Rigorous derivation of an effective model for periodic Schrödinger equations with linear band crossing of Dirac type

RIGOR OUS DERIV A TION OF AN EFFECTIVE MODEL F OR PERIODIC SCHR ÖDINGER EQUA TIONS WITH LINEAR BAND CR OSSING OF DIRA C TYPE ELENA D ANESI Abstra ct. In this pap er we consider a family of time–dep endent 1 − dimensional cubic Schrödinger equation (NLS) with p eriodic p otential. Exploiting semiclassical scaling and m ultiscale analysis, w e derive an effective nonlinear Dirac equation, which describ es the dynamics of solutions to NLS sp ectrally localized around Dirac p oints. 1. Intr oduction and main resul t In this pap er w e discuss the 1 − dimensional cubic Schrödinger equation i∂ t ψ = − ∂ 2 x ψ + V ( x ) ψ + κ | ψ | 2 ψ , ( t, x ) ∈ R × R (1.1) where V ( x ) is a smo oth, ev en, 1 − p erio dic p oten tial, κ = ± 1 , with initial datum ψ (0 , x ) = ψ 0 ( x ) , in some suitable frequency regime. In order to highlight the main features of (1.1), let us recall that the Schrödinger op erator H : = − ∂ 2 x + V ( x ) with the p oten tial V as b efore, has a purely absolutely contin uous sp ectrum whic h also displays a band structure, see Section (2.2). Heuristically , one then exp ects that, if the initial datum is spectrally localized in some suitable sense around a point, then the solution to the Cauc h y problem asso ciated with (1.1) is w ell approximated b y that of an effective problem, whic h dep ends on the corresp onding sp ectral bands. The study of sp ectrally localized wa v e pack ets in p erio dic media has b een extensiv ely in v es- tigated in the literature. Effective equations describing the env elope dynamics arise in many regimes (see, for instance, [8, 9, 3] and the monograph [10]). In this work we are in terested in the b eha viour of solutions lo calized “around” so called Dir ac p oints , where a linear band crossing of Dirac t yp e o ccurs. Near Dirac p oin ts, the dispersion relation resembles that of relativistic particles, and the effectiv e dynamics of w a v e pack ets are exp ected to b e gov erned by Dirac t yp e equations. In the 2 − dimensional setting the linear and nonlinear cases hav e b een discussed, see [7, 1], re- sp ectiv ely . The authors prov ed that, denoting b y ( k ∗ , µ ∗ ) a Dirac p oint and b y Φ − ( xk ∗ ) , Φ + ( x, k ∗ ) the tw o corresp onding Blo ch w a ves (see Section (2.3) for the definition), if the initial datum is lo calized around the t w o Bloch wa v es, that is v ε 0 ( x ) ∼ ε → 0 + √ ε  ψ 0 , − ( εx )Φ − ( x, k ∗ ) + ψ 0 , + ( εx )Φ + ( x, k ∗ )  , then the corresp onding solution of the linear/nonlinear p erio dic Sc hrödinger equations ev olv es as v ε ( t, x ) ∼ ε → 0 + e − iµ ∗ t √ ε  ψ − ( εt, εx )Φ − ( x, k ∗ ) + ψ + ( εt, εx )Φ + ( x, k ∗ )  , where ψ − , ψ + are the comp onen ts of a solution to an effectiv e linear/nonlinear Dirac equation, with initial datum ψ 0 , ± ( x ) , on a time interv al of order O ( ε − 2+ ) in the linear case and of order O ( ε − 1 ) in the nonlinear one. Concerning the linear equation we men tion also the pap er [4] where the stationary case is treated. 1 2 E. DANESI More recently , atten tion has turned to the stationary case in nonlinear regime. In [2] the existence of a particular type of standing w av es v ( x ) = e − itµ ∗ u ( x ) , called Dir ac solitons , in the 1 − dimensional case is studied. The authors prov ed that there exists a solution to the stationary NLS (1.1) of the following form u δ ( x ) = √ δ (Ψ − ( δ x )Φ − ( x, k ∗ ) + Ψ + ( δ x )Φ + ( x, k ∗ ) + r δ ( x )) where r δ is a low er order correction term as δ → 0 and Ψ( y ) = (Ψ − ( y ) , Ψ + ( y )) T is a spinor solving an effectiv e stationary nonlinear Dirac equation. W e recall that, in order to allow the existence of suc h solutions, b oth in dimension one and t wo, one has to open a sp ectral gap around the energy of the Dirac p oint, which reflects in considering a small p erturbation of the op erator H , that is H δ = H + δ W ( x ) , where W ∈ C ∞ ( R ) is real–v alued and satisfies some symmetries