ON A CLASS OF DERIVATIVE NONLINEAR SCHRÖDINGERTYPE EQUATIONS IN TWO SPATIAL DIMENSIONS


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1 ON A CLASS OF DERIVATIVE NONLINEAR SCHRÖDINGERTYPE EQUATIONS IN TWO SPATIAL DIMENSIONS JACK ARBUNICH, CHRISTIAN KLEIN, AND CHRISTOF SPARBER Absrac. We presen analyical resuls and numerical simulaions for a class of nonlinear dispersive equaions in wo spaial dimensions. These equaions are of (derivaive) nonlinear Schrödinger ype and have recenly been obained in [11] in he conex of nonlinear fiber opics. In conras o he usual nonlinear Schrödinger equaion, his new model incorporaes he addiional effecs of selfseepening and parial offaxis variaions of he group velociy of he laser pulse. We prove global inime exisence of he corresponding soluion for various choices of parameers, exending earlier resuls of []. In addiion, we presen a series of careful numerical simulaions concerning he (in)sabiliy of nonlinear ground saes and he possibiliy of finieime blowup. Conens 1. Inroducion 1. Ground saes 4 3. Numerical mehod for he imeevoluion 8 4. Global wellposedness wih full offaxis variaion (In)sabiliy properies of ground saes wih full offaxis variaion Wellposedness resuls for he case wih parial offaxis variaion Numerical sudies for he case wih parial offaxis variaion References 7 1. Inroducion This work is devoed o he analysis and numerical simulaions for he following class of nonlinear dispersive equaions in wo spaial dimensions: (1.1) ip ε u + u + (1 + iδ ) ( u σ u ) =, u = = u (, x ), where x = (, x ) R, δ = (δ 1, δ ) R is a given vecor wih δ 1, and σ > is a parameer describing he srengh of he nonlineariy. In addiion, for < ε 1, we denoe by P ε he following linear differenial operaor k (1.) P ε = 1 ε x j, k. j=1 Dae: June 1, 18. Mahemaics Subjec Classificaion. 65M7, 65L5, 35Q55. Key words and phrases. nonlinear Schrödinger equaion, derivaive nonlineariy, ground saes, finieime blowup, selfseepening, specral resoluion, RungeKua algorihm. This publicaion is based on work suppored by he NSF hrough gran no. DMS Addiional suppor by he NSF research nework KINe is also acknowledged. C.K. also acknowledges parial suppor by he ANRFWF projec ANuI and by he MarieCurie RISE nework IPaDEGAN. 1
2 J. ARBUNICH, C.KLEIN, AND C. SPARBER Indeed, we shall mainly be concerned wih (1.1) rewrien in is evoluionary form: (1.3) i u + Pε 1 u + Pε 1 (1 + iδ ) ( u σ u ) =, u = = u (, x ). Here and in he following, P s ε, for any s R, is he nonlocal operaor defined hrough muliplicaion in Fourier space using he symbol k ) s, P ε s (ξ) = (1 + ε ξj k, j=1 where ξ (ξ 1, ξ ) R is he Fourier variable dual o x = (, x ). For s = 1 his obviously yields a bounded operaor Pε 1 : L (R ) L (R ). In addiion, P ε is seen o be uniformly ellipic provided k =. The inclusion of P ε implies ha (1.1), or equivalenly (1.3), shares a formal similariy wih he wellknown BenjaminBonaMahoney equaion for unidirecional shallow waer waves [3, 4]. However, he physical conex for (1.1) is raher differen. Equaions of he form (1.1) have recenly been derived in [11] as an effecive descripion for he propagaion of high inensiy laser beams. This was par of an effor o remedy some of he shorcomings of he classical (focusing) nonlinear Schrödinger equaion (NLS), which is obained from (1.1) when ε = δ 1 = δ =, i.e. (1.4) i u + u + u σ u =, u = = u (, x ). The NLS is a canonical model for slowly modulaed, selffocusing wave propagaion in a weakly nonlinear dispersive medium. The choice of σ = 1 hereby corresponds o he physically mos relevan case of a Kerr nonlineariy, cf. [13, 35]. Equaion (1.4) is known o conserve, among oher quaniies, he oal mass M() u(, ) L = u L. A scaling consideraion hen indicaes ha (1.4) is L criical for σ = 1 and L  supercriical for σ > 1. I is well known ha in hese regimes, soluions o (1.4) may no exis for all R, due o he possibiliy of finieime blowup. The laer means ha here exiss a ime T <, depending on he iniial daa u, such ha lim u(, ) L = +. T In he physics lieraure his is referred o as opical collapse, see [13]. In he L criical case, here is a sharp dichoomy characerizing he possibiliy of his blowup: Indeed, one can prove ha he soluion u o (1.4) wih σ = 1 exis for all R, provided is oal mass is below ha of he nonlinear ground sae, i.e., he leas energy soluion of he form u(, x) = e i Q(x). Soluions u whose L norm exceeds he norm of Q, however, will in general exhibi a selfsimilar blowup wih a profile given by Q (up o symmeries), see [3, 31]. In urn, his also implies ha saionary saes of he form e i Q(x) are srongly unsable. For more deails on all his we refer he reader o [6, 13, 35] and references herein. In comparison o (1.4), he new model (1.3) includes wo addiional physical effecs. Firsly, here is an addiional nonlineariy of derivaive ype which describes he possibiliy of selfseepening of he laser pulse in he direcion δ R. Secondly, he operaor P ε describes offaxis variaions of he group velociy of he beam. The case k = is hereby referred o as full offaxis dependence, whereas for k = 1 he model incorporaes only a parial offaxis variaion. Boh of hese effecs become more pronounced for high beam inensiies (see [11]) and boh are expeced o have a significan influence on he possibiliy of finieime blowup. In his
3 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 3 conex, i is imporan o noe ha (1.3) does no admi a simple scaling invariance analogous o (1.4). Hence, here is no clear indicaion of sub or supercriical regimes for equaion (1.3). A leas formally, hough, equaion (1.3) admis he following conservaion law (1.5) M ε () Pε 1/ u(, ) 1/ L = Pε u L, generalizing he usual mass conservaion. In he case of fulloff axis dependence, (1.5) yields an apriori bound on he H 1 norm of u, ruling ou he possibiliy of finieime blowup. However, he siuaion is more complicaed in he case wih only a parial offaxis variaion. The laer was sudied analyically in he recen work [], bu only for he much simpler case wihou selfseepening, i.e., only for δ 1 = δ =. I was rigorously shown ha in his case, even a parial offaxis variaion (mediaed by P ε wih k = 1) can arres he blowup for all σ <. In paricular, his allows for nonlineariies larger han he L criical case, cf. Secion 6 for more deails. One moivaion for he presen work is o give numerical evidence for he fac ha hese resuls are indeed sharp, and ha one can expec finieime blowup as soon as σ. Furhermore, he curren work aims o exend he analysis of [] o siuaions wih addiional selfseepening, i.e., δ, and o provide furher insigh ino he qualiaive inerplay beween he laer effec and he one semming from P ε. From a mahemaical poin of view, he addiion of a derivaive nonlineariy makes he quesion of global wellposedness versus finieime blowup much more involved. Derivaive NLS and heir corresponding ground saes are usually sudied in one spaial dimension only, see e.g. [1, 8, 1, 14, 7, 8, 36, 37] and references herein. For σ = 1, he classical onedimensional derivaive NLS is known o be compleely inegrable. Neverheless, here has only very recenly been a breakhrough in he proof of global inime exisence for his case, see [15, 16]. In conras o ha, [8] gives srong numerical indicaions for a selfsimilar finieime blowup in derivaive NLS wih σ > 1. The blowup hereby seems o be a resul of he selfseepening effec in he densiy ρ = u, which generically undergoes a imeevoluion similar o a dispersive shock wave formaion in Burgers equaion. To our knowledge here is, however, no rigorous proof of his phenomenon is currenly available. In wo and higher dimensions, even he local inime exisence of soluions o derivaive NLS ype equaions seems o be largely unknown, le alone any furher qualiaive properies of heir soluions. In view of his, he presen paper aims o shine some ligh on he specific varian of wodimensional derivaive NLS given by (1.3). Excep for is physical significance, his class of models also has he advanage ha he inclusion of (parial) offaxis variaions via P ε are expeced o have a srong regularizing effec on he soluion, and hus allow for several sable siuaions wihou blowup. The organizaion of our paper is hen as follows: In Secion, we shall numerically consruc nonlinear ground sae soluions o (1.1), or equivalenly (1.3). For he sake of illusraion, we shall also derive explici formulas for 1D ground saes and compare hem wih wellknown resuls for he classical (derivaive) NLS. Cerain perurbaions of such ground saes will form he class of iniial daa considered in he numerical imeinegraion of (1.3). The numerical algorihm used o perform he respecive simulaions is deailed in Secion 3. In i, we also include several basic numerical ess which compare he new model (1.3) o he classical (derivaive) NLS. Analyical resuls yielding global wellposedness of (1.3) wih eiher full or parial offaxis variaions are given in Secions 4 and 6, respecively.
