( C 1( ( T (u, T (u ; H 1 ( which is the maximal solution of problem (3. The solution u is maximal in the sense that if T (u < (respectively T (u < + then u(t H 1 + as t T (u and t < T (u (6 (respectively u(t H 1 + as t T (u and t > T (u. In order to prove that T (u = + for all u H 1 (, we consider the functionals charge and energy defined in ( T (u, T (u respectively by Q(t = u(t, x dx E(t = 1 u x (t, x dx 1 u(t, x 4 dx. 4 Lemma. For all u H 1 ( and t ( T (u, T (u, if u is the solution given by Theorem 1, we have: Q (t = Q(t ( Q(t ( x ( ux E (t = 1 u(t, ξ dξ (t, x u(t, x 4 (7 dx. Proof. The differential equation for the charge Q in (7 is obtained by multiplying both sides of (1 by iu. Since u(t, x as x +, we get after an integration by parts on u t udx i u x dx + i u 4 ( dx = 1 F(u u dx. (8 By taking the real part in (8 we get 1 d u(t, x ( dx = 1 F(u(t, x u(t, x dx dt and the conclusion follows from the observation that x u(t, x u(t, ξ dξ = 1 ( d x u(t, ξ dξ. dx To obtain the second identity in (7, we first multiply both sides of (1 by u t. After integrating by parts on and taking the real part we get 1 d u x dx 1 d u 4 ( dx = Im uu t 1 F(u dx. dt 4 dt On the other hand, by taking the imaginary part of equation (8 we get Im u t udx = u x dx u 4 dx, and consequently E (t = u x dx u 4 dx + Im uu t F(udx.

4 14. CIPOLATTI AND O. KAVIAN Moreover, by multiplying the complex conjugate of equation (1 by iuf(u, and taking the imaginary part of the resulting expression, we obtain Im uu t F(udx = e u xx u F(udx + u 4 F(udx. Since, after an integration by parts, we have: ( u xx uf(u dx = the conclusion of the lemma follows. u x F(udx, emark 1. It follows immediately from the differential equation for the charge Q(t in (7 that the sphere S = { u H 1 (; u = } (9 is invariant under the flow of the equation (1 and attracts every orbit of (1 starting from u. Indeed, the equation for the charge Q(t has the explicit solution given by Q( Q(t = Q( + ( Q(e t, Q( = u (1 for T (u < t < T (u, and one sees that Q(t as t + (indeed when T (u = +. Moreover, if u >, it follows from (1 that the solution cannot exist in L ( for negative times t such that t 1 ( log u u <, and therefore T (u 1 ( log u u, (11 and in fact, as we will prove below, we have T (u = log[ u / u ]. We are now in a position to prove that the solutions are global for positive times. Theorem. For all u H 1 ( we have T (u = +. Furthermore, if u then T (u = +. Proof. Noting that E (t = u x dx u 4 dx + u 4 F(udx u x F(udx, and E(t = u x dx 1 u 4 dx from the second identity in (7, it follows that E (t E(t + u(t 4 F(u(tdx. Since F(u(t Q(t K = max{, u }, we obtain E (t E(t + K u(t 4 4. u x dx u 4 dx,

