Wellposedness for the fourth order nonlinear Schrödinger equations

Transcription

1 J. Mah. Anal. Appl Wellposedness for he fourh order nonlinear Schrödinger equaions Chengchun Hao a,,linghsiao a,, Baoiang Wang b a Academy of Mahemaics and Sysems Science, CAS, Beijing 8, PR China b Deparmen of Mahemaics, Peking Universiy, Beijing 87, PR China Received 7 February 5 Available online 9 Augus 5 Submied by P.J. McKenna Absrac We sudy he local smoohing effecs and wellposedness of Cauchy problem for he fourh order nonlinear Schrödinger equaions in D i u = 4 u + P α u α, α ū α,, R, where P is a polynomial ecluding consan or linear erms. 5 Elsevier Inc. All righs reserved. Keywords: Fourh order nonlinear Schrödinger equaions; Cauchy problem; Wellposedness; Smoohing effecs. Inroducion We consider he Cauchy problem for he fourh order nonlinear Schrödinger equaions i u = 4 u + P α u α, α ū α,, R,. u,= u,. * Corresponding auhor. addresses: hccwzj@yahoo.com.cn C. Hao, hsiaol@amss.ac.cn L. Hsiao, wb@publica.bj.cninfo.ne B. Wang. Suppored parially by NSFC Gran No. 436, NSAF Gran No X/$ see fron maer 5 Elsevier Inc. All righs reserved. doi:.6/j.jmaa.5.6.9

2 C. Hao e al. / J. Mah. Anal. Appl where u = u, is a comple valued wave funcion, and P is a comple valued polynomial defined in C 6 such ha P z = Pz,z,...,z 6 = a α z α,.3 β α γ α Z 6 for β and here eiss a α forsomeα Z 6 wih α =β. This class of nonlinear Schrödinger equaions comes from he infinie hierarchy of commuing flows arising from he D cubic nonlinear Schrödinger equaion. A large amoun of ineresing work has been devoed o he sudy of he Cauchy problem for dispersive equaions. One can see [,3 7] and references cied herein. In order o sudy he influence of higher order dispersion on soliary waves, insabiliy and he collapse phenomena, Karpman [9] inroduced a class of nonlinear Schrödinger equaions iψ + ΔΨ + γ Δ Ψ + f Ψ Ψ =, R n, R. This sysem wih differen nonlineariies was discussed by several auhors. In [6], by using he mehod of Fourier resricion norm, Segaa sudied a special fourh order nonlinear Schrödinger equaion in one-dimensional space and considered he hree-dimensional moion of an isolaed vore filamen, which was inroduced by Da Rios, embedded in inviscid incompressible fluid fulfilled in an infinie region. And he resuls have been improved in [8,7]. In [], Ben-Arzi e al. discussed he sharp space ime decay properies of fundamenal soluions o he linear equaion iψ εδψ + Δ Ψ =, ε {,, }. In [6], Guo and Wang considered he eisence and scaering heory for he Cauchy problem of nonlinear Schrödinger equaions wih he form iu + Δ m u + fu=,.4 u,= ϕ,.5 where m is an ineger. Pecher and von Wahl in [5] proved he eisence of classical global soluions of.4.5 for he space dimensions n 7m for he case m. In he presen paper we deal wih Eq.. in which he difficuly arises from he fac ha he nonlineariy of P involves he firs and second derivaives u, ū, u and ū. This could cause he so-called loss of derivaives so long as we make direc use of he sandard mehods, such as he energy esimaes, he space ime esimaes, ec. In [4], Kenig e al. made a grea progress on he nonlinear Schrödinger equaion of he form u = iδu + Pu,ū, u, ū, R, R n, and proved ha he Cauchy problem is locally well-posed for small daa in he Sobolev space H s R n and in is weighed version by pushing forward he linear esimaes associaed wih he Schrödinger group {e iδ } and by inroducing suiable funcion spaces