4 4 J. ARBUNICH, C.KLEIN, AND C. SPARBER In he former case, he picure is much more complee, which allows us o perform a numerical sudy of he (in)sabiliy properies of he corresponding nonlinear ground saes in Secion 5. In he case wih only parial offaxis variaions, he problem of global exisence is more complicaed and one needs o disinguish beween he cases where he acion of P ε is eiher parallel or orhogonal o he selfseepening. Analyically, only he former case can be reaed so far (see Secion 6). Numerically, however, we shall presen simulaions for boh of hese cases in Secion 7.. Ground saes In his secion, we focus on imeperiodic soluions o (1.1) given by (.1) u(,, x ) = e i Q(, x ). The funcion Q hen solves he following nonlinear parial differenial equaion: (.) P ε Q = Q + (1 + iδ )( Q σ Q), subjec o he requiremen ha Q(x) as x. A soluion Q o (.1) wih minimal energy will be called a ground sae. For he classical NLS, i.e. ε = δ 1 = δ =, he ground sae Q is he unique, radial and nonnegaive soluion o (.1), cf. [13, 35]. Ground saes will be an imporan benchmark for our numerical simulaions below. Remark.1. Noe ha in he ansaz (.1) we only allow for a simple imedependence exp(iω) wih ω = 1 in (.1). This is no a resricion for he usual D NLS, given is scaling invariance, bu i is a resricion for our model in which his invariance is broken..1. Explici ground sae soluions in 1D. In one spaial dimension, equaion (.) allows for explici formulas for he ground sae, which will serve as a basic illusraion for he combined effecs of selfseepening and offaxis variaions. Indeed, in one spaial dimension, equaion (1.1) simplifies o (.3) i(1 ε x) u + xu + (1 + iδ x )( u σ u) =. Seeking he ground sae soluion in he form (.1) hus yields he following ordinary differenial equaion: (.4) (1 + ε )Q + ( Q σ 1)Q + iδ( Q σ Q) =. To solve his equaion, we shall use he polar represenaion for Q(x) C Q(x) = A(x)e iθ(x), A(x), θ(x) R, and impose he requiremen ha lim x ± A(x) =. Plugging his ansaz ino (.4) and isolaing he real and imaginary par yields he following coupled sysem: ( 1 + ε ) A + (A σ 1)A Aθ ( (1 + ε )θ + δa σ) =, ( 1 + ε )( Aθ + θ A ) + (σ + 1)δA σ A =. Muliplying he second equaion by A and inegraing from o x gives (1 + ε )θ = (σ + 1)δAσ. (σ + 1) Using he above, we infer ha he ampliude solves (.5) ( 1 + ε ) A + (A σ 1)A + (σ + 1)δ 4(1 + ε )(σ + 1) A4σ+1 =,
5 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 5 while he phase is given aposeriori hrough (σ + 1)δ (.6) θ(x) = (1 + ε )(σ + 1) x A σ (y) dy. Afer some lenghy compuaion, similar o wha is done for he usual NLS, cf. [13], he soluion o (.5) can be wrien in he form (.7) A(x) = (σ + 1) ( σx 1 + K ε,δ cosh 1/(σ) ), 1+ε where K ε,δ = 1 + δ 1+ε >. In view of (.6), his implies ha he phase funcion θ is given by ( ( ) ) 1 + ε (.8) θ(x) = sgn(δ)(σ + 1) arcan 1 + K ε,δ e σx 1+ε, δ where we omied a physically irrelevan consan in he phase (clearly, Q is only unique up o muliplicaion by a consan phase). Noe ha in he case wih no selfseepening δ =, he phase θ(x) =. Thus, Q(x) A(x) and we find ( ) σx Q(x) = (σ + 1) 1/(σ) sech 1/σ. 1 + ε For ε =, his is he wellknown ground sae soluion o (1.4) in one spaial dimension, cf. [13, 35]. We noice ha he effec of adding he offaxis dispersion (ε > ) widens he profile, causing i o decay more slowly as x ± as can be seen in Fig. 1 on he lef. On he righ of Fig. 1, i is shown ha he maximum of he ground sae decreases wih σ bu ha he peak becomes more compressed Q Q x x Figure 1. Ground sae soluion o (.7) wih δ = : On he lef for σ = 1 and ε = (blue), ε =.5 (green) and ε = 1 (red). On he righ for ε = 1 and σ = 1 (blue), σ = (green) and σ = 3 (red). Remark.. The (σgeneralized) onedimensional derivaive NLS can be obained from (.3) by puing ε =, rescaling u(, x) = δ 1/(σ) ũ(, x),
6 6 J. ARBUNICH, C.KLEIN, AND C. SPARBER and leing δ. Denoing he rescaled ground sae by Q = Ãei θ(x), we ge from (.7) and (.8) he ground sae of he derivaive NLS, i.e., Ã(x) = ( (σ + 1) sech(σx) ) 1/(σ), θ(x) = (σ + 1) arcan(e σx ). The sabiliy of hese saes has been sudied in, e.g., [8, 7, 14]... Numerical consrucion of ground saes. In more han one spaial dimension, no explici formula for Q is known. Insead, we shall numerically consruc Q by following an approach similar o hose in [, 4]. Since we can expec Q o be rapidly decreasing, we use a Fourier specral mehod and approximae F(Q) Q(ξ 1, ξ ) = 1 π R Q(, x )e ix1ξ1 e ixξ d dx, by a discree Fourier ransform which can be efficienly compued via he Fas Fourier Transform (FFT). In an abuse of noaion, we shall in he following use he same symbols for he discree and coninuous Fourier ransform. To apply FFTs, we will use a compuaional domain of he form (.9) Ω = [ π, π]l x1 [ π, π]l x, and choose L x1, L x > sufficienly large so ha he obained Fourier coefficiens of Q decrease o machine precision, roughly 1 16, which in pracice is slighly larger due o unavoidable rounding errors. Now, recall ha for a soluion of he form (.1) o saisfy (1.1), he funcion Q needs o solve (.). In Fourier space, his equaion akes he simple form Q(ξ 1, ξ ) = Γ ε F( Q σ Q)(ξ 1, ξ ), where (1 δ 1 ξ 1 δ ξ ) Γ ε (ξ 1, ξ ) = 1 + ξ1 + ξ +. ε k i=1 ξ i For δ 1 = δ =, he soluion Q can be chosen o be real, bu his will no longer be rue for δ 1,. In he laer siuaion, we will decompose Q(, x ) = α(, x ) + iβ(, x ), and separae (.) ino is real and imaginary par, yielding a coupled nonlinear sysem for α, β. By using FFTs, his is equivalen o he following sysem for α and β: α(ξ 1, ξ ) Γ ( (α ε F + β ) ) σ α β(ξ 1, ξ ) Γ ε F (ξ 1, ξ ) =, ( (α + β ) ) σ β (ξ 1, ξ ) =. Formally, he sysem can be wrien as M( q) = where q = ( α, β) and solved via a Newon ieraion. One hereby sars from an iniial ierae q () and compues he nh ierae via he well known formula (.1) q (n) = q (n 1) J ( q (n 1)) 1 ( M q (n 1)) n N, where J is he Jacobian of M wih respec o q. Since our required numerical resoluion makes i impossible o direcly compue he acion of he inverse Jacobian, we insead employ a Krylov subspace approach as in [34]. Numerical experimens show ha when he iniial ierae q () is sufficienly close o he final soluion, we obain he expeced quadraic convergence of our scheme and reach a precision of order 1 1 afer only 4 o 8 ieraions.