5 ON A NONLINEA SCHÖDINGE EQUATION 15 By Gagliardo-Nirenberg inequality we have v 4 C v x 1/4 v 3/4, when v H 1 (, for some constant C >, and consequently (using Young s inequality αβ α + β E (t E(t + C u x E(t + C + 1 u x. (1 (Here and in what follows we denote by C various constants which may depend only on various norms of u. After integrating the inequality in (1 on (, t we obtain and therefore E(t E( + t u x (t C(1 + t + C E(sds + Ct + 1 t t u x (s ds, u x (s ds + 1 u(t 4 4. Another application of the Gagliardo-Nirenberg inequality, together with the use of the estimate u(t K, yield and therefore u(t 4 4 C u x (t C + 1 u x(t t u x (t C(1 + t + C u x (s ds. As a consequence of the Gronwall inequality we obtain u x (t C(1 + ηecη, for all t [, η] and we conclude that T (u = + (see (6. In the case of the standard subcritical (that is when the exponent p satisfies 1 < p < 5 nonlinear Schrödinger equation iv t v xx ± v p 1 v =, using the Gagliardo Nirenberg inequality, one shows that a uniform estimate in time on the L norm of v implies an estimate in time on the H 1 norm of the solution, and thus global existence of the solution v in H 1 (see for instance Th. Cazenave [1]. Here, in order to show global existence of u for negative times, assuming that u, we are going to use the same kind of arguments. Indeed by the expression (1 for Q(t, we know that u(t for t > T (u, and since, according to (7: E (t u(t 4 4 u x (t F(u(tdx u(t 4 4 u(t u x(t and by Gagliardo-Nirenberg inequality u(t 4 4 C u(t 3 u x (t C u x (t, it follows that E (t C u x (t C, for all t ( T (u, T (u. After integrating on [ η, ], for η < T (u, we get E( η E( + C η u x (s ds + Cη. (13

6 16. CIPOLATTI AND O. KAVIAN Since u x ( η E( η+ u( η 4 4, another application of Gagliardo-Nirenberg inequality, followed by the use of Young s inequality, yields u x ( η C(1 + η + C C(1 + η + C η η u x ( s ds + 1 u( η 4 4 u x ( s ds and again from Gronwall s inequality it follows that u(t remains bounded in H 1 ( on any interval [ η, ]. Hence T (u = + as claimed. As a matter of fact, a detailed inspection of the former argument shows that when u > then the negative blow-up time is precisely given by the estimate (11, that is T (u = log[ u / u ]. Proposition 1. Assume that u H 1 ( is such that u >. Then T (u = 1 ( log u u, and the solution u(t blows up in L norm, that is u(t + as t T (u and t > T (u. Proof. We already know that T (u log[ u / u ] so, in order to prove our claim, assume that we have a strict inequality in the estimate (11. In this case we would have u ( u e t > u ( u et (u for all t > T (u, and therefore u(t u u ( u et (u =: K. As we already noticed above, since we have E (t u(t 4 4 u x (t F(u(tdx u(t 4 4 u(t u x (t, and since u(t 4 4 C u(t 3 u x (t CK 3/ u x (t (by Gagliardo-Nirenberg inequality, it follows that E (t C(K u x (t C(K, Now using the same arguments as the ones used in the proof of (13 and (14, we conclude that u x ( η C(K for some constant depending on K and thus only on T (u and u, for all η < T (u. Indeed this is in contradiction with the fact that T (u is finite. emark. It should be noted that the structure of equation (1 implies a completely different behaviour than that of the purely cubic Schrödinger equation (. Also if in equation (1 we drop either terms of the right hand side, we get one of the equations: iv t v xx v v = iv (15 or (14 iw t w xx w w = if(ww. (16

7 ON A NONLINEA SCHÖDINGE EQUATION 17 Now one may see easily that solutions of both equations (15 and (16 exist for all times t, and that v(t = et v(, while w(t = w( 1 + t w(. These equalities are obtained by multiplication of (15 by v and (16 by w, integrating by parts over and taking the imaginary parts of the resulting relations. This means that the behaviour of solutions of equation (1 is an interpolation between that of v and w as t +. emark 3. More generally we may consider the equation iu t u xx u u = iαu iβu x u(t, ξ dξ, (17 for α > and β >. Then, proceeding in the same way as we did in the proof of lemma, one may establish easily that, Q(t := u(t satisfies and therefore Q(t = Q (t = αq(t βq(t, αq( βq( + (α βq(e αt. In this case one sees that Q(t α/β as t + (when u(. Indeed all the results concerning the global existence or the finite time blow up for negative times hold also in this case. 3. Global solution in L ( and the global attractor A. In this section we prove that the Cauchy problem for equation (5 is well posed in L ( and we show that the closed ball of radius in L ( is the minimal global attractor. In order to construct solutions in L (, we shall use the Strichartz inequality (see for instance Th. Cazenave [1], M. Keel & T. Tao [], K. Yajima [8],. Strichartz [5]. Also following a classical denomination in this context, we say that a pair (q, r is admissible if r and q = 1 1 r. The Strichartz inequality states that if (q, r and (q, r are two admissible pairs, and if for h L q (( T1, T, L r ( we define M(h by M(h(t, x := t T(t τh(τ, xdτ, then there exists a constant C > such that for any T 1, T one has M(h Lq (( T 1,T,L r ( C h L q (( T 1,T,L r (, (18 for a constant C independent of T 1, T > and of q, r. With the notation (4 for the nonlinearity u g(u, we write g = g 1 +g, where g 1 (u = i u u and g (u = u uf(u. Direct applications of Hölder inequality yield the following estimates.