3 48 C. Hao e al. / J. Mah. Anal. Appl where hese esimaes ac naurally. In he one-dimensional case, n =, he smallness assumpion on he size of he daa was removed by Hayashi and Ozawa [7] by using a change of variable o obain an equivalen sysem wih a nonlinear erm independen of u, where he new sysem can be reaed by he sandard energy mehod. Bu his mehod migh no be able o be applied o he fourh order nonlinear Schrödinger equaion including erms u and ū. Kao sudied in [] he Cauchy problem for he generalized Koreweg de Vries equaion u + 3 u + u u =,, R, by using he smoohing effec mehod. In he presen paper, we will discuss he local smoohing effecs of he uniary group {S} in order o overcome he loss of derivaives. More precisely, we will prove, in Secion, ha j S τfτ,dτ CT 3 j 4 F L L L + j 3 u, L. To consruc he work space, we have o sudy, in Secion 3, he boundedness properies of he maimal funcion [,T ] S. This idea is implici in he spliing argumen inroduced by Ginibre and Tsusumi [5] o deal wih uniqueness for he generalized KdV equaion. Finally, we will consider some special cases in Secion 4 o apply he esimaes we have obained. More precisely, we sudy he Cauchy problem.. wihou he resricion on he size of he iniial daa and ge he local wellposedness by he fied poin argumen. For convenience, we firs inroduce some noaions. S := e i 4 denoes he uniary group generaed by i 4 in L R. z denoes he conjugae of he comple number z. F u or û F u, respecively denoes he Fourier inverse, respecively ransform of u wih respec o all variables. S denoes he space of Schwarz funcions. And we denoe s, = H s, p = L p for p< and f l,,j = f H l R; j d = γ f / d j..6 γ l Throughou he paper, he consan C migh be differen from each oher. Now we sae he main resuls of his paper. Theorem. Case β 3. Given any polynomial P as in.3 wih β 3, hen, for any u H s+ R wih s 7/, here eiss a T = T u s+, > such ha he Cauchy problem.. has a unique soluion u defined in he ime inerval [,T] and saisfying and u C [,T]; H s R.7 s+/ u L R; L [,T]..8

4 C. Hao e al. / J. Mah. Anal. Appl Theorem. Case β =. Given any polynomial P as in.3 wih β =, hen, for any u H s+ R H 6 R; d wih s + /, here eiss a T = T u s+,, u 6,, > such ha he Cauchy problem.. has a unique soluion u defined in he ime inerval [,T] and saisfying and u C [,T]; H s R H 6 R; d.9 s+/ u L R; L [,T]... Local smoohing effecs We will prove he local smoohing effecs of Kao ype ehibied by he group {S} in his secion. Lemma. Local smoohing effec: homogeneous case. We have he following esimae 3/ Su L R;L [,T ] C u,. and he corresponding dual version 3/ S τfτ, dτ C f L R;L [,T ].. Proof.. and. can be derived from [, Theorem 4.] for which we omi he deails. More precisely,. yields, for [,T], ha 3/ S τfτ, dτ C f L R;L Now we urn o consider he inhomogeneous Cauchy problem: [,T ]..3 i u = 4 u + F,,, R,.4 u,=,.5 wih F S R R. We have he following esimae on he local smoohing effec in his inhomogeneous case. Lemma. Local smoohing effec: inhomogeneous case. The soluion u, of he Cauchy problem.4.5 saisfies j u, 3 j L R;L CT 4 F, [,T ] L R;L [,T ], j =,, 3..6

5 5 C. Hao e al. / J. Mah. Anal. Appl Proof. We formally ake Fourier ransform in boh variables and in Eq..4 and obain û = ˆFτ,ξ τ ξ 4. Consequenly, we have j u, e iτ e iξ ξ j τ ξ 4 ˆFτ,ξdξ dτ for j =,, 3. By he Plancherel heorem in he ime variable, we can ge j u, L F τ e iξ ξ j τ ξ ˆFτ,ξdξ 4 e iξ e iξ ξ j τ ξ ˆFτ,ξdξ 4 ξ j τ ξ 4 e i yξ L τ L e iyξ ˆF τ, y dy dξ ξ j τ ξ 4 dξ ˆF τ, y dy L τ L τ K j τ, y ˆF τ, y dy,.7 L τ where ˆF denoes he Fourier ransform of F in he ime variable, K j τ, l = e ilξ ξ j τ ξ 4 dξ, and he inegral is undersood in he principal value sense. In order o coninue he above esimae, we have o esimae K j τ, l ne. Since he proof for he case τ< is similar o he case τ>, we only give he proof for he case τ>. In fac, by changing he variables, we have, for τ>, ha K j τ, l = = e ilξ ξ j τ ξ 4 dξ e ilτ/4 η τ /4 η j τ τ /4 η 4 τ /4 dη