7 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 7 As a basic es case, we compue he ground sae of he sandard wodimensional focusing NLS wih σ = 1, using he iniial ierae q q () (x1, x ) = sech x1 + x on he compuaional domain (.9) wih Lx1 = Lx = 5. By choosing Nx1 = Nx = 9 many Fourier modes, we have afer seven ieraions of (.1) a residual smaller han 1 1. The obained soluion is given on he lef of Fig.. As expeced, he soluion is radially symmeric. Figure. Ground sae soluion o equaion (1.1) wih σ = 1 and δ = : On he lef for ε =, in he middle for ε = 1 and k = 1 (parial offaxis dependence), on he righ for ε = 1 and k = (full offaxis dependence). The numerical ground sae soluion hereby obained hen be used as an iniial ierae for he siuaion wih nonvanishing ε and δ, as follows: Sep 1: In he case wihou selfseepening δ1 = δ =, he ieraion is sraighforward even for relaively large values such as ε = 1. I can be seen in he middle of Fig., ha he ground sae for ε = 1 and k = 1 is no longer radially symmeric. As an effec of he parial offaxis variaion, he soluion is elongaed in he x1 direcion. In he case of full offaxis dependence, he ground sae for he same value of ε = 1 can be seen in Fig. on he righ. The soluion is again radially symmeric, bu as expeced less localized han he ground sae of he classical sandard NLS. This is consisen wih he explici formulas for Q found in he one dimensional case above. Sep : In he case wih selfseeping δ1 = δ = 1, smaller inermediae seps have o be used in he ieraions: We incremen δ, by firs varying only δ in seps of., always using he las compued value for Q as an iniial ierae for he slighly larger δ. The resuling soluion Q can be seen in Fig. 3. Noe ha he imaginary par of Q is of he same order of magniude as he real par. Sep 3: In order o combine boh effecs wihin he same model, we shall use he ground sae obained for ε = and δ 6= as an iniial ierae for he case of nonvanishing ε. In Fig. 4 we show on he lef he ground sae for ε = 1, k = 1, δ1 = and δ = 1, when he acion of Pε is orhogonal o he selfseepening. When compared o he case wih ε =, he soluion is seen o be elongaed in he x1 direcion. Nex, we simulae when Pε acs parallel o he selfseepening, ha is when ε = 1, k = 1, δ1 = 1 and δ =. The resul is shown in he middle of Fig. 4. In comparison o he former case, he imaginary par of he soluion is essenially roaed clockwise by 9 degrees. The elongaion effec in he x1 direcion is sill visible bu less pronounced. Sep 4: For σ > 1 he ground saes become increasingly peaked, as is seen from he 1D picure in Figure 1. Hence, o consruc ground saes for higher nonlinear
8 8 J. ARBUNICH, C.KLEIN, AND C. SPARBER Figure 3. Ground sae soluion Q o equaion (1.1) wih σ = 1, ε = δ1 = and δ = 1: On he lef, he real par of Q, on he righ is imaginary par. Figure 4. Real and imaginary pars of he ground sae soluion o equaion (1.1) wih σ = 1: On he lef for ε = 1, k = 1, δ1 = and δ = 1, in he middle for ε = 1, k = 1, δ1 = 1 and δ =, and on he righ for ε = 1, k =, δ1 = and δ = 1. powers in D, we will consequenly require more Fourier coefficiens o effecively resolve hese soluions. To his end, we work on he numerical domain (.9) wih Lx1 = Lx = 3 and Nx1 = Nx = 1 Fourier modes. We use he ground sae obained for σ = 1 as an iniial ierae for he case σ =, 3, and follow he same program as oulined above. 3. Numerical mehod for he imeevoluion 3.1. A Fourier specral mehod. In his secion, we briefly describe he numerical algorihm used o inegrae our model equaion in is evoluionary form (1.3). Afer a Fourier ransformaion, his equaion becomes σ u), \ u b = ipbε 1 (ξ) ξ u b (1 δ ξ)( u ξ R. Approximaing he above by a discree Fourier ransform (via FFT) on a compuaional domain Ω given by (.9), yields a finie dimensional sysem of ordinary
9 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 9 differenial equaions, which formally reads (3.1) û = L ε û + N ε (û). Here L ε = ipε 1 ξ is a linear, diagonal operaor in Fourier space, and N ε (û) has a nonlinear and nonlocal dependence on û. Since L ε can be large, equaion (3.1) belongs o a family of siff ODEs, for which several efficien numerical schemes have been developed, cf. [17, 1] where he paricular siuaion of semiclassical NLS is considered. Driscoll s composie RungeKua (RK) mehod [1] has proven o be paricularly efficien and hus will also be applied in he presen work. This mehod uses a siffly sable hird order RK mehod for he high wave numbers of L ε and combines i wih a sandard explici fourh order RK mehod for he low wave numbers of L ε and he nonlinear par N ε (û). Despie combining a hird order and a fourh order mehod, his approach yields fourh order inime convergence in many applicaions. Moreover, i provides an explici mehod wih much larger ime seps han allowed by he usual fourh order sabiliy condiions in siff regimes. Remark 3.1. The evoluionary form of our model (1.3) is in many aspecs similar o he wellknown DaveySewarson (DS) sysem, which is a nonlocal NLS ype equaion in wo spaial dimensions, cf. [9, 35]. In [1, 3, 5], he possibiliy of selfsimilar blowup in DS is sudied, using a numerical approach similar o ours. As a firs basic es of consisency, we apply our numerical code o he cubic NLS in D i.e. equaion (1.4) wih σ = 1. As iniial daa u we ake he ground sae Q, obained numerically as oulined in Secion above. We use N = 1 imeseps for imes 1. In his case, we know ha he exac imedependen soluion u is simply given by u = Qe i. Comparing his o he numerical soluion obained a = 1 yields an L difference of he order of 1 1. This verifies boh he code for he imeevoluion and he one for he ground sae Q which in iself is obained wih an accuracy of order 1 1. Thus, he imeevoluion algorihm evolves he ground sae wih he same precision as wih which i is known. For general iniial daa u, we shall conrol he accuracy of our code in wo ways: On he one hand, he resoluion in space is conrolled via he decrease of he Fourier coefficiens wihin (he finie approximaion of) û. The coefficiens of he highes wavenumbers hereby indicae he order of magniude of he numerical error made in approximaing he funcion via a runcaed Fourier series. On he oher hand, he qualiy of he imeinegraion is conrolled via he conserved quaniy M ε () defined in (1.5). Due o unavoidable numerical errors, he laer will numerically depend on ime. For sufficien spaial resoluion, he relaive conservaion of M ε () will overesimae he accuracy in he imeinegraion by 1 orders of magniude. 3.. Reproducing known resuls for he classical NLS. As already discussed in he inroducion of his paper, he cubic NLS in wo spaial dimensions is L  criical and is ground sae soluion Q is srongly unsable. Indeed, any perurbaion of Q which lowers he L norm of he soluion below ha of Q iself, is known o produce purely dispersive, global inime soluions which behave like he free ime evoluion for large 1. However, perurbaions ha increase he L norm above ha of Q are expeced o generically produce a (selfsimilar) blowup in finie ime. This behavior can be reproduced in our simulaions. To do so, we ake iniial daa of he form (3.) u (, x ) = Q(, x ).1e x 1 x, and work on he numerical domain Ω given by (.9) wih L x1 = L x = 3. We will use N = 5 imeseps wihin 5. We can see on he righ of Fig. 5 ha he L norm of he soluion decreases monoonically, indicaing purely dispersive
10 1 J. ARBUNICH, C.KLEIN, AND C. SPARBER behavior. The ploed absolue value of he soluion a = 5 confirms his behavior. In addiion, M ε () is conserved o beer han 1 13, indicaing ha he problem is indeed well resolved in ime Figure 5. Soluion o he classical NLS (1.4) wih σ = 1 and iniial daa (3.): on he lef u a = 5, and on he righ he L norm of he soluion as a funcion of. Remark 3.. Noe ha we effecively run our simulaions on Ω T, insead of R. As a consequence, he periodiciy will afer some ime induce radiaion effecs appearing on he opposie side of Ω. The reamen of (large) imes > 5 herefore requires a larger compuaional domain o suppress hese unwaned effecs. Nex, for iniial daa of he form (3.3) u (, x ) = Q(, x ) +.1e x 1 x, we again use N = 5 ime seps for. As can be seen in Fig. 6 on he righ, here is numerical indicaion for finieime blowup. The code is sopped a = 1.89 when he relaive error in he conservaion of M ε () drops below 1 3. The soluion for = 1.88 can be seen on he lef of Fig. 6. This is in accordance wih he selfsimilar blowup esablished by Merle and Raphaël, cf. [3, 31]. In paricular, we noe ha he resul does no change noably if a higher resoluion in boh x and is used Figure 6. Soluion o he classical NLS (1.4) wih σ = 1 and iniial daa (3.3): on he lef u a = 1.88 and on he righ he L norm of he soluion as a funcion of.
