Traveling waves for nonlinear Schrodinger equations with nonzero conditions at infinity II

David Chiron

Description

Traveling waves for nonlinear Schrodinger equations with nonzero conditions at infinity, II David CHIRON? and Mihai MARIS¸ † Abstract We present two constraint minimization approaches to prove the existence of traveling waves for a wide class of nonlinear Schrodinger equations with nonvanishing conditions at infinity in space dimension N ≥ 2. Minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves. Minimization of the action at constant kinetic energy can be used in all cases, but it gives no information about the orbital stability of the set of solutions. Keywords. nonlinear Schrodinger equation, nonzero conditions at infinity, traveling wave, Gross-Pitaevskii equation, cubic-quintic NLS, constrained minimization, Ginzburg- Landau energy. AMS subject classifications. 35Q51, 35Q55, 35Q40, 35J20, 35J15, 35B65, 37K40. 1 Introduction We study a class of special solutions to the nonlinear Schrodinger equation (1.1) i ∂? ∂t + ∆? + F (|?|2)? = 0 in RN , where ? is a complex-valued function on RN satisfying the ”boundary condition” |?| ?? r0 as |x| ?? ∞, r0 > 0 and F is a real-valued function on R+ such that F (r20) = 0.

• associated nonlinear


• any


• minimizers completely


• local minimizers


• has been


• there exist


• †institut de mathematiques de toulouse umr


• nonlinear potential


• ginzburg-landau energy

Sujets

• Antipolis
• Chiron
• Sabatier
• Has Been
• Existential quantification

Informations

• Publié par: pefav
• Nombre de lectures: 17
• Langue: English

Extrait

TravelingwavesfornonlinearSchr¨odingerequationswith nonzero conditions at infinity, II David CHIRON ∗ ARiMhaMi¸ISadn †

Abstract

We present two constraint minimization approaches to prove the existence of traveling wavesforawideclassofnonlinearSchr¨odingerequationswithnonvanishingconditionsat infinity in space dimensionN≥ of the energy at fixed momentum can2. Minimization be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves. Minimization of the action at constant kinetic energy can be used in all cases, but it gives no information about the orbital stability of the set of solutions. Keywords.nigvale,nonzeoncoroitndsnoinitatinfirt,ynonlinearScrho¨idgnreqeauit wave, Gross-Pitaevskii equation, cubic-quintic NLS, constrained minimization, Ginzburg- Landau energy. AMS subject classifications.35Q51, 35Q55, 35Q40, 35J20, 35J15, 35B65, 37K40.

1 Introduction

WestudyaclassofspecialsolutionstothenonlinearSchr¨odingerequation (1.1) ∂ i ∂ Φ t + ΔΦ +F(|Φ| 2  in)Φ = 0R N , where Φ is a complex-valued function onR N satisfying the ”boundary condition”|Φ−→|r 0 as|x−|∞→,r 0 >0 andFis a real-valued function onR + such thatF(r 02 ) = 0. Equation (1.1), with the considered non-zero conditions at infinity, arises in a great variety of physical problems such as superconductivity, superfluidity in Helium II, phase transitions and Bose-Einstein condensate ([1], [3], [4], [5], [18], [27], [29], [30], [31], [32], [44]). In nonlinear optics, it appears in the context of dark solitons ([35], [36]). Two important model cases for (1.1) have been extensively studied both in the physical and mathematical literature: the Gross-Pitaevskii equation (whereF(s) = 1−san)-csohedtSchrtic”nger¨odi”duclaeluqniib-c equation (whereF(s) =−α 1 +α 3 s−α 5 s 2 ,α 1 , α 3 , α 5 are positive andFhas two positive roots). In contrast to the case of zero boundary conditions at infinity (when the dynamics asso- ciated to (1.1) is essentially governed by dispersion and scattering), the non-zero boundary conditions allow a much richer dynamics and give rise to a remarkable variety of special solu- tions, such as traveling waves, standing waves or vortex solutions. Using the Madelung transformation Φ(x, t) = p ρ(x, t)e iθ(x,t) (which is well-defined in any region where Φ6is equivalent to a system of Euler’s equations for a= 0), equation (1.1) ∗ A.D.eiduno´nUeRMLaboratoireJiNedS-ecihpotnAa2166ni,Ursve´eit1680es0,iNecis,PipolalroarcV Cedex 02, France.e-mail: [e-mail]. † uesdeTouh´ematiqutdtMetaIsnitulPabaSarsve´eit9125inU,suolRMUe23106nne,arbodeNeortu1,81itre Toulouse Cedex 9, France.e-mail: [e-mail].

