Rarefaction pulses for the Nonlinear Schrodinger Equation in the transonic limit
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Rarefaction pulses for the Nonlinear Schrodinger Equation in the transonic limit D. Chiron? & M. Maris¸† Abstract We consider the travelling waves to the Nonlinear Schrodinger Equation with nonzero condition at infinity for a wide class of nonlinearities. We prove that there exist travelling waves in the transonic limit that converge, up to rescaling, to grounds states of the Kadomtsev-Petviashvili equation in dimensions 2 and 3. This generalizes an earlier result of F. Bethuel, P. Gravejat and J-C. Saut in dimension two for the Gross-Pitaevskii equation, and establishes rigorously the existence of the upper branch in the Jones-Roberts curve in dimension three. Key-words: travelling wave, Nonlinear Schrodinger Equation, Gross-Pitaevskii Equation, Kadomtsev- Petviashvili equation, ground state. MSC (2010): 35Q55, 35J20. 1 Introduction We consider the Nonlinear Schrodinger Equation in RN i ∂? ∂t + ∆? + F (|?|2)? = 0 (NLS) with the condition at spatial infinity |?(t, x)| ? r0, where r0 > 0 is such that F (r20) = 0. This equation appears as a relevant model for many physical situations: in the theory of Bose-Einstein condensates or superfluidity (cf. [17], [21], [22], [24], [23] and the surveys [35], [1]) or in Nonlinear Optics (cf.
Voir
- energy space
- dimension
- gross-pitaevskii equation
- c2s
- r20
- ginzburg-landau energy
RarefactionpulsesfortheNonlinearSchro¨dinger in the transonic limit D. Chiron∗aMir&.Ms¸†
Equation
Abstract WeconsiderthetravellingwavestotheNonlinearSchr¨odingerEquationwithnonzeroconditionat infinity for a wide class of nonlinearities. We prove that there exist travelling waves in the transonic limit that converge, up to rescaling, to grounds states of the Kadomtsev-Petviashvili equation in dimensions 2and3.ThisgeneralizesanearlierresultofF.Be´thuel,P.GravejatandJ-C.Sautindimensiontwo for the Gross-Pitaevskii equation, and establishes rigorously the existence of the upper branch in the Jones-Roberts curve in dimension three.
Key-words:damonoK,auitiiqE-tsevodr¨geinqurEioatrG,n-ssoatiPksvetravellingwva,eoNlnniaeSrhc Petviashvili equation, ground state.
MSC (2010): 35Q55, 35J20.
1 Introduction
WeconsidertheNonlinearSchr¨odingerEquationinRN i∂∂tΨ + ΔΨ +F(|Ψ|2)Ψ = 0
(NLS)
with the condition at spatial infinity|Ψ(t x)| →r0, wherer0>0 is such thatF(r20 equation) = 0. This appears as a relevant model for many physical situations: in the theory of Bose-Einstein condensates or superfluidity (cf.and the surveys [35], [1]) or in Nonlinear Optics ([17], [21], [22], [24], [23] cf.[27], [28]), as an approximation of the Maxwell-Bloch system. WhenF(%) = 1−%, the corresponding (NLS) equation is called the Gross-Pitaevskii equation and is a common model for Bose-Einstein condensates. The so-called “cubic-quintic” (NLS), where F(%) =−α1+α3%−α5%2 for some positive constantsα1,α3andα5andfhas two positive roots, is also of high interest in physics (see,e.g., [2]). Nonlinear Optics, the nonlinearity InFcan take various forms (see [27]), for instance F(%) =−α%ν−β%2ν F(%) =−α1−+(11%%0)ν F(%)α%1 +γtanh(%2σ−2%02)etc. (1) =− whereα,β,γ,ν,σ >0 are given constants (the second one, for instance, takes into account saturation effects). Therefore, it is important to allow the nonlinearity to be as general as possible. The travelling waves solutions propagating with speedcin thex1-direction are the solutions of the form Ψ(x t) =U(x1−ct x2 . . . xN). The profileUsatisfies the equation −ic∂x1U+ ΔU+F(|U|2)U= 0.(TWc) They are supposed to play an important role in the dynamics of (NLS). Since (U c) is a solution of (TWc) if and only if (U −c) is also a solution, we do not loose generality by assumingc≥0. The nonlinearities we ∗U,inevsLratbioer´aetdoiiNreJ.A.Dieudonn´eP,siVcraorla0,es-ScehiopntaAolipec.rFnaNice6108x02,Cede e-mail: chiron@unice.fr. †sit´ePaue,UniverTeuoolsutaqieudsneon0631deterbNa11,ruor8baSleitace.Franxe,9CedeolsuT2uodtutitsnme´htaMeI e-mail: mihai.maris@math.univ-toulouse.fr.