11 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 11 Remark 3.3. We wan o poin ou ha here are cerainly more sophisicaed mehods available o numerically sudy selfsimilar blowup, see for insance [4, 8, 35] for he case of NLS ype models, as well as [19, ] for he analogous problem in KdV ype equaions. However, hese mehods will no be useful for he presen work, since, as noed before, he model (1.1) does no admi a simple scaling invariance, which is he underlying reason for selfsimilar blowup in NLS and KdV ype models. As a resul, all our numerical findings concerning finieime blowup have o be aken wih a grain of sal. An apparen divergence of cerain norms of he soluion or overflow errors produced by he code can indicae a blowup, bu migh also jus indicae ha one has run ou of resoluion. The resuls repored in his paper herefore need o be undersood as being saed wih respec o he given numerical resoluion. However, we have checked ha hey remain sable under changes of he resoluion wihin he accessible limis of he compuers used o run he simulaions Timedependen change of variables in he case wih selfseepening. In he case of selfseepening, he abiliy o produce an accurae numerical imeinegraion in he presence of a derivaive nonlineariy (δ ) becomes slighly more complicaed. The inclusion of such a nonlineariy can lead o localized iniial daa moving (relaively fas) in he direcion chosen by δ. In urn, his migh cause he numerical soluion o hi he boundary of our compuaional domain Ω. To avoid his issue, we shall insead perform our numerical compuaions in a moving reference frame, chosen such ha he maximum of u(, x) remains fixed a he origin. More precisely, we consider he ransformaion x x y(), and denoe v() = ẏ(). Under his ransformaion (1.3) becomes (3.4) i u iv u + Pε 1 u + Pε 1 (1 + iδ ) ( u σ u ) =. The quaniy v() = (v 1 (), v ()) is hen deermined by he condiion ha he densiy ρ = u has a maximum a (, x ) = (, ) for all. We ge from (3.4) he following equaion for ρ: ρ = v ρ + i ( ūpε 1 u upε 1 ū ) + i ( ūpε 1 (ρ σ u) upε 1 (ρ σ ū) ) ūpε 1 δ (ρ σ u) upε 1 δ (ρ σ ū). Differeniaing his equaion wih respec o and x respecively, and seing = x = yields he desired condiions for v 1 and v. Noe ha he compuaion of he addiional derivaives appearing in his approach is expensive, since in pracice i needs o be enforced in every sep of he RungeKua scheme. Hence, we shall resric his approach solely o cases where he numerical resuls appear o be srongly affeced by he boundary of Ω. In addiion, we may always choose a reference frame such ha eiher one of he wo componens of δ is zero which consequenly allows us o se eiher v 1, or v = Basic numerical ess for a derivaive NLS in D. As an example, we consider he case of a cubically nonlinear, wodimensional derivaive NLS of he following form (3.5) i u + u + (1 + i x ) ( u σ u ) =, u = = u (, x ). which is obained from our general model (1.3) for ε =, δ 1 = and δ = 1. We ake iniial daa u given by (3.3). Here, Q is he ground sae compued earlier for his paricular choice of parameers, see Fig. 3. We work on he compuaional domain (.9) wih L x1 = L x = 3, using N x1 = N x = 1 Fourier modes and 1 5 imeseps for 5. We also apply a Krasny filer [6], which ses all
12 1 J. ARBUNICH, C.KLEIN, AND C. SPARBER Fourier coefficiens smaller han 1 1 equal o zero. The real and imaginary par of he soluion u a he final ime = 5 can be seen in Fig. 7 below. Noe ha hey are boh much more localized and peaked when compared o Fig. 3, indicaing a selffocusing behavior wihin u. Moreover, is Re u is no longer posiive due o phase modulaions. Figure 7. Real and imaginary par of he soluion o (3.5) a ime = 5 corresponding o iniial daa u = Q +.1e x 1 x, where Q is he ground sae in Fig. 3. Surprisingly, however, here is no indicaion of a finieime blowup, in conras o he analogous siuaion wihou derivaive nonlineariy (recall Fig. 6 above). Indeed, he Fourier coefficiens of u a = 5 are seen in Fig. 8 o decrease o he order of he Krasny filer. In addiion, he L norm of he soluion, ploed in he middle of he same figure, appears o exhibi a urning poin shorly before 4. Finally, he velociy componen v ploed on he righ in Fig. 8 seems o slowly converge o a some limiing value v. The laer would indicae he appearance of a sable moving solion, bu i is difficul o decide such quesions numerically. All of hese numerical findings are obained wih M ε () conserved up o errors of he order v Figure 8. Soluion o (3.5) wih perurbed ground sae iniial daa: The Fourier coefficiens of u a = 5 on he lef; he L  norm of he soluion as a funcion of ime in he middle, and he imeevoluion of is velociy v on he righ. I migh seem exremely surprising ha he addiion of a derivaive nonlineariy is able o suppress he appearance of finieime blowup. Noe however, ha in all he examples above we have used only (a special case of) perurbed ground saes Q as iniial daa. For more general iniial daa, he siuaion is radically differen, as
13 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 13 can be illusraed numerically in he following example: We solve (3.5) wih purely Gaussian iniial daa of he form u (, x ) = 4e (x 1 +x ) on a numerical domain Ω wih L x1 = L x =, using N x1 = N x = 1 Fourier coefficiens and N = 1 5 ime seps for.5. This case appears o exhibi finieime blowup, as is illusraed in Fig. 9. The conservaion of he numerically compued quaniy M ε () drops below 1 3 a =.1955 which indicaes ha ploing accuracy is no longer guaraneed. Consequenly we ignore daa aken for laer imes, bu noe ha he code sops wih an overflow error for u Figure 9. The modulus of he soluion o (3.5) for Gaussian iniial daa u = 4e x 1 x, a ime =.195. On he righ, he L norm of he soluion as a funcion of ime. Remark 3.4. These numerical findings are consisen wih analyical resuls for derivaive NLS in one spaial dimension. For cerain values of σ 1 and cerain velociies v he corresponding ground saes, or more general soliary wave soluions, are found o be orbially sable, see [8, 14, 7]. However, for general iniial daa and σ > 1 large enough, one expecs finieime blowup, see [8]. 4. Global wellposedness wih full offaxis variaion In his secion we will analyze he Cauchy problem corresponding o (1.3) in he case of full offaxis dependence, i.e. k =, so ha P ε = 1 ε. In his conex, we expec he soluion u of (1.3) o be very well behaved due o he srong regularizing effec of he ellipic operaor P ε acing in boh spaial direcions. To prove a global inime exisence resul, we rewrie (1.3), using Duhamel s formula (4.1) u() = S ε ()u + i Here, and in he following, we denoe by S ε ( s)pε 1 (1 + iδ ) ( u σ u ) (s) ds Φ(u)(). S ε () = e 1 ipε he corresponding linear propagaor, which is easily seen (via Plancherel s heorem) o be an isomery on H s (R ) for any s R. I is known ha in he case wih full offaxis variaion, S ε () does no allow for any Sricharz esimaes, see [5]. However, he acion of Pε 1 allows us o gain wo derivaives and offse he acion of he