assumptions (we refer to [2, Section 1.1] for all the details). Therefore, in the stationary case, the small parameter δ gives the amplitude of the sp ectral gap around the energy µ ∗ of the Dirac p oin t and u δ is actually a solutions of the NLS p erturb ed with a negligible p otential as δ → 0 . This is, to our knowledge, the only a v ailable result of this type, concerning the stationary NLS (1.1). The present w ork aims to fill the gap in the literature in the 1 − dimensional case, by rigorously justify the v alidit y of time–dep enden t nonlinear Dirac equations (NLD) as effective mo del for a class of NLS (1.1), in time scales of order ε − 1 . This is the conten t of the follo wing result. Theorem 1.1. L et V satisfies Assumption (2.3) and let ( π , µ ∗ ) b e a Dir ac p oint for the op er ator H : = − ∂ 2 x + V ( x ) . L et Φ − ( · , k ) , Φ + ( · , k ) , k ∈ [0 , 2 π ] b e the two c orr esp onding Blo ch waves as in Se ction (2.3) . L et c ♯ : = 2 i ⟨ ∂ x Φ − ( · , π ) , Φ − ( · , π ) ⟩ L 2 ([0 , 1]) = − 2 i ⟨ ∂ x Φ + ( · , π ) , Φ + ( · , π ) ⟩ L 2 ([0 , 1]) , (1.2) β 1 := Z 1 0 | Φ + ( x, π ) | 2 | Φ − ( x, π ) | 2 dx, β 2 := Z 1 0 Φ + 2 ( x, π )Φ 2 − ( x, π ) dx. Mor e over, let α ∈ C ([0 , T ]; H S ( R )) 2 b e a solution to the nonline ar Dir ac e quation (3.5) for some S > s + 3 , s > 1 2 . Final ly, denoting by Φ( x, π ) = (Φ − ( x, π ) , Φ + ( x, π )) , assume that the initial datum ψ 0 satisfies ∥ ψ 0 − √ εα 0 ( εx ) · Φ( x, π ) ∥ H s ≤ cε, (1.3) for some c > 0 . Then, for any T ∗ ∈ [0 , T ] , ther e exists ε 0 = ε 0 ( T ∗ ) ∈ (0 , 1) and a c onstant C > 0 such that for al l ε ∈ (0 , ε 0 ) , the solution ψ ∈ C ([0 , ε − 1 T ∗ ]; H s ( R )) to the Cauchy pr oblem asso ciate d with (1.1) exists and satisfies sup 0 ≤ t ≤ ε − 1 T ∗ ∥ ψ ( t, · ) − √ εe − itµ ∗ α ( εt, εx ) · Φ( x, π ) ∥ H s ≤ C ε. 1.1. Strategy. W e briefly describ e here the strategy adopted to prov e the main result. F ollo wing the ideas in [1], where the analogous 2 − dimensional case is studied, we find more conv enien t to w ork in the semiclassical setting. More in detail, giv en the function ψ , w e consider the rescaled function ψ ε ( t, x ) = ε − 1 2 ψ ( ε − 1 t, ε − 1 x ) . (1.4) Then, if ψ is a solution of (1.1) in a time interv al [0 , T ] , ψ ε is a solution, for t ∈ [0 , εT ] , of the semiclassical cubic Schrödinger equation given by iε∂ t ψ ε = − ε 2 ∂ 2 x ψ ε + V  x ε  ψ ε + εκ | ψ ε | 2 ψ ε , ψ ε (0 , x ) = ψ ε 0 ( x ) (1.5) where κ = ± 1 . Clearly , the viceversa holds to o. Notice that the factor ε in front of the nonlinear term is due to the scale ε − 1 2 in (1.4). As explained in [1], this is in order to “comp ensate” linear 3 and nonlinear effects and let the Dirac evolution app ear. W e fo cus then on (1.5), we p erform a multiscale expansion ψ ε N ( t, x ) : = e − iµ ∗ t ε N X n =0 ε n u n ( t, x, x ε ) , where w e require that, for an y n = 0 , . . . , N , u n ( t, x, · ) is π − pseudop erio dic in the v ariable y = x ε . Then, we formally plug this ansatz in (1.5). This yields i∂ t ψ ε N − H ε ψ ε N − εκ | ψ ε N | 2 ψ ε N = e − iµ ∗ t ε 3 N +1 X n =0 ε n X n where H ε : = − ε 2 ∂ 2 x + V ( x ε ) . By solving X n = 0 , for an y n ≤ N w e obtain an approximate solution, whic h formally solves the rescaled NLS up to an error of order O ( ε N +1 ) . W e observ e that in the 1 − dimensional case it is sufficient to take the expansion up to N = 1 . In particular, w e will prov e in Section (3) that the term u 0 is given by u 0 ( t, x, x ε ) = α − ( t, x )Φ − ( x ε , π ) + α + ( t, x )Φ + ( x ε , π ) where Φ ± ( · , π ) are the Blo c h wa v es as in Theorem (1.1) and α ± ( t, x ) are comp onents of the solution of the nonlinear Dirac equation (3.5). T o conclude, w e consider the time evolution of the H s -norm of the difference b et ween the exact and the appro ximate solution of (1.5) and w e estimate it combing (linear and nonlinear) functional inequalities and Gron w all’s Lemma. 