8 18. CIPOLATTI AND O. KAVIAN Lemma 3. For all v L ( L ( we have g 1 (v L 1 ( L ( and g (v L ( L (. Moreover, there exists C > such that g 1 (v g 1 (w 1 C ( v + w v w, g (v g (w C ( 1 + v + w v w, for all v, w L ( L (. Furthermore, there exists C > such that, for all η > : (i if v, w L 4 (, η; L ( L (, η; L (, then g 1 (v g 1 (w L 4/3 (,η;l 1 ( C η ( v 1/ L (,η;l ( + w L (,η;l ( v w L 4 (,η;l ( (ii if v, w L (, η; L (, then g (v g( (w L 1 (,η;l ( C η 1 + v L (,η;l ( + w L (,η;l ( v w L (,η;l (. ( Theorem 3. Let u L (. Then there exists a unique u C ( [, + ; L ( such that u L 4(, η; L ( for all η > and such that for all t u(t = T(tu + t (19 T(t sg ( u(s ds. (1 Furthermore, if u, then u C ( ; L ( L 4 loc( ; L (. Proof. Let u m H 1 ( be such that u m u as m. Then it follows from Theorems 1 and that there exists a unique u m C ( [, + ; H 1 ( C 1( [, + ; H 1 ( such that u m ( = u m and for all t u m (t = T(tu m + t T(t sg ( u m (s ds. The Sobolev imbedding theorem and Hölder inequality imply that for all η > and a, b [, ], we have u m L a(, η; L b (. Let Φ j (u m (t, j = 1,, be the function defined by Φ j (u m (t = t T(t sg j ( um (s ds. Then from the Strichartz inequality (18 (with (q, r := (4, for g 1 (u m, and (q, r := (, for g (u m, using inequalities (19 and ( it follows that, for all admissible pairs (q, r, there exists a constant C > (independent of q and r such that u m L q (,η;l r C ( u m + g 1 (u m L 4/3 (,η;l 1 + g (u m L 1 (,η;l. By using the estimates (19 and ( of Lemma 3, we obtain (for possibly another constant C > ( u m L q (,η;l r C u m + η 1/ Km u m L 4 (,η;l + + η(1 + Km u ( m L (,η;l,