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compressible inviscid fluid of densityρand velocity 2rθ this context it has been shown. In that, ifFisC 1 nearr 2 0 andF 0 (r 02 )<the sound velocity at infinity associated to (1.1) is0, v s =r 0 p −2F 0 (r 2 0 ) (see the introduction of [41]). Equation (1.1) has a Hamiltonian structure. DenotingV(s) = R r s 2 0 F(τ)dτ, it is easy to see that, at least formally, the ”energy” (1.2)E(Φ) = Z R N |rΦ| 2 Z R N dx+V(|Φ| 2 )dx is conserved. Another quantity which is conserved by the flow of (1.1) is the momentum, P(Φ) = (P 1 (Φ), . . . , P N of the momentum will be given in the next(Φ)). A rigorous definition section. If Φ is a function sufficiently localized in space, we haveP k (Φ) = R R N hiΦ x k ,Φidx, whereh∙,∙iis the usual scalar product inC'R 2 . In a series of papers (see, e.g., [3], [4], [27], [31], [32]), particular attention has been paid to the traveling waves of (1.1). These are solutions of the form Φ(x, t) =ψ(x+ctω), where ω∈S N−1 is the direction of propagation andc∈R ∗ is the speed of the traveling wave. They are supposed to play an important role in the dynamics of (1.1). We say thatψhas finite energy ifrψ∈L 2 (R N ) andV(|ψ| 2 )∈L 1 (R N the equation (1.1) is rotation invariant, ). Since we may assume thatω= (1,0, . . . ,0). Then a traveling wave of speedcsatisfies the equation ∂ψ + Δψ (1.3) ∂x ic 1 +F(|ψ| 2 )ψ in= 0R N . It is obvious that a functionψsatisfies (1.3) for some velocitycif and only ifψ(−x 1 , x 0 ) satisfies (1.3) withcreplaced by−c it suffices to consider the case. Hencec≥0.

In view of formal computations and numerical experiments, it has been conjectured that finite energy traveling waves of speedcexist only for subsonic speeds:c < v s . The nonexistence of traveling waves for supersonic speeds (c > v s [28] in the case of the) has been proved first in Gross-Pitaevskii equation, then in [41] for a wide class of nonlinearities. More qualitatively, the numerical investigation of the traveling waves of the Gross-Pitaevskii equation (F(s) = 1−s) has been carried out in [31]. The method used there was a continuation argument with respect to the speed, solving (1.3) by Newton’s algorithm. DenotingQ(ψ) =P 1 (ψ) the momentum ofψwith respect to thex 1 −direction, the representation of the solutions in the energy vs. momentum diagram gives the following curves (the straight line is the lineE=v s Q).

Figure 1: (E, P) diagrams for (GP): (a) dimensionN= 2; (b) dimensionN= 3.