1.2. Organization of the paper. The pap er is organized as follo ws. In Section (2) w e recall the main to ols on perio dic Sc hrö dinger operators in dimension one, with a focus on Floquet– Blo c h theory and Dirac p oints. Moreov er, w e describ e the functional background on which the analysis will b e carried on. In Section (3) we p erform the multiscale expansion of the solution to the semiclassical NLS. Section (4) is dedicated to the study of the well–posedness of the NLD. Finally , in Section (5) w e giv e the pro of of Theorem (1.1). 2. Preliminaries 2.1. Notation. W e fix the following notation: (1) w e denote by N + : = N \ { 0 } , (2) giv en t w o C 2 -v alued functions f = ( f 1 , f 2 ) and g = ( g 1 , g 2 ) , w e denote f · g = P 2 j =1 f j g j , (3) L 2 ( R ) -F ourier transform and its in verse are given by F ( f )( ξ ) = 1 √ 2 π Z R e − ixξ f ( x ) dx, F − 1 ( g )( x ) = 1 √ 2 π Z R e ixξ g ( ξ ) dξ , (4) giv en a measurable set X ⊆ R we denote ⟨ f , g ⟩ : = Z X f ¯ g dx and ∥ f ∥ 2 L 2 ( X ) : = ⟨ f , f ⟩ L 2 ( X ) and, for any s > 0 the Sob olev spaces H s ( X ) are defined in the usual wa y . 2.2. Flo quet–Blo c h theory. W e recall the main to ols of the Flo quet–Blo c h theory applied to the 1 − dimensional p erio dic Schrödinger op erator H : = − d 2 dx 2 + V ( x ) (2.1) where V is a smo oth, real–v alued, 1 − p erio dic p oten tial. F or a complete discussion on this topic w e refer the reader to [4, Chapter 2] and to [14, Section XI I I.16] for the general theory of p erio dic Sc hrö dinger op erators. Let us start b y introducing the pseudop erio dic spaces. 4 E. DANESI Definition 2.1. Given k ∈ R , w e define the space of k − pseudop erio dic L 2 functions as the set L 2 k ( R ) : = { f ∈ L 2 loc ( R ) : f ( x + 1 , k ) = e ik f ( x, k ) , ∀ x ∈ R } , endo w ed with inner pro duct ⟨ f , g ⟩ L 2 ([0 , 1]) and the asso ciated norm. The Sob olev spaces H s k ( R ) are then defined in the natural w a y , for all s ∈ N . Observ e that the pseudop erio dic condition is inv ariant under translations k 7→ k + 2 π . There- fore, it is natural to w ork with a fundamental dual p erio d cell, called Brillouin zone, B = [0 , 2 π ] . Then, for an y k ∈ [0 , 2 π ] we denote by H ( k ) the op erator in L 2 k ( R ) with the same action of H and domain H 2 k ( R ) and w e consider the one–parameter family of Flo quet–Blo ch eigenvalue pr oblems ( H ( k )Φ n ( x, k ) = µ n ( k )Φ n ( x, k ) , Φ n ( x + 1 , k ) = e ik Φ n ( x, k ) . (2.2) The op erator H ( k ) is self–adjoin t on its domain and has compact resolven t. Therefore, for each k ∈ [0 , 2 π ] this elliptic b oundary v alue problem originates a sequence of discrete eigenv alues µ 1 ( k ) ≤ µ 2 ( k ) ≤ · · · ≤ µ n ( k ) ≤ . . . . (2.3) The functions µ n : [0 , 2 π ] → R are called disp ersion b ands of the op erator H and they provide a description of the sp ectrum of the associated op erator; that is σ ( H ) = σ a.c. ( H ) = [ n ∈ N + µ n ([0 , 2 π ]) . Moreo v er, they satisfy some additional prop erties. F or an y n ∈ N + there holds (i) µ n ( k ) = µ n (2 π − k ) , k ∈ [0 , π ] (ii) µ n ( · ) is Lipschitz con tin uous and analytic in (0 , π ) ∪ ( π , 2 π ) , (iii) µ n ( · ) is monotone (with different monotonicity) in the interv als [0 , π ] and [ π , 2 π ] , (iv) for every k ∈ [0 , π ) ∪ ( π , 2 π ] the inequalities in (2.3) are strict. In addition, the corresp onding normalized eigenfunctions Φ n ( x, k ) are called Blo ch waves and the family { Φ n ( · , k ) , n ∈ N + , k ∈ [0 , 2 π ] } is complete in L 2 ( R ) , in the sense that any f ∈ L 2 ( R ) can b e decomp osed as f ( x ) = 1 2 π X n ≥ 1 Z B ⟨ f , Φ n ( · , k ) ⟩ L 2 ( R ) Φ n ( x, k ) dk , and, if f ∈ H 2 ( R ) , H f = 1 2 π X n ≥ 1 Z B ⟨ f , Φ n ( · , k ) ⟩ L 2 ( R ) µ n ( k )Φ n ( x, k ) dk . 2.3. Dirac p oin ts. As explained in the In tro duction, in this pap er w e are in terested in the case when the disp ersion bands exhibit a linear crossing of Dirac t yp e. W e giv e here the definit ion of Dirac point, i.e. a p oin t in the quasimomentum/energy plane at which there is a crossing of suc h type, and we discuss the necessary assumption on the p otential V for which the existence of this p oints is guaranteed. Definition 2.2. Let H b e as in (2.1) where V is 1 − p erio dic. W e sa y that a linear band crossing of Dirac t yp e occurs a the quasimomentum k ∗ ∈ [0 , 2 π ] and energy µ ∗ , or ( k ∗ , µ ∗ ) is a Dirac p oin t, if the follo wing holds: (1) there exists n ∗ ≥ 1 such that µ ∗ = µ n ∗ = µ n ∗ +1 ; (2) µ ∗ is an L 2 k ∗ eigen v alue of multiplicit y 2 ; 5 (3) there exists tw o spaces H 2 A , H 2 B suc h that H 2 k ∗ = H 2 A ⊕ H 2 B , H : H 2 A → L 2 A and H : H 2 → L 2 B ; (4) there exists an op erator S : H 2 A → H 2 B and S : H 2 B → H 2 A , S ◦ S = I , suc h that [ e − ik ∗ x H e ik ∗ x , S ] = 0 ; (5) there exists a function g 1 suc h that k er L 2 k ∗ ( H − µ ∗ I ) = S pan { g 1 ( x ) , g 2 ( x ) = S [ g 1 ]( x ) } , ⟨ g 1 , g 2 ⟩ L 2 ([0 , 1]) = δ ab , a, b = 1 , 2; (2.4) (6) there exists c ♯  = 0 , ζ 0 > 0 and t wo normalized eigenpairs (Φ − ( x, k ) , µ − ( k )) (Φ + ( x, k ) , µ + ( k )) and smo oth functions η ± ( k ) with η ± (0) = 0 , defined for | k − k ∗ | < ζ 0 and such that µ ± ( k ) − µ ∗ = ± c ♯ ( k − k ∗ )  1 + µ ± ( k − k ∗ )  . This is the general definition of a Dirac p oint. Let us no w fo cus on the Schrödinger perio dic op erator H such that V satisfies the follo wing Assumption 2.3. V ∈ C ∞ ( R ) , is real–v alued and there exists a real sequence ( V m ) m ∈ 2 N + ∈ l 2 suc h that V ( x ) = X m ∈ 2 N + V m cos(2 π mx ) , ∀ x ∈ R . Then, for almost ev ery V satisfying assumption (2.3), by [4, Theorems 3.6-3.7], there exists a Dirac p oint in sense of Definition (2.2) with k ∗ = π . R emark 2.4 . Let us observe that, by the prop erties of the disp ersion bands describ ed in Section (2.2), a Dirac p oin t can only app ear at quasimomentum k ∗ = π . Moreov er, since the Flo quet– Blo c h eigenv alue problem (2.2) is a second order 1 − dimensional ODE, at an y quasimomentum k the multiplicit y of an eigen v alue is at most t w o. Therefore, no more than t w o disp ersion bands can collapse at k . In this case it is p ossible to give a precise description of the spaces H 2 A , H 2 B and the op erator S . In order to lighten the presen tation we do not discuss them and w e refer to [4, Section 3.2] for all the details. R emark 2.5 . Let us observe that the choice of the famil y of Blo ch wa ves is unique but for a phase. In particular, it has b een prov ed in [2, Prop osition 2.12], that under some suitable assumptions on the p oten tial V , one can choose a family o f Blo ch wa ves with some additional symmetries. In addition, as prov en in [4, Prop osition 3.5], it is p ossible, in the 1 − dimensional case, to reparametrize the Blo c h w av es corresp onding to the Dirac p oin t in order to increase the regularit y of the disp ersion bands at the p oint k = π . In this pap er, w e will not need to exploit these other prop erties. Therefore, we do not need to sp ecify any particular c hoice of the family of Blo ch wa v es, but, for simplicity , w e adopt the same c hoice as in [2]. That is, given the Dirac p oint ( π , µ ∗ ) we denote by Φ + ( x, k ) , Φ − ( x, k ) the tw o normalized Blo c h functions suc h that (2.4) holds with g 1 ( x ) = Φ − ( x, k ) and g 2 ( x ) = Φ + ( x, k ) = Φ − ( − x, k ) . Moreo ver, w e also hav e that Φ ± ( x, π ) = Φ ∓ ( x, π ) . 2.4. F unctional bac kground. W e recall the functional space and the tools w e will use to handle the approximate solution, in particular the highly oscillatory Blo c h eigenfunctions, in the reparametrization given by (1.4). Definition 2.6. Let s ∈ [0 , + ∞ ) and 0 < ε ≤ 1 . W e define the scaled Sob olev space H s ε ( R ) : = { f ε ∈ L 2 ( R ) : ∥ f ε ∥ H s ε < + ∞} with ∥ f ε ∥ 2 H s ε : = Z R (1 + | εξ | 2 ) s | ˆ f ( ξ ) | 2 dξ . 6 E. DANESI Here ˆ f denotes the usual F ourier transform of f in L 2 ( R ) . Then, w e recall that if s > 1 2 , the space H s ε is a multiplicativ e algebra. Moreo v er, the following results hold. Lemma 2.7 (Gagliardo–Niren b erg inequalit y) . F or any s > 1 2 , ther e exists a c onstant C > 0 such that for any f ∈ H s ε ( R ) the fol lowing holds ∥ f ∥ L ∞ ≤ C ε − 1 2 ∥ f ∥ H s ε (2.5) Pr o of. The factor ε − 1 2 is obtained by scaling by the standard Gagliando–Nirenberg inequalit y . □ Lemma 2.8 (Moser–type Lemma) . L et R > 0 , s ∈ [0 , + ∞ ) and N ( z ) = κ | z | 2 z with κ ∈ R . Then, ther e exists a c onstant c m = c m ( R, s, κ ) > 0 such that if ∥ ( ε∂ ) γ f ∥ L ∞ ≤ R , ∀ γ ≤ s, and ∥ g ∥ L ∞ ≤ R , then ∥N ( f + g ) − N ( f ) ∥ H s ε ≤ c m ∥ g ∥ H s ε . F or the pro of of Lemma (2.8) we refer to [12, Lemma 8.1]. 3. Tw o–scale asymptotic exp ansion W e p erform the asymptotic expansion of the solution to (1.5). W e consider ψ ε a ( t, x, x ε ) = e − iµ ∗ t ε  u 0 ( t, x, x ε ) + εu 1 ( t, x, x ε )  , (3.1) where u n , n = 0 , 1 are π − pseudo-p erio dic function with resp ect to the v ariable y : = x ε . In order to identify the functions u n ( t, x, y ) , n = 0 , 1 w e plug (3.1) in to (1.5), treating x and y as indep endent v ariables. This yields i∂ t ψ ε a − H ε ψ ε a − εκ | ψ ε a | 2 ψ ε a = e − iµ ∗ t ε 4 X n =0 ε n X n where, denoting by H = − ∂ 2 y + V ( y ) X 0 =( µ ∗ − H ) u 0 X 1 =( µ ∗ − H ) u 1 + ( i∂ t + 2 ∂ x ∂ y − κ | u 0 | 2 ) u 0 X 2 + X 3 + X 4 = : ρ (Ψ ε ) = ε 2  ( i∂ t + 2 ∂ x ∂ y − κ | u 0 | 2 ) u 1 + ∂ 2 x u 0 − κ 2 ℜ ( ¯ u 0 u 1 ) u 0  + (3.2) + ε 3  ∂ 2 x u 1 − κ ( u 0 | u 1 | 2 + 2 ℜ ( ¯ u 0 u 1 ) u 1 )  − ε 4 κ | u 1 | 2 u 1 . Then, we solve X 0 = 0 . That is, u 0 is a linear com binations of Φ − ( y , π ) , Φ + ( y , π ) , solutions of the eigenv alue problem (2.2). Therefore, we ha ve that u 0 ( t, x, y ) = α − ( t, x )Φ − ( y , π ) + α + ( t, x )Φ + ( y , π ) (3.3) where α j , j = ± to b e chosen. W e lo ok now at the first order in ε and w e solv e X 1 = 0 , i.e. ( H − µ ∗ ) u 1 = ( i∂ t + 2 ∂ x ∂ y − k | u 0 | 2 ) u 0 . (3.4) Observ e that H is a F redholm op erator then by F redholm’s alternativ e, (3.4) admits solutions if and only if the right hand side is orthogonal to k er L 2 k ∗  − ∂ 2 y + V ( · ) − µ ∗  . Thus we imp ose ⟨ ( i∂ t + 2 ∂ x ∂ y − k | u 0 | 2 ) u 0 , Φ j ( · , π ) ⟩ L 2 y ([0 , 1]) = 0 , j = ± . T aking c ♯ as in (1.2), w e hav e 2 ⟨ ∂ x ∂ y u 0 , Φ j ( k , · ) ⟩ = ± ic ♯ ∂ x α j ( t, x ) , whereas, by orthogonality of Φ − ( y , π ) , Φ + ( y , π ) i ⟨ ∂ t u 0 , Φ j ( · , π ) ⟩ = i∂ t α j ( t, x ) . 7 W e no w focus on the last term, − κ ⟨| u 0 | 2 u 0 , Φ j ( · , π ) ⟩ . W e observ e that, due to the symmetries of the Blo ch functions, describ ed in Remark (2.5), Z 1 0 | Φ j ( y , π ) | 2 Φ j ( y , π )Φ j ′ ( y , π ) dy = Z 1 0 Φ 2 j ( y , π )Φ j ′ ( y , π )Φ j ( y , π ) dy = Z 1 0 | Φ j ( y , π ) | 2 Φ j ′ ( y , π )Φ j ( y , π ) dy = 0 for any j = ± and j ′ = − j . Moreov er, given β 1 := Z 1 0 | Φ − ( x, π ) | 2 | Φ + ( x, π ) | 2 dx, β 2 := Z 1 0 Φ − 2 ( x, π )Φ 2 + ( x, π ) dx, it holds β 2 ∈ R , as sho wn in [2, Remark 2.15]. Therefore, w e hav e that if j = − − κ ⟨| u 0 | 2 u 0 , Φ − ( · , π ) ⟩ = − κ  ¯ α − ( β 2 α 2 + + β 1 α 2 − ) + 2 β 1 α − | α + | 2  and, if j = + − κ ⟨| u 0 | 2 u 0 , Φ + ( · , π ) ⟩ = − κ  2 β 1 α + | α − | 2 + ¯ α + ( β 2 α 2 − + β 1 α 2 + )  . Summing up, if w e choose α = ( α − , α + ) to b e a solution of the equation i∂ t α = − ic ♯ σ 3 ∂ x α + κ G β 1 ,β 2 ( α ) α (3.5) where G β 1 ,β 2 ( α ) =  β 1 ( | α − | 2 + 2 | α + | 2 ) β 2 ¯ α − α + β 2 ¯ α + α − β 1 ( | α + | 2 + 2 | α − | 2 )  , then (3.4) admits solution given by u 1 ( t, x, y ) = ( H − µ ∗ ) − 1  ( i∂ t + 2 ∂ x ∂ y − κ | u 0 | 2 ) u 0  (3.6) Let us observe that a generic solution of (3.4) should b e of the form u 1 = ˜ u 1 + u ⊥ 1 where ˜ u 1 ∈ k er ( H − µ ∗ ) and u ⊥ 1 as in (3.6). How ever, w e can c ho ose to tak e ˜ u 1 = 0 , which simplifies the expression. Concerning the initial data, we assume α (0 , x ) : = α 0 ( x ) to b e in the Sc hw artz space S ( R ) 2 . Instead, the initial condition for u 1 cannot b e freely chosen, but it can b e derived from (3.6) and the initial condition for u 0 . R emark 3.1 . Let us conclude by noticing that in the 2 − dimensional case, discussed in [1], this expansion is carried on up to the term of order ε 2 . In our case, it is sufficient to stop at the first order. As will b e clear from the pro of of Theorem (1.1), this is due to the scaling factor app earing in the Gagliardo–Nirenberg inequality (2.5), which dep ends on the dimension of the ph ysical space R n . 4. Local well–posedness of the NLD Prop osition 4.1. F or any α 0 ∈ S ( R ) 2 and for any s > 1 2 , ther e exists a p ositive time T and a unique maximal solution α ∈ C  [0 , T ]; H s ( R )  2 ∩ C 1  [0 , T ]; H s − 1 ( R )  2 to (3.5) . Pr o of. Using the F ourier transform, we define for an y f ∈ H s ( R ) 2 , s ≥ 1 , the strongly contin uous unitary group of propagators  U ( t ) f  ( x ) = e − tc ♯ σ 3 ∂ x f ( x ) = F − 1  e − itc ♯ ξ σ 3 ˆ f ( ξ )  ( x ) 8 E. DANESI satisfying ∥ U ( t ) f ∥ H s = ∥ f ∥ H s for an y s ∈ R . Moreo v er, w e define the Banach space X = C  [0 , T ]; H s ( R )  2 , s > 1 2 , endow ed with the norm ∥ u ∥ X = sup 0 ≤ t ≤ T ∥ u ( t ) ∥ H s . By Duhamel’s formula, the solution to (3.5) can b e written as α ( t ) = U ( t ) α 0 − iκ Z t 0 U ( t − τ ) G β 1 ,β 2  α ( τ )  α ( τ ) dτ = : Φ( α ) . Let R > 0 and let T > 0 to b e c hosen later. W e no w prov e that, for an y α 0 suc h that α 0 ∈ B R (0) ⊆ H s ( R ) 2 , the map Φ : X → X is a contraction on Y = { u ∈ X : ∥ u ∥ X ≤ 2 R } . W e first recall that for any s > 1 2 , H s ( R ) 2  → L ∞ ( R ) 2 is a commutativ e algebra such that ∥ uv ∥ H s ≤ C s ∥ u ∥ H s ∥ v ∥ H s , ∀ u, v ∈ H s ( R ) 2 . This implies that, for any u ∈ H s ( R ) 2 ∥G β 1 ,β 1 ( u ) u ∥ H s ≤ C 2 s b ∥ u ∥ 3 , b = 3 β 1 + | β 2 | , and, moreov er, for an y u, v ∈ Y , given that G β 1 ,β 2 ( z ) z ∈ C ∞ ( C 2 ) ∥G β 1 ,β 2 ( u ) u − G β 1 ,β 2 ( v ) v ∥ H s ≤ C s,β 1 ,β 2 R 2 ∥ u − v ∥ H s . Therefore, for any α 0 ∈ H s ( R 2 ) 2 suc h that ∥ α 0 ∥ H s ≤ R and α ∈ X , b y Minko wski’s inequality , w e ha v e ∥ Φ( α ) ∥ H s ≤ ∥ U ( t ) α 0 ∥ H s + | κ | Z t 0   U ( t − τ ) G β 1 ,β 2  α ( τ )  α ( τ )   H s dτ ≤ R + T 8 C 2 s bR 3 . Th us, if T ≤ 1 8 C 2 s bR 2 w e hav e that Φ( Y ) ⊆ Y . Let no w α, β ∈ Y solutions to (3.5) such that α (0) = β (0) = α 0 . F rom previous estimates, w e hav e that   Φ( α ) − Φ( β )   H s ≤ Z t 0   U ( t − τ )  G β 1 ,β 2 ( α ) α − G β 1 ,β 1 ( β ) β    H s dτ ≤ C s,β 1 ,β 2 R 2 T ∥ α − β ∥ H s . Therefore, Φ is a con traction on Y pro vided that 0 < T < min { 1 8 C 2 s bR 2 , 1 2 C s,β 1 ,β 2 R 2 } . Then, for suc h T , by Banac h’s fixed p oint theorem, there exists a unique solution α ∈ X to (3.5). The regularity in time directly follows b y (3.5). □ R emark 4.2 . Let us observ e that b oth the mass and the energy are conserved quantities for the system (3.5). How ev er, differen tly from the fo cusing NLS discussed in Prop osition (5.2), the energy do es not hav e a definite sign, therefore it is not straightforw ard to extend the solution globally in time. There are cases in which it is still p ossible to obtain global solution, even for large initial data, but this discussion is b eyond the scop e of this pap er. F or a survey on this topic we refer to [11] and the reference therein. 5. Pr oof of the resul t Prop osition 5.1 (Existence of solution to semiclassical NLS) . L et V satisfies Assumption (2.3) and ψ ε 0 ∈ H s ε ( R ) for s > 1 2 . Then, for any ε ∈ (0 , 1) ther e exists a p ositive time T ε and a unique solution ψ ε ∈ C ([0 , T ε ]; H s ε ( R )) to the Cauchy pr oblem asso ciate d with (1.5) . Mor e over, ther e is p ersistenc e of r e gularity; that is, if ψ ε 0 ∈ H ˜ s ε ( R ) for some ˜ s ≥ s , then ψ ε ∈ C ([0 , T ε ]; H ˜ s ε ( R )) . Pr o of. W e define the linear Schrödinger propagator S ε ( t ) : = e − i t ε H ε 9 generated b y the op erator H ε = − ε 2 ∂ 2 x + V ( x ε ) . It has b een prov ed in [8, Lemma 4.3] that there exists a constant c l > 0 such that ∥ S ε ( t ) f ∥ H s ε ≤ c l ∥ f ∥ H s ε for all t ∈ R and s ∈ [0 , + ∞ ) . With this result, the pro of of the lo cal well–posedness of the semiclassical NLS exploits the algebra prop ert y of the space H s ε ( R ) , s > 1 2 and follows as in [15, Prop osition 3.8]. Let us observe that the time of existence T ε = O ( ε − 1 ) and dep ends only on c l , ∥ Ψ ε 0 ∥ H s ε . Concerning the p ersistence of regularit y , we observ e that, combining the Duhamel’s form ula and the Gron wall’s inequalit y we hav e ∥ ψ ε ∥ H s ε ≤ c l ∥ ψ ε 0 ∥ H s ε e c l ε ∥ ψ ε ∥ 2 L 2 t L ∞ where w e use the notation ∥ · ∥ L 2 t = ∥ · ∥ L 2 ([0 ,t ]) . Con versely , if ψ ε ∈ C t H s ε , then by Sob olev em b edding ψ ε ∈ L 2 t L ∞ , lo cally in time. Thus, one can contin ue a solution in H s ε for s > 1 2 if and only if the L 2 t L ∞ -norm remain finite. This is indep enden t on s . That is, if w e also hav e that the initial datum ψ ε 0 ∈ H ˜ s ε , then the solution ψ ε can b e contin ued in the regularity H ˜ s ε for the same amoun t of time as it can b e contin ued in H s ε . □ Prop osition 5.2 (Global well–posedness defo cusing NLS) . L et V satisfies Assumption (2.3) and ψ ε 0 ∈ H 1 ε ( R ) . Then, for any ε ∈ (0 , 1) the solution ψ ε to the Cauchy pr oblem asso ciate d with (1.5) with κ = 1 exists glob al ly in time. Pr o of. W e start by observing that the mass of the solution is a conserv ed quan tit y , that is ∥ ψ ε ( t ) ∥ L 2 = ∥ ψ ε 0 ∥ L 2 . This can b e obtained by multiplying the equation (1.5) by ¯ ψ ε , in tegrating on R and taking the imaginary part. Moreov er, b y multiplying the same equation b y ∂ t ¯ ψ ε and taking the real part we hav e d dt E ( ψ ε ( t )) = d dt  Z R | ( ε∂ x ) ψ ε | 2 dx + Z R V ( x ε ) | ψ ε | 2 dx + ε 2 Z R | ψ ε | 4 dx  = 0 , that is, also the energy of the solution is conserv ed. In particular w e deduce ∥ ψ ε ∥ H 1 ε = Z R | ( ε∂ x ) ψ ε | 2 dx = E ( ψ ε 0 ) − Z R V ( x ε ) | ψ ε | 2 dx − ε 2 Z R | ψ ε | 4 dx ≤ E ( ψ ε 0 ) + ∥ V ∥ L ∞ ∥ ψ ε 0 ∥ L 2 . This means that the H 1 ε -norm of ψ ε remains bounded for all time, with a constant that depends only on ∥ ψ ε 0 ∥ H 1 ε . Then, one can rep eat the argumen t in Prop osition (5.1) and extend the solution of all time t ∈ R . □ Lemma 5.3 (Estimate of the appro ximate sol and the remainder) . L et S ∈ ( 7 2 , + ∞ ) . L et α − , α + ∈ C  [0 , T ]; H S ( R )  ∩ C 1  [0 , T ]; H S − 1 ( R )  b e the c omp onents of the solution to the NLD (3.5) and ψ ε a as in (3.1) . Then, for any t ∈ [0 , T ] and for any γ ∈ N , γ ≤ S − 2 , the fol lowing estimates hold ∥ ψ ε a ( t ) ∥ H S − 1 ε ≤ c 1 , ∥ ( ε∂ ) γ ψ ε a ( t ) ∥ L ∞ x ≤ c 2 , ∥ ρ ( ψ ε a )( t ) ∥ H S − 3 ε ≤ c 3 ε 2 . for some c onstants c 1 , c 2 , c 3 > 0 indep endent on ε . Pr o of. W e start by addressing the regularity of u 0 , u 1 . Concerning the former, we recall that, b y Prop osition (4.1), α − , α + ∈ C  [0 , T ]; H S ( R )  ∩ C 1  [0 , T ]; H S − 1 ( R )  . Moreov er, we hav e that Φ − ( · , π ) , Φ + ( · , π ) ∈ C ∞ ([0 , 1]) (see [13, Thm IX.26]). Therefore, recalling the expression for u 0 stated in (3.3), w e deduce u 0 ( t ) ∈ H S ( R ) × C ∞ ([0 , 1]) , ∂ t u 0 ( t ) ∈ H S − 1 ( R ) × C ∞ ([0 , 1]) . Concerning u 1 , by the previous analysis and recalling the formula giv en in (3.6), w e ha v e that u 1 ( t, x, y ) =  H − µ ∗ ) − 1 g ( t, x, y ) 10 E. DANESI where H = − ∂ 2 y + V ( y ) and g ( t ) ∈ H S − 1 ( R ) × C ∞ ([0 , 1]) . W e thus ha ve u 1 ( t ) ∈ H S − 1 ( R ) × C ∞ ([0 , 1]) . Then, it follows that for any s ≤ S − 1 , u n ( t, · , · ε ) ∈ H s ε ( R ) n = 0 , 1 , that is, the first estimate holds true, for some constan t c 1 > 0 . Let us no w fo cus on the second inequality . W e observe that it is sufficien t to show that there exists a constant c > 0 such that ∥ ( ε∂ ) γ u 0 ( t ) ∥ L ∞ ≤ c. (5.1) Indeed, by the Gagliardo–Niren b erg inequality (2.5) w e easily get that ∥ ε ( ε∂ ) γ u 1 ( t ) ∥ L ∞ ≤ cε 1 2 ∥ u 1 ∥ H 1+ γ ε whic h is b ounded, b y the previous estimate, provided that γ + 1 < S − 1 . T o pro ve (5.1), we use the Leibniz’ rule and we estimate ∥ ( ε∂ ) γ u 0 ( t ) ∥ L ∞ ≤ X j = ± ∥ ( ε∂ ) γ ( α j ( t, · )Φ j ( · ε , π ) ∥ L ∞ ≲ X j = ±  X σ ≤ γ ∥ ( ε∂ ) σ α j ( t, · ) ∥ L ∞ ∥ ( ε∂ ) γ − σ Φ j ( · ε , π ) ∥ L ∞  ≲ X j = ±  ∥ Φ j ( · , π ) ∥ C γ X σ ≤ γ ∥ ∂ σ α j ( t, · ) ∥ L ∞  ≲ X j = ± ∥ α j ( t, · ) ∥ H 1+ γ ≤ c where we use the standard Gagliardo–Nirenberg inequalit y and the fact that α ( t ) ∈ H S ( R ) 2 . T o conclude, the last inequality of the Lemma follows by recalling the form ula for ρ ( ψ ε a ) giv en in (3.2) and observing that, given u 0 ( t ) , u 1 ( t ) ∈ H S , X n ∈ H S − 3 ε ( R ) for any n = 2 , 3 , 4 . □ Pr o of of The or em (1.1) . As explained in the Introduction, we pro v e Theorem (1.1) in the semi- classical setting, that is, for the rescaled function ψ ε defined by (1.4) and then w e come bac k to the standard NLS (1.1). Let ψ ε b e the solution of the Cauch y problem asso ciated with (1.5) given b y Prop osition (5.1) with initial datum ψ ε 0 suc h that ∥ ψ ε 0 − α 0 ( x ) · Φ( x ε , π ) ∥ H s ε ≤ cε where c is as in (1.3). W e now consider the difference b et ween the exact solution and the appro ximate one, that is φ ε = ψ ε − ψ ε a where ψ ε a is defined by (3.1). It satisfies iε∂ t φ ε = H ε φ ε + ε  N ( φ ε + ψ ε a ) − N ( ψ ε a )  − ρ ( ψ ε a ) , φ ε | t =0 = φ ε 0 , where H ε = − ε 2 ∂ 2 x + V ( x ε ) , the nonlinearity is given by N ( u ) = κ | u | 2 u and ρ ( ψ ε ) is as in (3.2). W e denote by w ( t ) = ∥ φ ( t ) ∥ H s ε . W e aim to pro v e that w ( t ) ≤ cε , uniformly in t . W e observ e that, by assumption (1.3) and Lemma (5.3) w (0) ≤ ∥ ψ ε 0 − u 0 (0) ∥ H s ε + ε ∥ u 1 (0) ∥ H s ε ≤ ˜ cε. (5.2) Moreo v er, b y Duhamel’s formula φ ε ( t ) = S ε ( t ) φ 0 − i Z t 0 S ε ( t − τ )   N ( φ ε + ψ ε a ) − N ( ψ ε a )  − ε − 1 ρ ( ψ ε a )  dτ . Therefore, w ( t ) ≤ c l w (0) + c l Z t 0   N ( φ ε + ψ ε a ) − N ( ψ ε a )   H s ε dτ + ε − 1 c l Z t 0 ∥ ρ ( ψ ε a ) ∥ H s ε dτ . 11 By estimate (5.2) and Lemma (5.3), for any T ∗ ∈ [0 , T ) w e hav e w ( t ) ≤ c l (˜ c + c 3 T ∗ ) ε + c l Z t 0   N ( φ ε + ψ ε a ) − N ( ψ ε a )   H s ε dτ . T o estimate the integral on the RHS w e would lik e to use the Mores–type Lemma (2.8). T o do so, w e observ e that w ( t ) is contin uous in time. Therefore, choosing M > max( ˜ c, c l (˜ c + c 3 T ∗ ) e c l c m T ∗ ) w e ha v e that for an y ε ∈ (0 , 1) there exists a p ositive time t ε M > 0 such that w ( t ) ≤ M ε for an y t ≤ t ε M . Then, the Gagliardo–Niren b erg inequality (2.5) yields ∥ φ ε ( t ) ∥ L ∞ ≤ C ε − 1 2 w ( t ) ≤ ε 1 2 C M for t ≤ t ε M . Hence, there exists ε 0 ∈ (0 , 1) suc h that for an y 0 < ε ≤ ε 0 and for any t ≤ t ε M ∥ φ ε ( t ) ∥ L ∞ ≤ c 2 , where the constan t c 2 is an Lemma (5.3). Therefore, w e can apply Lemma (2.8) with R = c 2 and we obtain w ( t ) ≤ c l (˜ c + c 3 T ∗ ) ε + c l c m Z t 0 w ( τ ) dτ , ∀ ε ∈ (0 , ε 0 ] , ∀ t ≤ t ε M . Then, Gronw all’s Lemma yields w ( t ) ≤ c l (˜ c + c 3 T ∗ ) e c l c m T ∗ ε, ε ∈ (0 , ε 0 ] , ∀ t ≤ t ε M . Since w e choose M ≥ c l (˜ c + c 3 T ∗ ) e c l c m T ∗ , we ha ve that the necessary assumptions to apply lemma (2.8) are fulfilled for any ε ∈ (0 , ε 0 ] and any t ≤ T ∗ . Thus, we ha ve that, for any t ∈ [0 , T ∗ ] ∥ ψ ε − e − i t ε µ ∗ α ( t, · ) · φ ( · ε , π ) ∥ H s ε ≤ ∥ ψ ε − ψ ε a ∥ H s ε + ∥ e − i t ε µ ∗ εu 1 ( t, · , · ε ) ∥ H s ε (5.3) ≤ w ( t ) + ε ∥ u 1 ( t, · , · ε ) ∥ H s ε ≤ C ε. T o conclude, let us observ e that if ψ ε is a solution of (1.5) in C ([0 , T ∗ ]; H s ε ( R )) then ψ ( t, x ) : = √ εψ ε ( εt, εx ) is a solution of (1.1) in C ([0 , ε − 1 T ∗ ]; H s ( R )) and for an y t ∈ [0 , ε − 1 T ∗ ] ∥ ψ ( t, x ) − √ εe − itµ ∗ α ( εt, εx ) · Φ( x, π ) ∥ H s x ≤ √ ε ∥ ψ ε ( εt, εx ) − e − itµ ∗ α ( εt, εx ) · Φ( x, π ) ∥ H s x = ∥ ψ ε ( εt, x ) − e − itµ ∗ α ( εt, x ) · Φ( x ε , π ) ∥ H s ε ≤ C ε where the constant C > 0 is as in (5.3) and th us do es not dep end on ε . □ A c kno wledgemen ts. The author ac knowledge that this study w as carried out within the INdAM - GNAMP A Pro ject CUP E53C25002010001. References [1] Arbunic h J., Sparb er C., Rigorous deriv ation of nonlinear Dirac equations for wa ve propagation in honeycom b structures, J. Math. Phys. 59 (2018), no. 1, Paper No. 011509, 18 pp. [2] Borrelli W., Danesi E., Dov etta S., T en tarelli L., Dirac solitons in one-dimensional nonlinear Schrödinger equations, pr eprint , . 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