9 ON A NONLINEA SCHÖDINGE EQUATION 19 where K m = max{, u m } is bounded by some constant depending on u. Let us introduce the space V η := L 4(, η; L ( L (, η; L (, which is a Banach space for the norm v Vη := v L4 (,η;l + v L (,η;l. By choosing (q, r respectively as (4, and (, in (, we obtain for η > small enough that {u m } m is bounded in V η. The same arguments imply (using again (19 and ( that, for m, n N, ( u m u n Vη C u m u n + C η 1/ Km u m u n L4 (,η;l + + η(1 + Km u m u n L (,η;l. Choosing η > possibly smaller, but still depending on u, it follows that {u m } m is a Cauchy sequence in V η, and in particular in C ( [, η]; L (. Therefore, if we denote by u the limit of (u m m in V η, we have in particular, by (1, that for all t [, η] u(t u = u + ( u. (3 e t One may see easily that u solves the integral equation (5. Since η depends only on u, it follows from (3 that one can iterate the arguments in order to cover [, + (or to cover if u. Since uniqueness and continuous dependence on initial data are shown by the same arguments, the proof of the Theorem is over. Once this is proved, let {S(t} t be the family of continuous operators defined by S(tu := u(t, where u C ( [, + ; L ( is the solution of (5 with initial data u. Then {S(t} t is a semigroup of nonlinear continuous operators in L (. Following the usual terminology in the theory of semigroups (see O.A. Ladyzhenskaya [3], page 4, we say that: Definition 1. A set A L ( is a global attractor for {S(t} t if, for any ε > and any bounded set B of L (, there exists a positive time τ(ε, B > and an ε-neighbourhood A ε of A such that S(tB A ε, for all t > τ(ε, B. In the case of the dynamical system defined by the semi-group {S(t} t, let A := { v L (; v }. Then we have as a direct consequence of the previous results: Corollary 1. For all t we have S(t(A = A, and A is the minimal global attractor for the semigroup {S(t} t acting on L (. emark 4. Note that A is the ball of radius and so is noncompact. Actually if instead of equation (1 one considers the more general version (17, then since the ratio α/β can be arbitrarily large, one has examples of simple dynamical systems which are a slight modification of the cubic nonlinear Schrödinger equation, and which have the arbitrarily large closed ball B(, α/β as their minimal global attractor.

10 13. CIPOLATTI AND O. KAVIAN 4. Standing waves and action of groups on the solution. In this section we determine a standing wave for the equation (1 and we discuss its stability. By a standing wave we mean a periodic (in time solution of the form u(t, x = e iωt ψ(x where ω is a fixed parameter and ψ : C is an appropriate function. Since a standing wave has conservation of charge, Q(t = Q(, we must search for ψ in the sphere S defined in (9. Without loss of generality, we may look for ψ having the form ψ(x = e iω(x ϕ(x, where Ω and ϕ are smooth real functions. By substituting in (1 and separating the real and imaginary parts, we get { ϕ + (Ω ωϕ = ϕ 3, Ω ϕ Ω (4 ϕ = ϕ ϕf(ϕ. Multiplying the second equation in (4 by ϕ and integrating we obtain Ω (xϕ(x = 1 ( F(ϕ(x F(ϕ(x + C, for some C. (5 Assuming that Ω (xϕ(x as x +, and knowing that ϕ S, it follows that C =. Moreover, since s s / s is negative for < s <, and F(ϕ <, we have Ω (xϕ(x = 1 F(ϕ(x F(ϕ(x and therefore necessarily we should have Ω (x for all x. Let us assume that Ω(x = βx with β >. Then the system (4 takes the form { ϕ + (β ωϕ = ϕ 3, β d dx F(ϕ = F(ϕ 1 (6 (F(ϕ, The second equation in (6 (which is (5 with C = has the general solutions F(ϕ(x, which does not yield a solution ϕ, or else F(ϕ(x = ceαx 1 + ce αx, where α = 1/β and c > because < F(ϕ < for a non trivial solution ϕ. Finally the relation ϕ(x = (F(ϕ(x implies that ϕ(x = α/ sech ( αx/+θ/, with e θ = c. It is well known (see for instance Th. Cazenave [1] that for any µ > the set of solutions of ϕ + µϕ = ϕ ϕ, which is the set of standing waves of the cubic Schrödinger equation is given by { e ia µ sech( µx+b; a, b }. Hence, by identification of parameters we get µ = 1/4, β = 1 and ω = 3/4 and we have the conclusion that ( x ψ(x = e iω(x ϕ(x = e ix sech. Lemma 4. For all s, θ, the function ( ( 3it x + s u(t, x = eiθ exp 4 ix sech is a solution of equation (1. (7