14 14 J. ARBUNICH, C.KLEIN, AND C. SPARBER gradien erm in he nonlineariy of (4.1). Using a fixed poin argumen, we can herefore prove he following resul. Theorem 4.1 (Full offaxis variaions). Le ε >, k = and σ > 1. Then for any δ R and any u H 1 (R ), here exiss a unique global inime soluion u C(R ; H 1 (R )) o (1.3), depending coninuously on he iniial daa. Moreover, u(, ) H 1 cons., R. Proof. Le T, M >. We aim o show ha u Φ(u) is a conracion on he ball X T,M = {u L ([, T ); H 1 (R )) : u L H1 M}. To his end, le us shorly denoe (4.) Φ(u)() = S ε ()u + N (u)(), where for g(u) = u σ u, we wrie N (u)() := i S ε ( s)pε 1 (1 + iδ )g(u(s)) ds. Now, le u, u X T,M and recall ha S ε () is an isomery on H 1 (R ). Using Minkowski s inequaliy, followed by CauchySchwarz yields ( N (u)() N (u )() ) H 1 ( ) ε g(u) g(u ) H 1 + δ g(u) g(u ) L (s) ds. To bound he wo erms on he righ hand side, we firs noe ha (4.3) g(u) g(u ) C σ ( u σ + u σ ) u u. In general spaial dimensions d N, he dual of Sobolev s embedding hen yields g(u) g(u ) H 1 g(u) g(u ) dσ H (σ+1) ( u σ H + 1 u σ H ) u u 1 H 1, provided σ (d ) + <, when d =. In addiion, if we impose 1 < σ d (d ) +, hen we have H 1 (R d ) H d(σ 1) 4σ (R d ) and H d(σ 1) 4σ (R d ) L 4σ (R d ), again by Sobolev s embedding. This allows us o also esimae g(u) g(u ) L ( u 4σ L + 4σ u 4σ ) L u u 4σ H 1 ( u 4σ H + 1 u 4σ H ) u u 1 H 1. Togeher wih a Hölder s inequaliy in, we can consequenly bound N (u) N (u ) L H 1 ε T M σ (1 + δ M σ ) u u L H 1. By choosing T > sufficienly small, Banach s fixed poin heorem direcly yields a unique local inime soluion u C([, T ], H 1 (R d )). Sandard argumens (see, e.g., [33]) hen allow us o exend his soluion up o a maximal ime of exisence T max = T max ( u H 1) > and we also coninuous dependence on he iniial daa. Nex, we shall prove ha (4.4) P 1/ ε u() L = P 1/ ε u L, for all [, T ] and T < T max. For ε >, his conservaion law yields a uniform bound on he H 1 norm of u, since c ε P 1/ ε ϕ L ϕ H 1 C ε P 1/ ε ϕ L, C ε, c ε >.
15 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 15 We consequenly can reapply he fixed poin argumen as many imes as we wish, hereby preserving he lengh of he maximal inerval in each ieraion, o yield T max = +. Since he equaion is imereversible modulo complex conjugaion, we obain a global H 1 soluion for all R, provided (4.4) holds. To prove (4.4), we adap and (slighly) modify an elegan argumen given in [3], which has he advanage ha i does no require an approximaion procedure via a sequence of sufficienly smooh soluions (as is classically done, see e.g. [6]): Le [, T ] for T < T max. We firs rewrie Duhamel s formula (4.1), using he coninuiy of he semigroup S ε o propagae backwards in ime (4.5) S ε ( )u() = u + S ε ( )N (u)(). As S ε ( ) is uniary in L, we have Pε 1/ u() L = S ε ( )Pε 1/ u() L. The laer can be expressed using he above ideniy: P 1/ ε u() L = = P 1/ ε u L + Re S ε ( )P 1/ ε N (u)(), P 1/ ε u L + S ε ( )P 1/ ε N (u)() L P 1/ ε u L + I 1 + I. We wan o show ha I 1 + I =. In view of (4.) we can rewrie I 1 = Im = Im S ε ( s)p 1/ ε (1 + iδ )g(u)(s) ds, P 1/ ε u L (1 + iδ )g(u)(s), Sε (s)u By CauchySchwarz we find ha his quaniy is indeed finie, since L ds. I 1 T (1 + iδ )g(u) L L S ε( )u L L <. Denoing for simpliciy G ε ( ) = Pε 1 (1 + iδ )g(u)( ), we find afer a lenghy compuaion (see [] for more deails) ha he inegral I = Re Pε G ε (s), in (u)(s) L ds. We can express in (u)(s) using he inegral formulaion (4.5) and wrie ( (4.6) I = Re Pε G ε (s), is ε (s)u ds + Pε G L ε (s), iu(s) ds). L Nex, we noe ha he paricular form of our nonlineariy implies Re P 1/ ε G ε ( ), ip 1/ ε u( ) L = Im (1 + iδ )g(u)( ), u( ) L = Im u( ) σ+ L Re g(u)( ), (δ )u( ). σ+ L Here, he firs expression in he las line is obviously zero, whereas for he second erm we compue Re g(u), (δ )u = u σ Re ( u(δ )u ) dx L R d = 1 (σ + 1) R d (δ )( u σ+ ) dx =, for H 1 soluions u. In summary, he second erm on he righ hand side of (4.6) simply vanishes and we find I = Im This finishes he proof of (4.4). (1 + iδ )g(u)(s), Sε (s)u L ds = I 1.
16 16 J. ARBUNICH, C.KLEIN, AND C. SPARBER Remark 4.. This proof can easily be exended o he analogous model problem in arbirary space dimensions d 1, provided σ (d ) + for δ = (no selfseepening), 1 < σ { d (d ) +, d 4 d, d > 4 for δ > (wih selfseepening). For δ = his is he usual H 1 subcriical resricion. 5. (In)sabiliy properies of ground saes wih full offaxis variaion In his secion, we shall perform numerical simulaions o sudy he (in)sabiliy properies of nonlinear ground saes Q in he case wih selfseepening and of fulloff axis variaion k =. In view of Theorem 4.1, we know ha here canno be any srong insabiliy of Q, i.e., insabiliy due o finieime blowup. Neverheless, we shall see ha here is a wealh of possible scenarios, depending on he precise choice of parameers, σ, δ, and on he way we perurb he iniial daa. To be more precise, we shall consider iniial daa o equaion (1.3) wih k =, given by (5.1) u (, x ) = Q(, x ) ±.1e x 1 x, where Q is again he ground sae consruced numerically as described in Secion. We will use N x1 = N x = 1 Fourier modes, a numerical domain Ω of he form (.9) wih L x1 = L x = 3, and a ime sep of = 1. Evidence for he sabiliy of he ground sae Q is obained via (sable) oscillaions of u(, ) L. Indeed, whenever boh perurbaions of (5.1) have oscillaing L norms, his generically indicaes ha he imedependen soluion u oscillaes around some sable imeperiodic sae plus a (small) remainder which will radiae away as ± The case wihou selfseepening. Les firs address he case δ 1 = δ = for nonlinear srenghs σ = 1,, 3: For σ = 1, we find ha he perurbed ground sae is unsable, and ha he iniial pulse disperses owards infiniy as can be seen in Fig. 1. The modulus of he soluion a = 1 in he same figure on he righ shows ha he iniial pulse disperses wih an annular profile. A perurbaion in (5.1) leads o he same qualiaive behavior and a corresponding figure is omied Figure 1. Soluion o equaion (1.3) wih σ = 1, ε = 1, k =, δ =, and iniial daa (5.1) wih he + sign: On he lef he L norm of he soluion as a funcion of, and on he righ he modulus of he soluion for = 1.