6 = = τ 4 j 3 4 C. Hao e al. / J. Mah. Anal. Appl e ilτ/4 η τ 4 j+ η j τ η 4 dη e ilτ/4 η η j η 4 dη. We disinguish he discussion ino hree cases according o he value of j. In he case j =, we have K τ, l = τ / = τ / [ = τ / 4 + e ilτ/4 η η η 4 dη e ilτ/4 η 4 + η + 4 e ilτ/4 ξ ξ dξ + 4 e ilτ/4 η η dη η + η e ilτ/4 ξ ξ dξ ] e ilτ/4 η + η η dη [ = τ / e 4 i ilτ/4 + e ilτ/4 sgn lτ /4 ] e ilτ/4 η + η η dη, + η dη η where we used he fac ha he Fourier ransform of / i.e., he kernel of he Hilber ransform is equal o i sgnξ. For he esimae of he second erm in he righ-hand side of he above, we inroduce an auiliary funcion. Le ϕ C R wih p ϕ [, ], ϕ in[, ] and ϕ, we have e iη ϕη + η η dη = F η η ϕη + η = F η η = ϕη i sgn Fη + η = i ϕη Fη + η ϕη sgn yfη + η y dy

7 5 C. Hao e al. / J. Mah. Anal. Appl and ϕη F η + η y dy C iη ϕη e + η η dη η dη C. + η Thus, we can ge τ / K τ, l C..8 τ,l Ne, we consider he case j =. We have as above K τ, l = τ /4 = τ /4 [ = τ /4 4 e ilτ/4 η η η 4 dη e ilτ/4 η 4 + η + 4 e ilτ/4 ξ ξ dξ + 4 e ilτ/4 η + η dη ] η + η dη e ilτ/4 ξ ξ dξ [ = τ /4 e 4 i ilτ/4 e ilτ/4 sgn lτ /4 and consequenly K τ, l [ τ /4 + ] + η dη = + π τ /4. ] e ilτ/4 η + η dη Thus, we obain τ /4 K τ, l C..9 τ,l For he case j = 3, we have K 3 τ, l = e ilτ/4 η η 3 η 4 dη

8 = C. Hao e al. / J. Mah. Anal. Appl e ilτ/4 η 4 + η + 4 η η + = [ 4 i e ilτ/4 + e ilτ/4 + sgn lτ /4 + Similar o he case j =, we can ge K 3 τ, l C. τ,l + η dη η ] e ilτ/4 η + η η dη.. Combining.8,.9, and., i yields K j τ, l C.. τ,l τ 3 j 4 Now we urn o he esimae of j u,. Due o.7, he above esimaes, he Young inequaliy, he Sobolev embedding heorem and he Hölder inequaliy, we obain, for [,T], ha j u, L C C C τ, The desired resul follows hen. Kj τ, y ˆF τ, y L dy τ τ, 3 j τ 4 K j τ, y τ 3 j 4 ˆF τ, y L dy τ τ 3 j 4 K j τ, F CT 3 j 4 F L L. L H 3 j 4 C F L 4 5 j L In general, he soluion u, of.4 may no vanish a =. However, by using he Parseval ideniy, we are able o show u,= u, = e iξ e iξ e iξ τ ξ 4 ˆFτ,ξdτ dξ τ ξ 4 ˆF s, ξ e iτs ˆF s, ξ ds dτ dξ e iτs dτ ds dξ τ ξ 4

9 54 C. Hao e al. / J. Mah. Anal. Appl e iξ e iξ ˆF s, ξ e iys ˆF s, ξ sgnse isξ4 ds dξ e iξ e isξ4 sgns ˆF s, ξ dξ ds Sssgns ˆF s, ds, which, combined wih., implies 3/ u, L R. Thus, by., he funcion w, w := u Su = i y S τfτdτ, dy e isξ4 ds dξ is he soluion of.4.5 and saisfies he esimae.6. Hence, for j =,, 3, j S τfτ,dτ = j w, L CT 3 j 4 F L L.. L 3. Esimaes for he maimal funcion We sar by saing an L -coninuiy resul for he maimal funcion [,T ] S. Lemma 3.. For any s>and any ρ>/4, i holds Su / C + T ρ u s,. 3. T j <j+ j= Proof. This is a special case of [3, Corollary.8]. For he convenience of readers, we give a direc and simple proof here. Denoe = T wih. We have Su = ST u T /4 e i e iξ ξ 4 T dξ u T /4 η η4 dη u