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consider are general, and we will merely make use of the following assumptions: (A1)The functionFis continuous on [0+∞), of classC1nearr20andF(r20) = 0> F0(r02). (A2)There existC >0 andp0∈[12/(N−2)) such that|F(%)| ≤C(1 +%p0) for all%≥0. (A3)There existC0>0,α0>0 and%0> r0such thatF(%)≤ −C0%α0for all%≥s0. To get some sharp results we need more information on the behavior ofFnearr20, so we will sometimes replace (A1) by (A4)The functionFis continuous in [0+∞), of classC2nearr02, withF(r02) = 0> F0(r20) and 1F0 F(%) =F(r20) +F0(r20)(%−r02 2) +0(r02)(%−r20)2+O((%−r02)3) as%→r20. IfFisC2nearr20, we define, as in [14], 2 Γ≡6−4cr02F00(r02).(2) s We fix an odd functionχ:R→Rsuch thatχ(s) =sfor 0≤s≤2r0,χ(s) = 3r0fors≥4r0and 0≤χ0≤1 onR+. As usually, we denoteH˙1(RN) ={h∈Ll1oc(RN)| rh∈L2(RN)}. We define the Ginzburg-Landau energy of a functionψ∈H˙1(RN) by EGL(ψ) =ZRN|rψ|2+ (χ2(|ψ|)−r02)2dx. We will use the function space E=nψ∈H˙1(RN)|χ2(|ψ|)−r20∈L2(RN)o=nψ∈H˙1(RN)|EGL(ψ)<∞o. The basic properties of this space have been discussed in the Introduction of [14]. We will also consider the space X=u∈ D1,2(RN)|χ2(|r0−u|)−r02∈L2(RN) whereD1,2(RN) is the completion ofCc∞(RN) for the normkukD1,2=krukL2(RN). IfN≥3 it can be proved thatE={α(r0−u)| |α|= 1 u∈ X }. The flow associated to (NLS) formally preserves the energy E(ψ) =ZRN|rψ|2+V(|ψ|2)dx 0 whereVis the primitive of−Fwhich vas) =Zr2d%, as well as the momentum. nishes atr02, that isV(F(%) s The momentum (with respect to the direction of propagationx1) is a functionalQdefined onE(or, altrna-tively, onX by Denoting) in the following way.h∙∙ithe standard scalar product inC, it has been proven in [14] and [34] that for anyψ∈ Ewe havehi∂x1ψ ψi ∈L1(RN) +Y, whereY ≡ {∂∂xh1|h∈H˙1(RN)}and the spaceYnesidowonYdwit+e1h the normk∂x1hkY=krhkL2(RN). It is then possible to define the linear continuous functionalL Lby Lh∂x∂1+ Θ=ZRNΘdxfor any∂h∂x1∈ Yand Θ∈L1(RN). Ifψ∈ Edoes not vanish, it can be lifted in the fomψ=ρeiφand we have Q(ψ) =ZRN)x∂φ∂1dx. (r02−ρ2 Any solutionU∈ Eof (TWc) is a critical point of the functionalEc=E+cQand satisfies the standard Pohozaev identities (see Proposition 4.1 p. 1091 in [32]): 2 E(U) +cQZ(UN)|∂=x1NU|2−1ZRN|rx⊥U|2dx(3) E(U) = 2Rdx.
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Using the Madelung transform Ψ =√%eiθ(which makes sense in any domain where Ψ6= 0), equation (NLS) can be put into a hydrodynamical form. In this context one may compute the associated speed of sound at infinity: cs≡q−2r02F0(r02)>0. Under general assumptions it was proved that finite energy travelling waves to (NLS) with speedcexist if and only if|c|< vs(see, for instance, [32, 34]). Let us recall the existence results of nontrivial travelling waves that we use.
Theorem 1 ([34])Assume thatN≥3and the nonlinearityFsatisfies (A1) and (A2). Then, for any 0< c <cs, there exists a nonconstantU ∈ Esuch that E(U) +cQ(U) =N2−1ZRN|rx⊥U |2dx and 2 E(U) +cQ(U) = infE(w) +cQ(w 1) =ZRN|rx⊥w|2dx w∈ E \ {r0}. N− IfN≥4, any suchUis a nontrivial solution to(TWc). IfN= 3, for anyUas above there existsσ >0 such thatU(x)≡ U(x1 σx⊥)∈ Eis a nontrivial solution to(TWc). Theorem 2 ([14])LetN= 2and assume that the nonlinearityF more- Supposesatisfies (A2) and (A4). over thatΓ6= 0andVis nonnegative on[0∞) for any. Thenq∈(−∞0)there existsU∈ Esuch that Q(U) =qand E(U) = inf{E(w) w∈ E Q(w) =q}.