11 ON A NONLINEA SCHÖDINGE EQUATION 131 emark 5. Due to the invariances under translation and multiplication by e is for the equation (1, we cannot expect to have orbital stability of the standing wave (7. To see this, we observe first that whenever u is a solution to equation (1, then for any given c, and appropriate values of α, β the function v defined to be v(t, x := e iαx e iβt u(t, x ct is again a solution of (1, provided α = c, β = c 4. This is due to the fact that v is a solution to (1 if, and only if, one has (α βu(t, x ct i(c + αu x (t, x ct. As a matter of fact, the traveling waves for equation (1 are obtained by taking u to be the standing wave x u(t, x = e3it/4 e sech( ix. Next observe that the group S 1 acts on the solutions of (1 by the following: if u(t, x is a solution, then for (s, e iθ S 1, the function U s,θ (t, x := e iθ u(t, x+s is also a solution and the mapping (s, e iθ U s,θ is the action of the group S 1 on the solutions. This implies in particular that we can construct a family of traveling waves related to (7. More precisely, for every c the function u c (t, x = x ct ei(3+4c+c t/4 e sech( i(c+x/ (8 is a solution with initial datum u c (, x = ψ c (x = e i(c+x/ sech(x/. It is easily seen that ψ c ψ as c. Since u c (t S(tψ as t + for every c, we cannot have orbital stability for u(t. This instability was noticed through physical arguments in the paper by L. Vázquez & al. [7]. The natural question concerning the stability of (7 may be viewed as to whether the following distance function: sup t inf s,θ eiθ τ s u(t S(tψ H (9 can be controlled in some way or another, where H denotes a suitable Banach space and τ s is the translation operator defined by τ s v(x := v(x + s. However, unfortunately, we are not able to give a satisfactory result when taking for instance H := L (, due to the fact that the linearization of equation (1 around the standing wave solution does not show a well understood structure. Nevertheless, if we restrict (9 to the family of travaling waves in (8 we get stability in H 1 (. More precisely, Proposition. Given any ε >, there exist c > and K > such that for c c one has inf u c(t e iθ τ s S(tψ H 1 < ε. s,θ sup t Proof. First of all, we remark that inf u c(t e iθ τ s S(tψ L u c (t e iθ τ ct S(tψ L. s

12 13. CIPOLATTI AND O. KAVIAN A straightforward calculation gives where and so u c (t e iθ τ ct S(tψ L = 4 [1 cos F(c := sech (ξcos(cξdξ = ( c ] t 4 + θ F(c, (3 π c sinh ( π c inf u c(t e iθ τ s S(tψ L 4( 1 F(c. s,θ Note that 1 F(c = O(c as c. Moreover, since x u c (t, x = 1 [ ( x ct u c(t, x tanh e iθ τ s x S(tψ(x = 1 ( [ ( x + s eiθ τ s S(tψ(x tanh another straightforward (and cumbersome calculation yields x u c (t x e iθ τ ct S(tψ L = 1 ( (c ( c ( cos t c 4 + θ 3 and so, inf s,θ x( uc (t e iθ τ s S(tψ L c ] + (c + i + i, ], 6 + 4c + F(c 3 ( 1 3 F(c + c ( 1 F(c + 13 ( 1 F(c. 3 (31 Since 1 F(c = O(c as c, we get the conclusion by (3 and (31. EFEENCES [1] Th. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, No. 6, io de Janeiro, See also: Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society,Providence, I, 3. [] M. Keel & T. Tao, Endpoint Strichartz estimates Amer. J. Math. 1 (1998, no. 5, pp [3] O.A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge Univesity Press, Cambridge, [4] A. Pazy, Semigroups of linear operators and applications to partial differential equations Applied Mathematical Sciences, No. 44, Springer-Verlag, [5]. Strichartz, estrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations Duke Math. J. 44 (1977, pp [6]. Temam, Infinite-dimensional dynamical systems in mechanics and physics Applied Mathematical Sciences, No. 68, Springer-Verlag, [7] L. Vázquez & al. Dissipative optical solitons Physical eview A, Vol. 49, No. 4, 1994, pp [8] K. Yajima, Existence of solutions for Schrödinger evolution equations Comm. Math. Phys. 11 (1987, pp [9] V.E. Zakharov & A.B. Shabat, Exact theory of two-dimenional self-focusing and onedimensional self-modulation of waves in nonlinear media Sov. Phys. J.T.T.P., 34 (197, pp eceived August 5; revised July 6. address: cipolatti@im.ufrj.br address: otared.kavian@math.uvsq.fr