10 T /4 C. Hao e al. / J. Mah. Anal. Appl e i = S v T /4, e i y T /4 η η4 dηu y dy T /4 z η η 4 dηu T /4 z dz where v = u T /4. As he same way as he proofs in [3, Theorem.7], we can check ha Su / C u s, j= j <j+ holds for s>. Thus, we obain Su / T j <j+ j= = C j= j T /4 y<j+ T /4 k= k <k+ S vy / S v / CT /8 v s, T /8 u T /4 s, C + T s/4 u s,. For he case ρ s/4, we have he desired resul since + T s/4 + T ρ. For he case /4 <ρ<s/4, le s = 4ρ. Then, following he above process and using he Sobolev embedding heorem, we obain Su / C + T ρ u s, C + T ρ u s,. j= T j <j+ Thus, he proof of Lemma 3. is compleed. I is clear ha 3. yields Su / d C + T ρ u s,. 3. [,T ]

11 56 C. Hao e al. / J. Mah. Anal. Appl To esimae he maimal funcion [,T ] Su in he L - and l -norms, we shall use he following weighed inequaliy. Lemma 3.. We have he esimae [,T ] Su d C + T u 7, + u 4,,, 3.3 where l,,j is defined as in.6. Proof. We firs derive an esimae which will be used in he ne derivaion. Taking [,T] such ha f we have for any f d, T f= f + f f = f + f s ds. Thus, we can ge namely, and f T f [,T ] T f d + f s ds T f d + We urn o prove 3.3. Noicing ha f d C f + C f T f d + f d, f d. 3.4 Su = Su + 4iS 3 u, 3.5 we have, from 3.4 and he Fubini heorem, ha [,T ] Su d

12 T = T C T C T + C C. Hao e al. / J. Mah. Anal. Appl Su d d + Su d d + Su + T Su d + C Su d d S 4 u d d u + Su + 4 S 3 u d S 4 u + S 4 u d 4 u + S 4 u + 4 S 7 u d C u + u + T 3 u + T 4 u + T 4 u + T 7 u C + T u 7, + u 4,,. Thus, he desired resul is obained. 4. Wellposedness We will give he proofs of he main heorems in his secion. Proof of Theorem.. Similar o he proof in [4, Theorem 4.], we shall only consider he case s = 7/. The general case follows by combining his resul wih he fac ha he highes derivaives involved in ha proof always appear linearly and wih some commuaor esimaes see [] for he cases where s k + /, k Z +. For u H 9/ R, we denoe by u = T v = T u v he soluion of he linear inhomogeneous Cauchy problem i u = 4 u + P α v α, α v α,, R, 4. u,= u. 4. In order o consruc T as a conracion mapping in some space, we use he inegral equaion u = T v = Su i We inroduce he following work space: S τp α vτ α, α vτ α dτ. 4.3

13 58 C. Hao e al. / J. Mah. Anal. Appl { ZT D = w : [,T] R C: 4 w L L T δ ; w 7 D;, [,T ] / + T ρ w + w + w d D}, where δ</4 is a consan. We noice ha P α v α, α v α = 4 vr + 4 vr + R, 4.4 where R j = R j α v α, α v α for j =,, and R = R α v α 3, α v α 3. Thus, from he inegral equaion 4.3, he Hölder inequaliy, he Moser inequaliy, 4.4 and., we can ge, for v Z D T, ha 4 u L L = T T 4 u, d / / 4 Su d + T / 4 Su +, T / u 9, + C T T T / u 9, + CT /4 4 vr + C / R d T L L T / u 9, + CT /4 4 v L L 4 S τpτdτ d S τ Pτdτ d / / S τ 4 vr + 4 vr + R τ dτ d + CT /4 4 vr L L R j L L j=, + CT / R. By using he commuaor esimaes [] abou he las erm and he Sobolev embedding heorem, we may bound his by /