17 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 17 The siuaion is found o be differen for σ =, where Q appears o be sable, see Fig. 11. The L norm of he soluion hereby oscillaes for boh signs of he perurbaion Figure 11. L norm of he soluion o equaion (1.3) wih σ =, ε = 1, k =, δ =, and iniial daa (5.1): On he lef for he sign, and on he righ for he + sign. Finally, for σ = 3 we find ha he behavior depends on how we perurb he iniial ground sae Q. Perurbaions wih a + sign in (5.1) again exhibi an oscillaory behavior of he L norm, see he lef of Fig. 1. However, a perurbaion yields a monoonically decreasing L norm of he soluion. The laer is is again dispersed wih an annular profile Figure 1. L norm of he soluion o equaion (1.3) wih ε = 1, k =, σ = 3, δ = and iniial daa (5.1): On he lef for he sign, on he righ for he + sign. 5.. The case wih selfseepening. in his subsecion, we shall perform he same numerical sudy in he case wih selfseepening, i.e. δ 1,. For σ = 1, he corresponding ground sae Q seems o remain sable, since boh ypes of perurbaions yield an oscillaory behavior of he L norm in ime, see Fig. 13. This is in sharp conras o he σ = 1 case wihou selfseepening depiced in Fig. 1 above. In addiion, we see ha he soluion no longer displays an annular profile.
18 18 J. ARBUNICH, C.KLEIN, AND C. SPARBER Figure 13. Soluion o equaion (1.3) wih ε = 1, k =, σ = 1, δ 1 =, δ = 1, and iniial daa (5.1): On he lef he L norm for he sign, in he middle u ploed a he final ime, and on he righ he L norm for he soluion wih he + sign. This sable behavior is los in he case of higher nonlineariies. More precisely, for boh σ = and 3 we find ha he behavior of he soluion u depends on he sign of he considered Gaussian perurbaion. On he one hand, for he + perurbaion in (5.1), boh σ = and σ = 3 yield an oscillaory behavior of he L norm, see Fig. 14. On he oher hand, he perurbaion for boh nonlineariies produce a soluion wih decreasing L norm in ime (alhough for σ = his decrease is no longer monoonically) Figure 14. Soluion o equaion (1.3) wih ε = 1, k =, σ = 3, δ 1 =, δ =.1, and iniial daa (5.1): On he lef, he L norm for he perurbaion, in he middle u ploed a he final ime, and on he righ he L norm for he soluion wih he + sign. Remark 5.1. Our numerical findings are reminiscen of recen resuls for he (generalized) BBM equaion, see [4]. In here, i is found ha for p 5, he regime where he underlying KdV equaion is expeced o exhibi blowup, soliary waves can be boh sable and unsable and are sensiive o he ype of perurbaion considered. The main difference o our case is of course ha hese earlier sudies are done in only one spaial dimension. 6. Wellposedness resuls for he case wih parial offaxis variaion From a mahemaical poin of view, he mos ineresing siuaion arises in he case where here is only a parial offaxis variaion. To sudy such a siuaion, we shall wihou loss of generaliy assume ha P ε acs only in he direcion, i.e. In his case (1.1) becomes P ε = 1 ε. (6.1) i(1 ε ) u + u + (1 + iδ )( u σ u) =, u = = u (, x ).
19 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 19 When δ = (δ, ) and σ = 1, his is precisely he model proposed in [11, Secion 4.3]. Moivaed by his, we shall in our analysis only consider he case where he regularizaion P ε and he derivaive nonlineariy ac in he same direcion. Numerically, however, we shall also rea he orhogonal case where, insead, δ = (, δ), see Secion 7 below Change of unknown and Sricharz esimaes. In [], which reas he case wihou selfseepening, he following change of unknown is proposed in order o sreamline he analysis: (6.) v(,, x ) := P 1/ ε u(,, x ). Rewriing he evoluionary form of (6.1) wih δ = (δ, ) in erms of v yields (6.3) i v + P 1 ε v + (1 + iδ x1 )P 1/ ε ( P 1/ ε v σ P 1/ ε v) =, subjec o iniial daa v = = v (, x ) P 1/ ε u (, x ). Insead of (1.5), one finds he new conservaion law (6.4) v(, ) 1/ L = Pε u(, ), 1/ L = Pε u L = v L, where we recall ha P 1/ ε only acs in he direcion, via is Fourier symbol P s/ 1 (ξ) = (1 + ξ 1) s/, ξ 1 R. This suggess o work in he mixed Sobolevype spaces L p (R x ; H s (R x1 )), which for any s R are defined hrough he following norm: f L p x H := P s/ x s 1 1 f ( ( ) p ) 1 p L p := P s/ x L 1 f(, x ) d dx. R R We will also make use of he mixed spaceime spaces L q L p x Hx s 1 (I) for some ime inerval I (or simply L q L p x Hx s 1 when he inerval is clear from conex), which we shall equip wih he norm ( ) 1 F L q Lp x Hx s (I) := F () q q d 1 L p x H. x s 1 The proof of (global) exisence of soluions o (6.3) will require us o use he 1 ip dispersive properies of he associaed linear propagaor S ε () = e ε, which in conras o he case k = allows for Sricharz esimaes. However, in comparison o he usual Schrödinger group e i, hese dispersive properies are considerably weaker. In he following, we say ha a pair (q, r) is Sricharz admissible, if (6.5) q = 1 1, for r, 4 q. r Now, le (q, r), (γ, ρ) be wo arbirary admissible pairs. I is proved in [, Proposiion 3.4] ha here exis consans C 1, C > independen of ε, such ha (6.6) S ε ( )f L q Lr x H γ C 1 f L, as well as (6.7) S ε ( s)f (s) ds I L q Lr x H q C F L γ. γ x H Here, one should noe he loss of derivaives in he direcion. Lρ
20 J. ARBUNICH, C.KLEIN, AND C. SPARBER 6.. Global exisence resuls. Using he Sricharz esimaes saed above, we shall now prove some L based global exisence resuls for he soluion v o (6.3). In urn, his will yield global exisence resuls (in mixed spaces) for he original equaion (6.1) via he ransformaion v = Pε 1/ u. To his end, we firs recall ha in he case wihou selfseepening δ =, he resuls of [] direcly give: Proposiion 6.1 (Parial offaxis variaion wihou selfseepening). Le σ <. Then for any iniial daa u L (R x ; H 1 (R x1 )) here exiss a unique global inime soluion u C(R ; L (R x ; H 1 (R x1 ))) o (6.8) i(1 ε ) u + u + u σ u =, u = = u (, x ). Our numerical findings in he nex secion indicae ha his resul is indeed sharp, i.e., ha for σ global exisence in general no longer holds. Nex, we shall ake ino accoun he effec of selfseepening, and rewrie (6.3) using Duhamel s formula: (6.9) v() = S ε ()v + i Φ(v)(). S ε ( s)pε 1/ (1 + iδ x1 )( Pε 1/ v σ Pε 1/ v)(s) ds To prove ha Φ is a conracion mapping, he following lemma is key. Lemma 6.. Le g(z) = z σ z wih σ N. For [, T ] denoe (6.1) N (v)() := i S ε ( s)pε 1/ (1 + iδ x1 )g(p 1/ v(s)) ds, and choose he admissible pair (γ, ρ) = ( 4(σ+1) σ, (σ + 1) ). Then for ε, δ >, i holds: N (v) N (v ) L γ Lρ x H γ ε (σ+1) (1 + δ)t 1 σ ( v σ L γ Lρ x H γ + v σ ε L γ Lρ x H γ Proof. Firs i is easy o check ha ( 4(σ + 1) ) (γ, ρ) =, (σ + 1) σ ) v v L γ Lρ x H γ. is admissible in he sense of (6.5). Moreover, since γ > 4 we have γ < 1, from which we infer ha H 1 γ (R) is indeed a normed Banach algebra, a fac o be used below. Using he Sricharz esimae (6.7) we have N (v) N (v ) C P 1/ ε P 1/ ε L γ Lρ x H γ ( )( 1 + iδ x1 g(p 1/ ε v) g(pε 1/ v ) ) L γ Lρ γ x H For simpliciy we shall in he following denoe u = Pε 1/ v, u = Pε 1/ v in view of (6.). Keeping and x fixed we can esimae ( )( 1 + iδ x1 g(p 1/ ε v) g(pε 1/ v ) ) γ H ε 1 ( 1 + iδ x1 )( g(u) g(u ) ) ε 1 (1 + δ) g(u) g(u ) H 1 γ, γ H 1.