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The rigorous proof of the existence of traveling waves has been a long lasting problem and was considered in a series of papers, see [9], [8], [14], [7], [43]. At least formally, traveling waves are critical points of the functionalE−cQ it is a natural idea to look for such. Therefore, solutions as minimizers of the energy at fixed momentum, the speedcbeing then the Lagrange multiplier associated to the minimization problem. In the case of the Gross-Pitaevskii equation, in view of the above diagrams, this method is expected to give the full curve of traveling waves ifN= 2 and only the lower part ifN= 3 (since clearly, minimizers ofEat fixedQcan not lie on the upper branch). On a rigorous level, minimizing the energy at fixed momentum was used in [8] to construct a sequence of traveling waves with speedsc n −→0 in dimensionN≥3. Minimizing the energyEat fixed momentumQhas the advantage to provide orbitally stable traveling waves, which is intimately related to the concavity of the curveQ7→E. On the other hand, ifQ7→Eis convex, as it is the case on the upper branch in figure 1 (b), one expects orbital instability. More recently, the curves describing the minimum of the energy at fixed momentum in dimension 2 and 3 have been obtained in [7], where the existence of minimizers ofEunder the constraintQ=constantis also proved for anyq >0 ifN= 2, respectively for anyq∈(q 0 ,∞) (withq 0 >0) ifNin [7] depend on the special algebraic structure of the= 3. proofs  The Gross-Pitaevskii nonlinearity and it seems difficult to extend them to other nonlinearities. The existence of minimizers has been proved by considering the corresponding problem on tori (R/2nπZ) N , proving a priori bounds for minimizers on tori, then passing to the limit as n−→∞ this method gives the existence of minimizers on. AlthoughR N , it does not imply the precompactness of all minimizing sequences, and therefore leaves the question of the orbital stability of minimizers completely open.

The existence of traveling waves for (1.1) under general conditions on the nonlinearity, in any space dimensionN≥3 and for any speedc∈(0, v s ) has been proved in [43] by minimizing the actionE−cQ Theunder a Pohozaev constraint. method in [43] cannot be used in space dimension two (there are no minimizers under Pohozaev constraints). Although the traveling waves obtained in [43] minimize the actionE−cQamong all traveling waves of speedc, the constraint used to prove their existence is not conserved by the flow of (1.1) and consequently it seems very difficult to prove their orbital stability (which is expected at least for small speeds c).

In the present paper we adopt a different strategy. If the nonlinear potentialVis non- negative, we consider the problem of minimizing the energy at fixed momentumQ=qand we show that in any space dimensionN≥2 there exist minimizers for anyq∈(q 0 ,∞), with q 0 ≥0. The minimizers are traveling waves and their speeds are the Lagrange multipliers associated to the variational problem. These speeds tend to zero asq∞→−. IfN= 2 andF has a good behavior nearr 2 0 (more precisely, if assumption (A4) below is satisfied with Γ6= 0), we prove thatq 0 waves that we obtain tend to= 0 and that the speeds of the traveling v s asq−→nonlinearities we obtain the properties of the minimum of the energy general 0. For vs. momentum curve and this is in agreement with the results in [31], [32] and [7]. We also prove the precompactness of all minimizing sequences for the above mentioned problem, which implies the orbital stability of the set of traveling waves obtained in this way. IfVachieves negative values (this happens, for instance, in the case of the cubic-quintic NLS), the infimum of the energy on the set of functions of constant momentum is always −∞. In this case we minimize the functionalE−Qin the set of functionsψsatisfying R R N |rψ| 2 dx=k space dimension. InN≥2 we prove that minimizers exist for anykin some interval (k 0 , k ∞  Moreover, if) and, after scaling, they give rise to traveling waves.N= 2 and Fbehaves nicely nearr 2 0 we havek 0 = 0 and the speeds of traveling waves obtained in this way tend tov s ask−→0. Letresult of [43], which holds for any us emphasize that the N≥3,

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does not require any sign assumption on the potentialV. In space dimension two, even in the case whenVtakes negative values it is still possible to find local minimizers of the energy under the constraintQ=qifq Ifis not too large.F satisfies assumption (A4) below with Γ60 this can be done for any= qin some interval (0, q ∞ ) and the speeds of traveling waves obtained in this way tend tov s asp−→ we0. Moreover, get the precompactness of all minimizing sequences, and consequently the orbital stability of the set of local minimizers. Our results cover as well nonlinearities of Gross-Pitaevskii type and of cubic-quintic type. To the best of our knowledge, all previous results in the literature about the existence of traveling waves for (1.1) in space dimension two are concerned only with the Gross-Pitaevskii equation and the proofs make use of the specific algebraic properties of this nonlinearity. The main disadvantage of the present approaches is that although we get minimizers for any momentum in some interval (q 0 ,∞) or (0, q ∞ or for any kinetic energy in some interval) (k 0 , k ∞ ), the speeds we obtain are Lagrange multipliers, so we cannot guarantee that these speeds cover a whole interval. However, in all cases it can be proved that we get an uncountable set of speeds. One might ask whether there is a relationship between the families of traveling waves obtained from different minimization problems. In dimensionN≥3 we prove that all traveling waves that we obtain in the present paper also minimize the actionE−cQunder the Pohozaev constraint considered in [43]. The converse is, in general, not true. For instance, in the case of the Gross-Pitaevskii equation in dimensionN≥3, it was proved in [7, 21] that there are no traveling waves of small energy, and we generalize that result in the present paper; this implies that there isc 0 < v s such that there are no traveling waves of speedc∈(c 0 , v s ) which minimize the energy at fixed momentum. However, ifN≥the existence of traveling waves as3 minimizers ofE−cQunder a Pohozaev constraint has been proved in [43] for anyc∈(0, v s ). This is in agreement with the energy-momentum diagram of figure 1 (b), where the traveling waves with speedcclose to the speed of soundv s  Weare expected to be on the upper branch. also prove that all minimizers of the energy at fixed momentum are (after scaling) minimizers ofE−Qat fixed kinetic energy. is an open question whether the converse is true or not. It An affirmative answer to this question would imply that the set of speeds of traveling waves which minimize the energy at fixed momentum is an interval. Let us mention that traveling waves for (1.1) in dimensionN= 2 with general nonlinearities as those studied in dimension one in [15] have been studied numerically in [17]. The numerical algorithms in [17] allow to perform the constrained minimizations used is this paper. It appears that forN= 2, even when the potentialVis nonnegative, it is not true in general that minimizingEat fixedQor minimizingE−Qat fixed kinetic energy provides a single interval of speeds; for instance, it may provide the union of two disjoint intervals.

We will consider the following set of assumptions: 2 (A1)The functionFis continuous on [0,∞),C 1 in a neighborhood ofr 0 ,F(r 2 0 ) = 0 and F 0 (r 2 )<0. 0 (A2)There existC >0 andp 0 < N 2 −2 (withp 0 <∞ifN= 2) such that|F(s)| ≤ C(1 +s p 0 ) for anys≥0. (A3)There existC, α 0 >0 andr ∗ > r 0 such thatF(s)≤ −Cs α 0 for anys≥r ∗ . (A4)FisC 2 nearr 02 and F(s) =F 0 (r 02 )(s−r 2 0 )+ 1 2 F 00 (r 02 )(s−r 2 0 ) 2 +O((s−r 2 0 ) 3 ) forsclose tor 2 0 .

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If (A4) holds, we define

(1.4)

2 Γ := 6− 4 v r 2 0 4 F 00 (r 0 ). s

If (A1) and (A3) are satisfied, it is explained in the introduction of [43] how it is possible ˜ to modifyFinfinity in such a way that the modified functionis a neighborhood of Fsatisfies also (A2) and (1.1) has the same traveling waves as the equation obtained from it by replacing ˜ FwithF (A1) and (A2) hold, we get traveling waves as minimizers of some functionals. If under constraints. However, if (A1) and (A3) are verified but (A2) is not, the above argument implies only the existence of such solutions, and not the fact that they are minimizers. If (A1) is satisfied, we denoteV(s) = R r s 2 0 F(τ)dτanda= q − 1 2 F 0 (r 2 0 ).Then the sound velocity at infinity associated to (1.1) isv s = 2ar 0 and using Taylor’s formula forsin a neighborhood ofr 02 we have (1.5)V(s 1 =) 2 V 00 (r 2 0 )(s−r 2 0 ) 2 + (s−r 02 ) 2 ε(s−r 2 0 ) =a 2 (s−r 02 ) 2 + (s−r 2 0 ) 2 ε(s−r 02 ), whereε(t)−→0 ast−→0.Hence for|ψ|close tor 0 ,V(|ψ| 2 ) can be approximated by the 2 2 Ginzburg-Landau potentiala 2 (|ψ| 2 −r 0 ) . Energy and function spaces.We fix an odd functionϕ∈C ∞ (R) such thatϕ(s) =sfor s∈[0,2r 0 ], 0≤ϕ 0 ≤1 onRandϕ(s) = 3r 0 fors≥4r 0  denote. WeW(s) =V(s)−V(ϕ 2 ( √ s)), so thatW(s) = 0 fors∈[0,4r 2 0 and (A2) are satisfied, it is not hard to]. If assumptions (A1) see that there existC 1 , C 2 , C 3 >0 such that s)| ≤C 1 (s−r 02 ) 2 for anys≤9r 2 0 ; (1.6) i | n V itucal,r ( arp|V(ϕ 2 (τ))| ≤C 1 (ϕ 2 (τ)−r 2 0 ) 2 for anyτ;

(1.7)|V(b)−V(a)| ≤C 2 |b−a|max(a p 0 , b p 0  any) fora, b≥2r 2 0 ; (1.8)|W(b 2 )−W(a 2 )| ≤C 3 |b−a|  a 1 1 {a>2r 0 } +b 2p 0 +1 1 {b>2r 0 }  for anya, b≥0. 2p 0 + Givenψ∈H l 1 oc (R N ) and Ω an open set inR N , the modified Ginzburg-Landau energy ofψin Ω is defined by (1.9)E Ω GL (ψ) = Z Ω |rψ| 2 dx+a 2 Z Ω  ϕ 2 (|ψ|)−r 2 0  2 dx. We simply writeE GL (ψ) instead ofE G R L N (ψ modified Ginzburg-Landau energy will play). The a central role in our analysis. We denoteH ˙ 1 (R N ) ={ψ∈L l 1 oc (R N )| rψ∈L 2 (R N )}and ∈H ˙ 1 (R N )|ϕ 2 (|ψ|)−r 02 ∈L 2 (R N )} (1.10) E = ={ { ψ ψ∈H 1 (R N )|E GL (ψ)<∞}. ˙ LetD 1,2 (R N ) be the completion ofC c ∞ for the normkvk=krvk L 2 (R N ) and let =u∈ D 1,2 (R N )|ϕ 2 (|r 0 −u|)−r 2 0 ∈L 2 (R N )} (1.11) X { H ˙ 1 (R N )|u∈L 2 ∗ (R N ), E GL (r 0 −u)<∞}ifN≥3. ={u∈

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IfN≥3 andψ∈ E, there exists a constantz 0 ∈Csuch thatψ−z 0 ∈L 2 ∗ (R N ), where 2 ∗ = N 2 − N 2 (see, for instance, Lemma 7 and Remark 4.2 pp. 774-775 follows in [24]). It thatϕ(|ψ|)−ϕ(|z 0 |)∈L 2 ∗ (R N ). On the other hand, the fact thatE GL (ψ)<∞implies ϕ(|ψ|)−r 0 ∈L 2 (R N ), thus necessarilyϕ(|z 0 |) =r 0 , that is|z 0 |=r 0  it is easily seen. Then that there existα 0 ∈[0,2π) andu∈ X, uniquely determined byψ, such thatψ=e iα 0 (r 0 −u). In other words, ifN≥3 we haveE={e iα 0 (r 0 −u)|α 0 ∈[0,2π), u }∈ X. It is not hard to see that forN≥2 we have (1.12)E={ψ:R N −→C|ψis measurable,|ψ| −r 0 ∈L 2 (R N ),rψ∈L 2 (R N )}. Indeed, we have|ϕ 2 (|ψ|)−r 2 0 | ≤4r 0 | |ψ|−r 0 |, henceϕ 2 (|ψ|)−r 2 0 ∈L 2 (R N ) if|ψ|−r 0 ∈L 2 (R N ). Conversely, letψ∈ E. IfN= 2, it follows from Lemma 2.1 below that|ψ| 2 −r 02 ∈L 2 (R 2 ) and we have| |ψ| −r 0 |= |ψ|+ 1 r 0 | |ψ| 2 −r 2 0 | ≤ r 1 0 | |ψ| 2 −r 02 |.IfN≥3, we know thatϕ(|ψ|)−r 0 ∈L 2 (R N ) and 0≤ |ψ| −ϕ(|ψ|)≤ |ψ|1 {|ψ|≥2r 0 } ≤2(|ψ| −r 0 )1 {|ψ|≥2r 0 } ≤ 2 2 ∗ 2 −1 | |ψ| − | 2 ∗ r 0 2 1 {|ψ|≥2r 0 } and r 0 the last function belongs toL 2 (R N  Moreover,) by the Sobolev embedding. one may find bounds fork |ψ| −r 0 k L 2 (R N ) in terms ofE GL (ψ) (see Corollary 4.3 below). Proceeding as in [25], section 1, one proves thatE ⊂L 2 +L ∞ (R N ) and thatEendowed with the distance

(1.13)d E (ψ 1 , ψ 2 ) =kψ 1 −ψ 2 k L 2 +L ∞ (R N ) +krψ 1 − rψ 2 k L 2 (R N ) +k |ψ 1 | − |ψ 2 | k L 2 (R N ) is a complete metric space. We recall that, given two Banach spaces of distributions onR N , the Banach norm on the spaceX+Yis defined bykwk X+Y = inf{kxk X +kyk Y |w=x+y, x∈ X, y∈Y}. We will also consider the following semi-distance onE:

(1.14)d 0 (ψ 1 , ψ 2 ) =krψ 1 − rψ 2 k L 2 (R N ) +k |ψ 1 | − |ψ 2 | k L 2 (R N ) . Ifψ 1 , ψ 2 ∈ Eandd 0 (ψ 1 , ψ 2 ) = 0, then we have|ψ 1 |=|ψ 2 |a.e. onR N andψ 1 −ψ 2 is a constant (of modulus not exceeding 2r 0 ) a.e. onR N . In space dimensionN= 2,3,4, the Cauchy problem for the Gross-Pitaevskii equation has beenstudiedbyPatrickG´erard([24,25])inthespacenaturallyassociatedtothatequation, namely E={ψ∈H l 1 oc (R N )| rψ∈L 2 (R N ),|ψ| 2 −r 02 ∈L 2 (R N )} endowed with the distance

d E (ψ 1 , ψ 2 ) =kψ 1 −ψ 2 k L 2 +L ∞ (R N ) +krψ 1 − rψ 2 k L 2 (R N ) +k |ψ 1 | 2 − |ψ 2 | 2 k L 2 (R N ) . Global well-posedness is shown (see section 6) ifN∈ {2,3}or ifN= 4 and the initial data is small. IfN= 2,3 or 4 it can be proved thatE=Eand the distancesd E andd E are equivalent onE. Notation.Throughout the paper,L N is the Lebesgue measure onR N andH s is the s−dimensional Hausdorff measure onR N . Forx= (x 1 , . . . , x N )∈R N , we denotex 0 = (x 2 , . . . , x N )∈R N−1 . We writehz 1 , z 2 ifor the scalar product of two complex numbersz 1 , z 2 . Given a functionfdefined onR N andλ, σ >0, we denote (1.15)f λ,σ (x) =f  λ x 1 x σ , 0  . If 1≤p < N, we writep ∗ for the Sobolev exponent associated top, that is p 1 ∗ = p 1 − 1 N . Main results.Our most important results can be summarized as follows.

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Theorem 1.1Assume thatN≥2, (A1) and (A2) are satisfied andV≥0on[0,∞). For q≥0, let

Then:

E min (q) = inf{E(ψ)|ψ∈ E, Q(ψ) =q}.

(i) The functionE min is concave, increasing on[0,∞),E min (q)≤v s qfor anyq≥0, the right derivative ofE min at0isv s , andE min (q)→−∞and E mi q n (q) −→0asq−→∞. (ii) Letq 0 = inf{q >0|E min (q)< v s q}.For anyq > q 0 , all sequences(ψ n ) n≥1 ⊂ E satisfyingQ(ψ n )−→qandE(ψ n )−→E min (q)are precompact ford 0 (modulo translations). The setS q ={ψ∈ E |Q(ψ) =q, E(ψ) =E min (q)}is not empty and is orbitally stable (for the semi-distanced 0 ) by the flow associated to (1.1). (iii) Anyψ q ∈ S q a traveling wave for (1.1) of speedis c(ψ q )∈[d + E min (q), d − E min (q)], where we denote byd − andd +  Wethe left and right derivatives. havec(ψ q )−→0asq−→∞. (iv) WhenN≥3, we haveq 0 >0. Moreover, ifN= 2and assumption (A4) is satisfied, we haveq 0 = 0if and only ifΓ6= 0, in which casec(ψ q )−→v s asq−→0. IfVachieves negative values, the infimum ofEon the set{ψ∈ E |Q(ψ) =q}is−∞ for anyq. In this case we prove the existence of traveling waves by minimizing the functional I(ψ) =−Q(ψ) + R R N V(|ψ| 2 )dx(or, equivalently, the functionalE−Q) under the constraint R R N |rψ| 2 dx=k.More precisely, we have the following results: Theorem 1.2Assume thatN≥2 Forand (A1), (A2) are satisfied.k≥0, let I min (k) = inf n I(ψ)|ψ∈ E, Z R N |rψ| 2 dx=k o . Then, there isk ∞ ∈(0,∞]such that the following holds: (i) For anyk > k ∞ ,I min (k) =−∞. The functionI min is concave, decreasing on[0, k ∞ ), I min (k)≤ −k/v s 2 , for anyk≥0, the right derivative ofI min at0is−1/v s 2 , and I mi k n (k) → ∞ − − ask→−∞. (ii) Letk 0 = inf{k >0|I min (k)<−k/v s 2 } ∈[0, k ∞ ] any. Fork∈(k 0 , k ∞ ), all sequences(ψ n ) n≥1 ⊂ Esatisfying R R N |rψ n | 2 dx−→kandI(ψ n )−→I min (k)are pre- compact ford 0  If(modulo translations).ψ k ∈ Eis a minimizer forI min (k), there exists c=c(ψ k )∈[ p −1/d + I min (k), p −1/d − I min (k)]such thatψ k ( ∙ c )is tant traveling a non cons wave of (1.1) of speedc(ψ k ). (iii) We havek ∞ <∞if and only if (N= 2andinfV <0). Ifk ∞ =∞, the speeds of the traveling waves obtained from minimizers ofI min (k)tend to0ask−→∞. (iv) ForN≥3, we havek 0 >0. IfN= 2and assumption (A4) is satisfied we havek 0 = 0 if and only ifΓ6= 0the speeds of the traveling waves obtained from minimizers, in which case ofI min (k)tend tov s ask−→0. In space dimension two, the tools developed to prove Theorem 1.2 enable us to find min- imizers ofEat fixed momentum on a subset ofEeven ifV Weachieves negative values. have:

Theorem 1.3Assume thatN= 2and that (A1), (A2) are satisfied. Let E in ] m (q) = inf n E(ψ)|ψ∈ E, Q(ψ) =qand Z R 2 V(|ψ| 2 )dx≥0 o .

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