14 C. Hao e al. / J. Mah. Anal. Appl T / u 9, + CT /4 4 v L v L + v + v L L + v + v + v γ 3 L + CT v 3 γ 3 L 7 + v, 7, T / u 9, + CT /4 4 v L v L + v + v L L + v γ CT v 3 γ 3, 7 + v, 7, T / u 9, + CT /4 T δ D + D γ 3 + CT D 3 + D γ 3 T δ, where we ake T so small ha 4.5 T / δ u 9, + CT /4 D + D γ 3 + CT δ D 3 + D γ in he las sep. By he properies of he Sobolev spaces cf. [], he inegral equaion 4.3, he group properies.3 and 4.5, we have u 7, u + 7/ u u + + 7/ u 7, + S τpτdτ + 7/ u S τpτdτ S τpτ dτ + 3/ u 7, + T P + P L L C u 7, + CT v 3 γ v, 7, + C 4 v L L R j L L + CT / R j=, S τ Pτdτ C u 7, + CT v 3 γ v, 7, + C 4 v L v L + v + v L L + v γ 3 7, + CT / v 3 γ v, 7, C u 7, + CT D3 + D γ 3 + CT δ D + D γ 3 + CT / D 3 + D γ 3

15 6 C. Hao e al. / J. Mah. Anal. Appl D, where in he las sep, we have chosen D u 9,, and T small enough such ha CT D + D γ 3 + CT δ D + D γ 3 + CT / D + D γ Similar o he derivaion of 4.5 and 4.7, we obain, from 3., ha / u + u + u d + + / Su + S u + S u d [ S τpτdτ + ] S τpτdτ d C + T ρ u 7, + C + Tρ / j= S τpτdτ S τ j Pτdτ C + T ρ{ u 7, + C 4 v L L v + v + v L L + v γ 3 7, + C T / + T v 3 γ 3 } 7 + v, 7, + T ρ D, 4. where ρ>/4and 4.9 has been used. Therefore, choosing a D as in 4.8 and hen aking a T sufficienly small such ha boh 4.6 and 4.9 hold, we obain ha he mapping T = T u : Z D T ZD T is well defined. For convenience, we denoe { Λ T w = ma DT δ 4 w L L ; w 7, ; [,T ] + T ρ w + w + w d 3, / }. 4.

16 C. Hao e al. / J. Mah. Anal. Appl To show ha T is a conracion mapping, we apply he esimaes obained in 4.5, 4.7 and 4. o he following inegral equaion: T v T w = S τ [ P α v α, α v α P α w α, α w α ] τ dτ, and obain, for v,w ZT D, ha Λ T T v T w CT δ Λ T v w [Λ T v + Λγ T v + Λ T w + Λγ T w ] CT δ D + D γ Λ T v w, 4. where he consan C depends only on he form of P and he linear esimaes.,.3,.6 and 3.. Thus, we can choose <T saisfying 4.6, 4.9 and CT δ D + D γ /. 4.3 Therefore, for hose T, saisfying 4.6, 4.9 and 4.3, he mapping T u is a conracion mapping in ZT D. Consequenly, by he Banach conracion mapping principle, here eiss a unique funcion u ZT D such ha T u u = u which solves he Cauchy problem. By he mehod given in [4, Theorem 4.], we can prove he persisence propery of u in H 7/, i.e., u, C [,T]; H 7/, he uniqueness and he coninuous dependence on he iniial daa of soluion. For simpliciy, we omi he res of he proof. Proof of Theorem.. For simpliciy, we assume P α u α, α ū α = u. I will be clear, from he argumen presened below, ha his does no represen any loss of generaliy. And as in he proof of Theorem., we consider he case s = + /. For u H 5/ R H 3 R; d, we denoe by u = T v = T u v he soluion of he linear inhomogeneous Cauchy problem i u = 4 u + v,, R, 4.4 u,= u. 4.5 We will consider he inegral equaion u = T v = Su i in he following work space: S τ v dτ, 4.6

17 6 C. Hao e al. / J. Mah. Anal. Appl ZT {v E = : [,T] R C: v 3 E; v, 6,, E; v L L T δ ; + T v, L L where <δ</4 is a consan and he norm is defined as { v Z E = ma v 3 T, ; v 6,, ; ET δ v L L ; + T } v, L L. } E, We noice ha v = v v + R α v α. From he inegral equaion 4.3,., he Sobolev embedding heorem, 4.4,., he Moser inequaliy and he commuaor esimaes[], we can ge, as in 4.5, ha u L L = T T u, d / / Su d + T T / u 5, + CT /4 v v L L + C T / u 5, + CT /4 v L L T / u 5, + CT /4 T δ + T 4 E + CT E S τ vτ dτ d / R d / v, d + CT v 3, T δ, if we ake T sufficienly small such ha 4.7 T / δ u 5, + CT /4 + T 4 E + CT δ E. 4.8 We can rewrie 4.6 as u = S u i S τ v dτ. 4.9 From 3.3 and 3.5, we have

18 + T C. Hao e al. / J. Mah. Anal. Appl u, d C u 9, + u 6,, + C u 9, + u 6,, + C + S S τ v dτ 4,, C u 9, + u 6,, + CT v, + C u 9, + u 6,, + CT v, + C u 9, + u 6,, + CT v, + 4 j= S τ v dτ d S τ v dτ 9, S τ v 4,, dτ 4 j= S τ j+ v dτ j+ S τ v j+5 4iτS τ v dτ C u 9, + u 6,, + CT v, + 4 j= [ j+ v + 4τ j+5 v ] dτ C u 9, + u 6,, + CT v, + T v 6,, v 9, + CT v, C u 9, + u 6,, + CT + T v, + T v 6,, v 9, C u 5, + u 6,, + CT + TE E, where we have chosen E u 5, + u 6,, and T so small ha 4. CT + TE. 4. Similar o 4., we have

19 64 C. Hao e al. / J. Mah. Anal. Appl u 6,, = 6 j u j= [ 6 j S u + 4T j+3 S u j= + S τ j v dτ C u 6,, + T u 9, + T + T v, ] v 6,, v 9, C u 6,, + T u 9, + TE + T E E. 4. As in he derivaion of 4.7, we can ge u 3 E. 4.3, The esimaes 4.7, 4., 4. and 4.3 yield u ZT E. Thus, we can fi E and T as above such ha T is a conracion mapping from ZT E o iself. By he sandard argumen used in he proof of Theorem., we can complee he proof for which we omi he deails. Acknowledgmens The auhors hank he referees for valuable commens and suggesions for he original manuscrip. References [] M. Ben-Arzi, H. Koch, J.C. Sau, Dispersion esimaes for fourh order Schrödinger equaions, C. R. Acad. Sci. Paris Sér. I Mah [] J. Bergh, J. Löfsröm, Inerpolaion Spaces, an Inroducion, Springer-Verlag, 976. [3] T. Cazenave, An Inroducion o Nonlinear Schrödinger Equaions, Teos de Méodos Maemáicos, vol. 6, Universidade Federal do Rio de Janeiro. [4] P. Consanin, J.C. Sau, Local smoohing properies of dispersive equaions, J. Amer. Mah. Soc [5] J. Ginibre, Y. Tsusumi, Uniqueness for he generalized Koreweg de Vries equaions, SIAM J. Mah. Anal [6] B.L. Guo, B.X. Wang, The global Cauchy problem and scaering of soluions for nonlinear Schrödinger equaions in H s, Differenial Inegral Equaions [7] N. Hayashi, T. Ozawa, Remarks on nonlinear Schrödinger equaions in one space dimension, Differenial Inegral Equaions [8] Z.H. Huo, Y.L. Jia, The Cauchy problem for he fourh-order nonlinear Schrödinger equaion relaed o he vore filamen, J. Differenial Equaions

20 C. Hao e al. / J. Mah. Anal. Appl [9] V.I. Karpman, Sabilizaion of solion insabiliies by higher-order dispersion: fourh order nonlinear Schrödinger-ype equaions, Phys. Rev. E R336 R339. [] T. Kao, On he Cauchy problem for he generalized Koreweg de Vries equaion, in: Sudies in Applied Mah., in: Adv. Mah. Supp. Sud., vol. 8, Academic Press, 983, pp [] T. Kao, G. Ponce, Commuaor esimaes and he Euler and Navier Sokes equaions, Comm. Pure Appl. Mah [] C.E. Kenig, G. Ponce, L. Vega, Oscillaory inegrals and regulariy of dispersive equaions, Indiana Univ. Mah. J [3] C.E. Kenig, G. Ponce, L. Vega, Well-posedness of he iniial value problem for he Koreweg de Vries equaion, J. Amer. Mah. Soc [4] C.E. Kenig, G. Ponce, L. Vega, Small soluions o nonlinear Schrödinger equaions, Ann. Ins. H. Poincaré, Anal. Non Linéaire [5] H. Pecher, W. von Wahl, Time dependen nonlinear Schrödinger equaions, Manuscripa Mah [6] J. Segaa, Well-posedness for he fourh order nonlinear Schrödinger ype equaion relaed o he vore filamen, Differenial Inegral Equaions [7] J. Segaa, Remark on well-posedness for he fourh order nonlinear Schrödinger ype equaion, Proc. Amer. Mah. Soc