21 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 1 where in he las inequaliy we have used he fac ha H 1 γ (R) H γ (R). Nex, use again (4.3) which ogeher wih he algebra propery of H 1 γ (R) for σ N implies g(u) g(u ) H 1 γ ( u σ H 1 γ + u σ ε (σ+1)( v σ H γ H 1 γ + v σ ) u u H γ H 1 γ ) v v H γ. I consequenly follows afer Hölder s inequaliy in x, ha we obain N (v) N (v ) L ρ x H γ ( ε (σ+1) (1 + δ) v σ L ρ x H γ + v σ L ρ x H γ ) v v L ρ x H γ. The resul hen follows afer applying ye anoher Hölder s inequaliy in. This lemma allows us o prove he following global exisence resul for (6.1). Theorem 6.3 (Parial offaxis variaion wih parallel selfseepening). Le σ = 1 and δ = (δ, ) for δ R. Then for any u L (R x ; H 1 (R x1 )) here exiss a unique global soluion u C(R ; L (R x ; H 1 (R x1 ))) o (6.1). Here, he resricion σ = 1 is due o he fac ha his is he only σ N (required for he normed algebra propery above) for which 1 σ >. Proof. We seek o show ha v Φ(v) is a conracion mapping in a suiable space. To his end, we denoe, as before, Φ(v)() = S ε ()v + N (v)(), where N (v) is given by (6.1). Le T, M > and denoe Y T,M ={v L ([, T ); L (R )) L 8 ([, T ); L 4 (R x ; H 1 4 (Rx1 ))) : v L L + v L 8 L4 x H 1 4 M}. The Sricharz esimaes (6.6) and (6.7) ogeher wih Lemma (6.) imply ha for any admissible pair (q, r) and soluions v, v Y T,M ha Φ(v) Φ(v ) L q Lr x H q S ε ()(v v ) L q Lr x H q C 1 v v L + C P 1/ ε + N (v) N (v ) ( 1 + iδ x1 )( g(p 1/ ( ) C σ,ε v v L + T 1/ M v v L 8 1. L 4 x H 4 L q Lr x H q ε v) g(pε 1/ v ) ) L 7 L 3x H 4 Choosing M = M( v L ) and T sufficienly small, i is clear ha Φ is a conracion on Y T,M. Banach s fixed poin heorem and a sandard coninuiy argumen hus yield he exisence of a unique maximal soluion v C([, T max ), L (R )) where T max = T max ( v L ). Coninuous dependence on he iniial daa follows by classical argumens. The conservaion propery (6.4) for v follows similarly as in he proof of Proposiion 4. in [] and we shall herefore only skech is main seps below. By he uniary of S ε ( ) in L we obain v() L = v L + Re S ε ( )N (v)(), v L + S ε ( )N (v)() L =: v L + I 1 + I.
22 J. ARBUNICH, C.KLEIN, AND C. SPARBER To show ha I 1 + I =, we use (4.) and rewrie I 1 = Im P 1/ ε (1 + iδ x1 )g(p 1/ ε v)(s), S ε (s)v L ds. By dualiy in and Hölder s inequaliy in and x we find ha his quaniy is indeed finie, since I 1 Pε 1/ (1 + iδ x1 )g(pε 1/ v) L γ Lρ γ x H S ε ( )v L γ Lρ x H γ <. Once again we find, afer a lenghy compuaion (see [] for more deails), ha I = Re P 1/ ε (1 + iδ x1 )g(p 1/ ε v)(s), in (u)(s) L ds. We express in (u)(s) using he inegral formulaion (6.9) and wrie I = Re + P 1/ ε (1 + iδ x1 )g(p 1/ ε v), is ε (s)u L ds Im Pε 1/ v(s) σ+ L δ Re g(p 1/ σ+ ε v), x1 Pε 1/ v (s) ds L Here he second ime inegral vanishes enirely, and, as in he full offaxis case, he laer erm in he inegrand vanishes due o Re g(pε 1/ v), x1 Pε 1/ v = L x1 ( Pε 1/ v σ+ ) d dx =. (σ + 1) In summary, we find ha I = Im R R P 1/ ε g(p 1/ ε v)(s), S ε (s)u L ds = I 1, which finishes he proof of (6.4). We can hus exend v o become a global soluion by repeaed ieraions o conclude T max = +. Finally, we use he fac ha v = Pε 1/ u o obain a unique global inime soluion u C(R ; L (R x ; H 1 (R x1 ))) which finishes he proof. Remark 6.4. I is possible o rea he criical case σ = using he same ype of argumens as in [7] (see also []). Unforunaely, his will only yield local inime soluions up o some ime T = T (u ) >, which depends on he iniial profile u (and no only is norm). Only for sufficienly small iniial daa u L x H < 1, does 1 one obain a global inime soluion. Bu since i is hard o deec small nonlinear effecs numerically, we won be concerned wih his case in he following. We also menion he possibiliy of obaining (no necessarily unique) global weak soluions for derivaive NLS, which has been done in [1] in one spaial dimension. 7. Numerical sudies for he case wih parial offaxis variaion In his secion we presen numerical sudies for he model (6.1) wih ε = 1 and differen values of he selfseepening parameer δ, as well as σ >. We will always use N x1 = N x = 1, Fourier coefficiens on he numerical domain Ω given by (.9) wih L x1 = L x = 3. The ime sep is = 1 unless oherwise noed. The iniial daa is he same as in (5.1), i.e. a numerically consruced ground sae Q perurbed by adding and subracing small Gaussians, respecively.
23 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS The case wihou selfseepening. We shall firs sudy he paricular siuaion furnished by equaion (6.8) wih ε = 1. I is obained from he general model (1.1) in he case wihou selfseepening δ 1 = δ = : In he case σ = 1, he ground sae perurbaion in (5.1) wih a + sign is unsable and resuls in a purely dispersive soluion wih monoonically decreasing L norm, see Fig. 15. The modulus of he soluion a ime =.5 is shown on he righ of he same figure. Ineresingly, he iniial hump appears o separae ino four smaller humps and we hus loose radial symmery of he soluion. The siuaion is qualiaively similar for perurbaions corresponding o he sign in (5.1) and we hus omi a corresponding figure Figure 15. Soluion o (6.8) wih ε = 1, σ = 1, and iniial daa (5.1) wih a + sign: On he lef he L norm in dependence of ime, on he righ he modulus of u a =.5. The siuaion changes significanly for σ =, as can be seen in Fig. 16. While he L norm of he soluion obained from iniial daa (5.1) wih he sign is again decreasing, he + sign yields a monoonically increasing L norm indicaing a blowup a.64. The modulus of he soluion a he las recorded ime Figure 16. Timedependence of he L norm of he soluion o (6.8) wih ε = 1, σ =, and iniial daa (5.1): On he lef, he case wih a perurbaion; on he righ he case wih + sign. =.645 is shown in Fig. 17 on he lef. I can be seen ha i is srongly compressed in he x direcion. The corresponding Fourier coefficiens are shown
24 4 J. ARBUNICH, C.KLEIN, AND C. SPARBER on he righ of he same figure. They also indicae he appearance of a singulariy in he x direcion. Figure 17. Soluion o (6.8) for ε = 1, σ = and iniial daa (5.1) wih he + sign: On he lef he modulus of he soluion a he las recorded ime =.645; on he righ he corresponding Fourier coefficiens of û. These numerical findings indicae ha he global exisence resul saed in Theorem 6.3 is indeed sharp. I also shows ha he wodimensional model wih parial offaxis variaion essenially behaves like he classical onedimensional focusing NLS in he unmodified x direcion (i.e., he direcion in which P ε does no ac). Recall ha for he classical onedimensional (focusing) NLS, finieime blowup is known o appear as soon as σ. 7.. The case wih selfseepening parallel o he offaxis variaion. In his subsecion, we include he effec of selfseepening and consider equaion (6.1) wih ε = 1, δ =, and δ 1 >. For σ = 1, he ground sae appears o be sable agains all sudied perurbaions. Indeed, he siuaion is found o be qualiaively similar o he case wih full offaxis perurbaions (excep for a loss of radial symmery) and we herefore omi a corresponding figure. When σ =, he ground sae no longer appears o be sable. However, we also do no have any indicaion of finieime blowup in his case. Indeed, given a perurbaion in he iniial daa (5.1), i can be seen on he lef of Fig. 18 ha he L norm of he soluion simply decreases monoonically in ime. Noice, Figure 18. Soluion o (6.1) wih ε = 1, σ =, and δ = (.3, ) : On he lef, he L norm of he soluion obained for iniial daa (5.1) wih he sign, on he righ for he +, and in he middle u a = 5 for he sign perurbaion.