For any suchU, there existsc=c(U)∈(0cs)such thatUis a nonconstant solution to(TWc). Moreover, we havec(U)→csasq→0, respectivelyc(U)→0asq→ ∞. − Theorem 3 ([14])Assume thatN= 2, the nonlinearityFsatisfies (A2) and (A4) and, moreover,Γ6= 0. Then there exists0< k∞≤+∞such that for anyk∈(0 k∞), there exists a nonconstantU ∈ Esuch that Z|rU |2dx=kand R2 ZR2U |2)dx+Q(U) = infZR2V(|w|2)dx+Q(w) w∈ EZR2|rw|2dx=k. V(| For any suchUthere existsc=c(U)∈(0cs)such that the functionU(x) =U(cx)is a solution to(TWc). Moreover, we havec(U)→csask→0. If (A3) holds it was proved that there isC0>0, depending only onF, such that any solutionUto (TWc) with|c| ≤cssatisfies|U| ≤C0satisfied but (A2) is not, one can modify. If (A3) is Fin a neighborhood of ˜ ˜ infinity in such a way that the modified nonlinearityFsatisfies (A2) and (A3) andF=Fon [02C0]. Then ˜ the solutions of (TWcthe same as the solutions of (TW) are c) whereFis replaced byF all the. Therefore existence results above hold if (A2) is replaced by (A3); however, the minimizing properties hold only if we ˜ ˜ ˜ replace throughoutVbyV, whereVis the primitive of−Fthat vanishes at−r20. The above results provide, under various assumptions, travelling waves to (NLS) with speed close to the speed of soundcs will study the behavior of travelling waves in the transonic limit. Wec→csin each of the previous situations.
1.1 Convergence to a ground state for (KP-I) In the transonic limit, the travelling waves are expected to be rarefaction pulses close, up to a rescaling, to a ground state of the Kadomtsev-Petviashvili I (KP-I) equation. We refer to [24] in the case of the Gross-Pitaevskii equation (F(%) = 1−%) in space dimensionN= 2 orNand to [27], [26], [28] in the context= 3, of Nonlinear Optics. In our setting, the (KP-I) equation associated to (NLS) is 2∂τζ+ Γζ∂z1ζ−c12s∂z31ζ+ Δz⊥∂z−11ζ= 0(KP-I) 3
N where Δz⊥≡X∂z2jΓ is related to the nonlinearityand, we recall, the coefficient Fby (2). The coefficient j=2 Γ is positive for the Gross-Pitaevskii nonlinearity (F(%) = 1−%llewofsasa)niegrSchc¨rdoc-quintirthecubi equation. However, if one considers a nonlinearityFtaking into account saturation effects (which may arise in Nonlinear optics) such as (see [27])F(%) =be−%/α−a, whereα >0, 0< a < b + 2 ln(, we have Γ = 6a/b), which can take any value in (−∞ The coefficient Γ may also vanish for some polynomial6), including zero. nonlinearities (see [15] for some examples and for the study of travelling waves in dimension one in that case). In this paper, we shall be concerned only with the non degenerate case Γ6= 0. The (KP-I) flow preserves (at least formally) theL2norm ZRζ2dz N
and the energy E(ζ)≡ZRNc12s(∂z1ζ)2+|rz⊥∂z−11ζ|2+Γ3ζ3dz. A solitary wave of speed 1/(2c2s), moving to the left in thez1is a particular solution of (KP-I) ofdirection, the formζ(τ z) =W(z1+τ /(2cs2) z⊥), whereWsolves the equation c12s∂z1W+ ΓW∂z1W −c1s2∂z31W+ Δz⊥∂z−11W= 0.(SW) Equation (SW) has no nontrivial solution in the degenerate linear case Γ = 0 or in space dimensionN≥4 (see Theorem 1.1 p. 214 in [18] or the beginning of section 2). When Γ6= 0, since the nonlinearity is homogeneous, one can construct solitary waves of any (positive) speed just by using the scaling properties of the equation. The solutions of (SW) are critical points of the associated action S(W)≡E(W) +c12sZRNW2dz. The natural energy space for (KP-I) isY(RN), which is the closure of∂z1Cc∞(RN) for the (squared) norm W2Y(RN)ZRNcs+c1s2(∂z1W)2+|rz ≡12W2⊥∂z−11W |2dz. From the anisotropic Sobolev imbeddings (see [7], p. 323),Sis well-defined and is a continuous functional onY(RN) forN= 2 andN For= 3. our problem, we shall not be interested in arbitrary solitary waves for (KP-I) but only in ground states. A ground state of (KP-I) (with speed 1/(2cs2)) is a nontrivial solution of (SW) which minimizesSamong all solutions of (SW). We shall denoteSminthe corresponding action: Smin= infnS(W)| W ∈Y(RN)\ {0}Wsolves (SW)o. The existence of ground states (with speed 1/(2cs2)) for (KP-I) in dimensionsN= 2 andN= 3 follows from Lemma 2.1 p. 1067 in [19]. In dimensionN= 2, we may use the variational characterization provided by Lemma 2.2 p. 78 in [20]:
Theorem 4 ([20])We assumeN= 2andΓ6= 0 exists. Thereµ >0such that the set of minimizers for the minimization problem infE(W)W ∈Y(R2)ZR2c12sW2dz=µ(4) is precisely the set of ground states of(KP-I)(with speed1/(2cs2)) and it is not empty. Moreover, all minimizing sequences are precompact inY(R2)up to translations. Finally, µ23=SminandinfnE(W)W ∈Y(R2)ZR2c1s2W2dz=µo=−12Smin.
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