25 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 5 ha here is sill an effec of selfseepening visible in he modulus of he soluion u, depiced in he middle of he same figure. The behavior of he L norm in he case of a + perurbaion is shown on he righ of Fig. 18. I is no longer monoonically decreasing bu sill converges o zero. For σ = 3, a perurbaion of (5.1) is found o be qualiaively similar o he case σ = and we herefore omi a figure illusraing his behavior. However, he siuaion radically changes if we consider a perurbaion wih he + sign, see Fig. 19. The L norm of he soluion indicaes a blowup for.1555, where he code sops wih an overflow error. In his paricular simulaion we have used Figure 19. L norm of he soluion o (6.1) wih ε = 1, σ = 3, δ 1 = (.1, ), and iniial daa (5.1) wih he + sign. On he righ he modulus of he Fourier coefficiens of he soluion a ime = ime seps for [,.17] and N x1 = 1, N x = 11 Fourier modes (since he maximum of he soluion hardly moved, i was no necessary o use a comoving frame). The soluion is sill well resolved in ime a =.155 since M ε () remains numerically conserved up o he order of Bu despie he higher resoluion in x used for his simulaion, he Fourier coefficiens indicae a loss of resoluion in he x direcion. The modulus of he soluion a he las recorded ime is ploed in Fig.. Noe ha u is sill regular in he direcion in which Pε 1 acs, bu i has become srongly compressed in he x direcion. Figure. The modulus of he soluion o equaion (6.1) wih ε = 1, σ = 3, δ 1 = (.1, ), and iniial daa (5.1) wih he + sign, ploed a ime =.155.
26 6 J. ARBUNICH, C.KLEIN, AND C. SPARBER 7.3. The case wih selfseepening orhogonal o he offaxis variaion. Finally, we shall consider he same model equaion (6.1) wih ε = 1, bu his ime we le δ 1 = for nonvanishing δ >. This is he only case, for which we do no have any analyical exisence resuls a presen. For σ = 1, i can be seen ha a sign in he iniial daa (5.1) yields a purely dispersive soluion wih monoonically decreasing L norm, see Fig. 1 which also shows a picure of u a =. The + sign again leads o oscillaions of he L norm in ime, indicaing sabiliy of he ground sae. The siuaion for σ = is qualiaively very similar and hence we omi he corresponding figure Figure 1. Soluion o (6.1) wih ε = 1, σ = 1, and δ 1 = (, 1). On he lef he L norm of he soluion for iniial daa (5.1) wih he sign, on he righ for he one wih + sign, and in he middle u a ime = for he sign. For σ = 3 and a sign in he iniial daa (5.1), we again find a purely dispersive soluion. However, he behavior of he soluion obained from a perurbaion of Q wih he + sign is less clear. As one can see in Fig., he soluion is iniially focused up o a cerain poin afer which is L norm decreases again Figure. Soluion o (6.1) wih ε = 1, σ = 3, δ 1 = (,.1), and iniial daa (5.1) wih he + sign: On he lef he L norm of u as a funcion of ime, on he righ he Fourier coefficiens of û a =.5. This simulaion is done wih N x1 = 1, N x = 11 Fourier modes and N = 1 4 ime seps for [,.5]. The relaive conservaion of he numerically compued quaniy M ε () is beer han 1 1 during he whole compuaion indicaing an excellen resoluion in ime. The spaial resoluion is indicaed by he Fourier coefficiens of he soluion near he maximum of he L norm as shown on he righ of Fig.. Obviously, a much higher resoluion is needed in he x direcion,
27 DERIVATIVE NLS TYPE EQUATIONS IN TWO SPATIAL DIMENSIONS 7 bu even near he maximum of he L norm he modulus of he Fourier coefficiens decreases o he order of 1 5. The modulus of he soluion a ime =.5 can be seen in Fig. 3. I shows a srong compression in he x direcion bu neverheless remains regular for all imes. This is in sark conras o he analogous siuaion wih parallel selfseepening and offaxis variaions, cf. Figures 19 and above. Figure 3. The modulus of he soluion o (6.1) wih ε = 1, σ = 3, δ 1 = (,.1), and iniial daa (5.1) wih he + sign, ploed a =.5. References 1. D. M. Ambrose and G. Simpson. Local exisence heory for derivaive nonlinear Schrödinger equaions wih nonineger power nonlineariies. SIAM J. Mah. Anal. 47 (15), no. 3, P. Anonelli, J. Arbunich, and C. Sparber, Regularizing nonlinear Schrödinger equaions hrough parial offaxis variaions. SIAM J. Mah. Anal. (18), o appear. 3. T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equaions for long waves in nonlinear dispersive sysems. Phil. Trans. Royal Soc. London A 7 (197), J. L. Bona, W. R. McKinney, and J. M. Resrepo, Sable and unsable soliarywave soluions of he generalized regularized longwave equaion. J. Nonl. Sci. 1 (), no. 6, R. Carles, On Schrödinger equaions wih modified dispersion. Dyn. Parial Differ. Equ. 8 (11), no. 3, T. Cazenave, Semilinear Schrödinger equaions. Couran Lecure Noes in Mahemaics vol. 1, American Mahemaical Sociey, T. Cazenave, F. Weissler, Some remarks on he nonlinear Schrödinger equaion in he criical case. In: Lecure Noes Mah. vol. 1394, Springer, New York, 1989, M. Colin and M. Oha, Sabiliy of soliary waves for derivaive nonlinear Schrödinger equaion. Ann. Ins. H. Poincaré Anal. Non Lineaire 3 (6), no. 5, A. Davey and K. Sewarson, On hreedimensional packes of waer waves. Proc. R. Soc. Lond. Ser. A 338 (1974), no. 1613, T. Driscoll, A composie RungeKua Mehod for he specral soluion of semilinear PDEs. J. Compu. Phys. 18 (), E. Dumas, D. Lannes, and J. Szefel, Varians of he focusing NLS equaion. Derivaion, jusificaion and open problems relaed o filamenaion. In: CRM Series in Mahemaical Physics, pp Springer, N. Hayashi and T. Ozawa. On he derivaive nonlinear Schrödinger equaion. Phys. D 55 (199), no. 1, G. Fibich, The nonlinear Schrödinger equaion; Singular soluions and opical collapse. Springer Series on Appl. Mah. Sciences vol. 19, Springer Verlag, Z. Guo, C. Ning, and Y. Wu, Insabiliy of he soliary wave soluions for he genenalized derivaive Nonlinear Schrödinger equaion in he criical frequency case. Preprin arxiv: R. Jenkins, J. Liu, P. Perry, and C. Sulem, Solion resoluion for he derivaive nonlinear Schrödinger Equaion. Preprin arxiv: