On the Dynamics of solitons in the nonlinear Schroedinger equation
aa r X i v : . [ m a t h - ph ] A p r On the Dynamics of solitons in the nonlinearSchroedinger equation
Vieri Benci ∗ Marco Ghimenti † Anna Maria Micheletti † November 28, 2018
Abstract
We study the behavior of the soliton solutions of the equation i ∂ψ∂t = − m ∆ ψ + 12 W ′ ε ( ψ ) + V ( x ) ψ where W ′ ε is a suitable nonlinear term which is singular for ε = 0 . Weuse the “strong” nonlinearity to obtain results on existence, shape,stability and dynamics of the soliton . The main result of this paper(Theorem 1) shows that for ε → V ( x ). Mathematics subject classification . 35Q55, 35Q51, 37K40, 37K45,47J30.
Keywords . Soliton dynamics, Nonlinear Schroedinger Equation, or-bital stability, concentration phenomena, stress tensor.
Contents ∗ Dipartimento di Matematica Applicata, Universit`a degli Studi di Pisa, Via F. Buonar-roti 1/c, Pisa, ITALY and Department of Mathematics, College of Science, King SaudUniversity, Riyadh, 11451, SAUDI ARABIA. e-mail: [email protected] † Dipartimento di Matematica Applicata, Universit`a degli Studi di Pisa, ViaF. Buonarroti 1/c, Pisa, ITALY. e-mail: [email protected],[email protected] The dynamics of solitons 9 ρ ε . . . . . . . . . . . . . . . . . . . . 203.3 The equation of dynamics as ε → Roughly speaking a solitary wave is a solution of a field equation whoseenergy travels as a localized packet and which preserves this localization intime. A soliton is a solitary wave which exhibits some strong form of stabilityso that it has a particle-like behavior. In this paper we study the dynamicsof solitons arising in the nonlinear Schroedinger equation (NSE): i ∂ψ∂t = − m ∆ ψ + 12 W ′ ε ( ψ ) + V ( x ) ψ. (1)A solution of our equation can be written as follows ψ ( t, x ) = Ψ ε ( t, x ) + ϕ ( t, x ) (2)where ϕ ( t, x ) can be considered as a wave and Ψ ε ( t, x ) is our soliton: a bumpof energy concentrated in a ball centered at the point q = q ε ( t ) with radius R ε → ε → ε occursin the equation as a parameter. The main purpose of this paper is to showthat for ε and ϕ (0 , x ) sufficiently small, our soliton behaves as a classicalparticle in a potential V ( x ). More exactly, we prove that the decomposition(2) holds for all times and the bump follows a dynamics which approachesthe dynamics of a pointwise particle moving under the action of the potential V (Theorem 1); in particular the position q ε ( t ) of the soliton approaches theposition of the particle uniformly on bounded time intervals (Corollary 2).The attention of the mathematical community on the dynamics of solitonof NSE began with the pioneering paper of Bronski and Jerrard [11]; thenFr¨ohlich, Gustafson, Jonsson, and Sigal faced this problem using a differentapproach [15]. In the last years, several others works appeared following thefirst approach ([24], [25], [29], [30], [31]) or the second one ([1], [2], [3], [4],[16], [22], [28]). 2n this paper, we have studied equation (1) which gives a different problemwith respect to the ones mentioned above. Actually, in equation (1), unlikelythe other papers, the parameter ε appears in the nonlinear term and it will bechosen in such a way that | ψ | approaches the delta-measure as ε → . Thusequation (1) describes the dynamics of a soliton when its support is small withrespect to the other relevant elements (namely V ( x ), the initial conditionsand its L -size). Also the method employed here is different from thoseof the paper quoted above and it exploits and develops some ideas of [9].Basically, we use the “strong” nonlinearity to obtain results on existence,shape, stability and dynamics of the soliton (Theorem 1). Finally, we noticethat this method applies to a large class of nonlinearities and we do not makeany assumption on the nondegenaracy or the uniqueness of the ground statesolution (see the discussion in Section 1.2). In the next we will use the following notations:Re( z ) , Im( z ) are the real and the imaginary part of zB ρ ( x ) = B ( x , ρ ) = { x ∈ R N : | x − x | ≤ ρ } B ( x , ρ ) C = R N r B ( x , ρ ) S σ = { u ∈ H : || u || L = σ } J ( u ) = Z (cid:18) |∇ u | + W ( u ) (cid:19) dxJ c = { u ∈ H : J ( u ) < c }| ∂ α V ( x ) | = sup i ,...,i α (cid:12)(cid:12)(cid:12)(cid:12) ∂ α V ( x ) ∂x i . . . ∂x i α (cid:12)(cid:12)(cid:12)(cid:12) where α ∈ N , i , . . . , i α ∈ { , . . . , N } I σ = inf u ∈ H , R u = σ J ( u ) = c | · | is the euclidean norm both of a vector or of a matrixΓ = (cid:26) U ∈ H , J ( U ) = inf || U || L =1 J ( u ) (cid:27) is the set of ground state solutions First, we focus on the “concentration” properties of a soliton solution of eq.(1) without the potential term V . We consider the following Cauchy problem3elative to the NSE: i ∂ψ∂t = − m ∆ ψ + 12 W ′ ε ( ψ ) (3) ψ (0 , x ) = U ε ( x − ¯ q ) e i ¯ p · x (4)where, with some abuse of notation, we have set W ε ( ψ ) = 1 ε N +2 W (cid:0) ε N/ | ψ | (cid:1) ; W ′ ε ( ψ ) = 1 ε N/ W ′ (cid:0) ε N/ | ψ | (cid:1) ψ | ψ | , (5)and W : R + → R is a real function which satisfies suitable assumptions (see(ii) below). U denotes a ground state solution of the equation − m ∆ U + 12 W ′ ( U ) = ω U (6)namely a function such that Z (cid:18) |∇ U | m + W ( U ) (cid:19) dx = c with c = inf || u || L =1 u ∈ H Z (cid:18) |∇ u | m + W ( u ) (cid:19) dx (7)It is well known that we can choose U radially symmetric and positive.Direct computations show that, by virtue of (5), the function U ε ( x ) = 1 ε N/ U (cid:16) xε (cid:17) . satisfies the equation − m ∆ U ε + 12 W ′ ε ( U ε ) = ω ε U ε . (8)where ω ε = ω ε . Moreover U ε ( x ) is a ground state solution of (8). In many cases, the groundstate solution U is unique up to translations and change of sign, but we donot need this assumption.Notice that the choice of W ε given by (5) implies that k U ε k L = 14or every ε > ψ q,ε ( t, x ) = U ε ( x − ¯ q − ¯ vt ) e i (¯ p · x − ωt ) where ¯ v = ¯ pm (9)with ω = ω ε + 12 m ¯ v Thus ψ q,ε ( t, x ) behaves as a particle of “radius” ε living in the point q = ¯ q + ¯ vt (10)Since (cid:13)(cid:13) ψ q,ε ( t, · ) (cid:13)(cid:13) L = 1 for every ε > , if ε → , we have that (cid:12)(cid:12) ψ q,ε ( t, x ) (cid:12)(cid:12) → δ ( x − ¯ q − ¯ vt ) in D ′ (cid:0) R N (cid:1) ∀ t ∈ R , where δ ( x − x ) denotes the Dirac measure concentrated in the point x . The energy E ε ( ψ ) of the configuration ψ is given by E ε ( ψ ) = Z (cid:20) m |∇ ψ | + W ε ( ψ ) (cid:21) dx, so the energy of ψ q,ε is E ε (cid:0) ψ q,ε (cid:1) = Z (cid:18) |∇ U ε | m + W ε ( U ε ) (cid:19) dx + 12 m ¯ v (11)Thus ψ q,ε ( t, x ) behaves as a particle of mass m : ¯ p can be interpreted asits momentum, m ¯ p = m ¯ v as its kinetic energy and Z |∇ U ε | m + W ε ( U ε ) dx = c ε as the internal energy; here c is a constant defined as follows c := Z |∇ U | m + W ( U ) dx The aim of this paper is to study the dynamics of the solitons in thepresence of a potential V ( x ) namely to investigate the problem i ∂ψ∂t = − m ∆ ψ + W ′ ε ( ψ ) + V ( x ) ψψ (0 , x ) = ψ ,ε ( x ) ( P )5here ψ ,ε satisfies the following assumptions ψ ,ε ( x ) = U ε ( x − ¯ q ) e i ¯ p · x + ϕ ,ε ( x ) , ϕ ,ε ∈ H ( R N ); (12) (cid:13)(cid:13) ψ ,ε (cid:13)(cid:13) L = 1 (13) E ε (cid:0) ψ ,ε (cid:1) ≤ c ε + M (14)with M > ε ; here E ε ( ψ ) denotes the energy in the presenceof the potential V : E ε ( ψ ) = Z (cid:20) m |∇ ψ | + W ε ( ψ ) + V ( x ) | ψ | (cid:21) dx. We make the following assumptions:(i) W : R + → R is a C function which satisfies the following assumptions: W (0) = W ′ (0) = W ′′ (0) = 0 ( W ) | W ′′ ( s ) | ≤ c | s | q − + c | s | p − for some 2 < q ≤ p < ∗ ( W ) W ( s ) ≥ − c | s | ν , c ≥ , < ν < N for s large ( W ) ∃ s ∈ R + such that W ( s ) < W )(ii) V : R N → R is a C -function with bounded derivatives which satisfiesthe following assumptions:0 ≤ V ( x ) ≤ V < ∞ ; ( V )The main result of this paper is the following theorem which describesthe shape and the dynamics of the soliton Ψ ε ( t, x ): Theorem 1
Assume (i) and (ii); then the solution of problem ( P ) has thefollowing form ψ ε ( t, x ) = Ψ ε ( t, x ) + ϕ ε ( t, x ) (15) where Ψ ε ( t, x ) is a function having support in a ball B R ε ( q ε ) , with radius R ε → and center q ε = q ε ( t ) . Moreover, k| Ψ ε ( t, x ) | − U ε ( x − q ε ( t )) k L → as ε → niformly in t, where U ε = ε N/ U (cid:0) xε (cid:1) and U is a ground state solution of (6).The dynamics is given by the following equations: ˙ q ε ( t ) = m ε ( t ) p ε ( t ) + K ε ( t )˙ p ε ( t ) = −∇ V ( q ε ( t )) + F ε ( q ε ) + H ε ( t ) (17) with initial data (cid:26) q ε (0) = ¯ q + o (1) p ε (0) = ¯ p + o (1) (18) where • (a) q ε ( t ) is the barycenter of the soliton and it has the following form: q ε ( t ) = Z x | Ψ ε | dx Z | Ψ ε | dx • (b) m ε ( t ) = m Z | Ψ ε ( t, x ) | dx = m + o (1) can be interpreted as the mass of the soliton, • (c) p ε ( t ) is the momentum of the soliton and it has the followingform: p ε ( t ) = Im Z ∇ Ψ ε ( t, x ) Ψ ε ( t, x ) dx ; • (d) K ε ( t ) and H ε ( t ) are errors due to the fact that the soliton is not apoint and sup t ∈ R ( | H ε ( t ) | + | K ε ( t ) | ) → as ε → . • (e) F ε ( q ε ) is the force due to the pressure of the wave ϕ ε on the solitonand F ε → in the space of distributions, more exactly we have that ∀ τ , τ , (cid:12)(cid:12)(cid:12)(cid:12)Z τ τ F ε ( q ε ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( ε ) (1 + | τ − τ | ) where c ( ε ) → as ε → . orollary 2 Let q and p be the solution of the following Cauchy problem: ˙ q ( t ) = m p ( t )˙ p ( t ) = −∇ V ( q ( t )) (19) with initial data (cid:26) q (0) = q ε (0) p (0) = p ε (0) (20) where q ε ( t ) and p ε ( t ) are as in Th. 1. Then, as ε → q ε ( t ) , p ε ( t )) → ( q ( t ) , p ( t )) uniformly on compact sets. Let us discuss the set of our assumptions.
Remark 3
The conditions ( W ) and ( V ) are assumed for simplicity; in factthey can be weakened as follows W (0) = W ′ (0) = 0 , W ′′ (0) = E and E ≤ V ( x ) ≤ V ∞ < + ∞ . In fact, in the general case, the solution of the Schroedinger equation is mod-ified only by a phase factor.
Remark 4
In [6] the authors prove that if (ii) holds equation (3) admitsorbitally stable solitary waves having the form (4). In particular the authorsshow that, under assumptions ( W ), ( W ), ( W ) and ( W ), for any σ thereexists a minimizer U ( x ) = U σ ( x ) of the functional J ( u ) = Z (cid:18) |∇ u | + W ( u ) (cid:19) dx on the manifold S σ := { u ∈ H , || u || L = σ } . Such a minimizer satisfieseq.(6) where ω is a Lagrange multiplier. Remark 5
By our assumptions, the problem ( P ) has a unique solution ψ ∈ C ( R , H ( R N )) ∩ C ( R , L ( R N )) . (21)8 et us recall a result on the global existence of solutions of the Cauchy problem( P ) (see [13, 18, 23]). Assume ( W ), ( W ) and ( W ) for W . Let D ( A ) (resp. D ( A / ) ) denote the domain of the self-adjoint operator A (resp. A / ) where A = − ∆ + V : L ( R N ) → L ( R N ) . If V ≥ , V ∈ C and | ∂ V | ∈ L ∞ and the initial data ψ (0 , x ) ∈ D ( A / ) then there exists the global solution ψ of ( P ) and ψ ( t, x ) ∈ C (cid:0) R , D ( A / ) (cid:1) ∩ C ( R , H − ( R N )) . Furthermore, if ψ (0 , x ) ∈ D ( A ) then ψ ( t, x ) ∈ C ( R , D ( A )) ∩ C ( R , L ( R N )) . In this case, since D ( A ) ⊂ H ( R N ) , (21) is satisfied. In this section, after having stated the main dynamical properties of our sys-tem (Subsection 2.1), we define explicitely the splitting (2) and the equationswhich rules the dynamics of the soliton. In all this section, it is not necessaryto assume ε to be small. Equation ( P ) is the Euler-Lagrange equation relative to the Lagrangian den-sity L = Re( i∂ t ψ ¯ ψ ) − m |∇ ψ | − W ε ( ψ ) − V ( x ) | ψ | . (22)Sometimes it is useful to write ψ in polar form ψ ( t, x ) = u ( t, x ) e iS ( t,x ) . (23)Thus the state of the system ψ is uniquely defined by the couple of variables( u, S ) ∈ R + × [ R / (2 π Z )]. Using these variables, the action S = R L dxdt takes the form S ( u, S ) = − Z (cid:20) m |∇ u | + W ε ( u ) + (cid:18) ∂ t S + 12 m |∇ S | + V ( x ) (cid:19) u (cid:21) dxdt (24)and equation ( P ) splits in the two following equations:9 m ∆ u + W ε ( u ) + (cid:18) ∂ t S + 12 m |∇ S | + V ( x ) (cid:19) u = 0 (25) ∂ t (cid:0) u (cid:1) + ∇ · (cid:18) u ∇ Sm (cid:19) = 0 (26)Noether’s theorem states that any invariance for a one-parameter groupof the Lagrangian implies the existence of an integral of motion (see e.g. [17]or [7]). They are derived by a continuity equation.Now we describe the first integrals which will be relevant for this paper,namely the energy, the “hylenic charge” and the momentum. Energy.
The energy, by definition, is the quantity which is preserved bythe time invariance of the Lagrangian; it has the following form E ε ( ψ ) = Z (cid:20) m |∇ ψ | + W ε ( ψ ) + V ( x ) | ψ | (cid:21) dx. (27)Using (23) we get: E ε ( ψ ) = Z (cid:18) m |∇ u | + W ε ( u ) (cid:19) dx + Z (cid:18) m |∇ S | + V ( x ) (cid:19) u dx (28)Thus the energy has two components: the internal energy (which, sometimes,is also called binding energy ) J ε ( u ) = Z (cid:18) m |∇ u | + W ε ( u ) (cid:19) dx (29)and the dynamical energy G ( u, S ) = Z (cid:18) m |∇ S | + V ( x ) (cid:19) u dx (30)which is composed by the kinetic energy m R |∇ S | u dx and the potentialenergy R V ( x ) u dx . By our assumptions, the internal energy is boundedfrom below and the dynamical energy is positive. Hylenic charge.
Following [7] the hylenic charge , is defined as thequantity which is preserved by the invariance of the Lagrangian with respectto the action ψ e iθ ψ. L norm, namely: C ( ψ ) = Z | ψ | dx = Z u dx. Momentum.
The momentum is constant in time if the Lagrangian isspace-translation invariant; this happens when V is a constant. In generalwe have the equation ∂ t (cid:0) u ∇ S (cid:1) = − u ∇ V + ∇ · T (31)where T is the stress tensor whose components are given by T jk = Re (cid:0) ∂ x j ψ∂ x k ¯ ψ (cid:1) − δ jk (cid:20) m ∆ | ψ | − W ′ ε ( ψ ) ¯ ψ + W ε ( ψ ) (cid:21) (32)= (cid:20) ∂ x j u ∂ x k u + (cid:0) ∂ x j S ∂ x k S (cid:1) u − m δ jk ∆ (cid:0) u (cid:1)(cid:21) + δ jk (cid:18) W ′ ε ( u ) u − W ε ( u ) (cid:19) (33) Proof:
It is well known that the stress tensor has the following form (seee.g. [7] or [8]) T jk = − Re ∂ L ∂ (cid:0) ∂ x j ψ (cid:1) · ∂ x k ¯ ψ ! + δ jk L Using equation ( P ) to eliminate i∂ t ψ, we get L = Re( i∂ t ψ ¯ ψ ) − m |∇ ψ | − W ε ( ψ ) − V ( x ) | ψ | = Re (cid:20)(cid:18) − m ∆ ψ + 12 W ′ ε ( ψ ) + V ( x ) ψ (cid:19) ¯ ψ (cid:21) − m |∇ ψ | − W ε ( ψ ) − V ( x ) | ψ | = Re (cid:18) − m ∆ ψ ¯ ψ (cid:19) − m |∇ ψ | + 12 W ′ ε ( ψ ) ¯ ψ − W ε ( ψ ) . Moreover, we have that∆ ψ ¯ ψ = ∆ ψ ψ + ∆ ψ ψ = ∇ · ( ψ ∇ ψ ) + ∇ · ( ψ ∇ ψ ) − |∇ ψ | = 12 ∆ | ψ | + 12 ∆ | ψ | − |∇ ψ | = 12 ∆ | ψ | − |∇ ψ | L = − m ∆ | ψ | + 12 W ′ ε ( ψ ) ¯ ψ − W ε ( ψ )We recall that, if z = a + ib is a complex number, by definition, ∂L∂z = ∂L∂a + i ∂L∂b . So Re ∂ L ∂ (cid:0) ∂ x j ψ (cid:1) · ∂ x k ¯ ψ ! = − Re (cid:0) ∂ x j ψ∂ x k ¯ ψ (cid:1) and the conclusion follows from direct computation. (cid:3) If V = const , eq. (31) is a continuity equation and the momentum P ( ψ ) = Z u ∇ S dx = Z Im (cid:0) ¯ ψ ∇ ψ (cid:1) dx (34)is constant in time. Notice that, by equation (26), the density of momentum u ∇ S is nothing else than the flow of hylenic charge.Let us consider the soliton (9); in this case we have E ε (cid:0) ψ q,ε (cid:1) = c ε + 12 m ¯ p where c is defined by (7), C ( ψ ε ) = 1and P ( ψ ε ) = ¯ p. Now, let us see the rescaling properties of the internal energy and the L norm of a function u ( x ) having the form u ( x ) := 1 ε N/ v (cid:16) xε (cid:17) . We have || u || L = 1 ε N Z v (cid:16) xε (cid:17) dx = Z v ( ξ ) dξ = || v || L . and 12 ε ( u ) = Z (cid:20) m |∇ u | + W ε ( u ) (cid:21) dx = Z (cid:20) m |∇ u | + 1 ε N +2 W ( ε N/ u ) (cid:21) dx = Z (cid:20) m ε N (cid:12)(cid:12)(cid:12) ∇ x v (cid:16) xε (cid:17)(cid:12)(cid:12)(cid:12) + 1 ε N +2 W (cid:16) v (cid:16) xε (cid:17)(cid:17)(cid:21) dx = Z (cid:20) m ε N +2 |∇ ξ v ( ξ ) | + 1 ε N +2 W ( v ( ξ )) (cid:21) ε N dξ = 1 ε Z m |∇ ξ v ( ξ ) | + W ( v ( ξ )) dξ = 1 ε J ( u ) . In this section we want describe a method to split a solution of eq. ( P ) in awave and a soliton as in (2).If our solution has the following form ψ ε ( t, x ) = u ε e iS ε = [ U ε ( x − ξ ( t )) + w ε ( t, x )] e iS ε ( t,x ) where w is sufficiently small, then a possible choice is to identify the solitonwith U ( x − ξ ( t )) e iS ( t,x ) and the wave with w ε ( t, x ) e iS ( t,x ) . However, we wantto give a definition which localize the soliton, namely to assume the functionΨ ε ( t, x ) to have compact support in space.Roughly speaking, the soliton can be defined as the part of the field ψ ε where some density function ρ ε ( t, x ) is sufficiently large (e.g, after a suitablenormalization, ρ ε ( t, x ) ≥ ρ ε ( t, x ) explicitely. We just require that ρ ε ( t, x ) satisfies the follwing assumptions: • ρ ε ∈ C ( R N +1 ) and ρ ε ( t, x ) → | x | → ∞• ρ ε satisfies the continuity equation ∂ t ρ ε + ∇ · J ρ ε = 0 (35)for some J ρ ε ∈ C ( R N +1 )In order to fix the ideas you may think of ρ ε ( t, x ) as a smooth approxi-mation of u ε ( t, x ) . An explicit definition of ρ ε ( t, x ) is given in Section 3.2.However, in other problems, it might be more useful to make different choices13f it such as the energy density. We have postponed the choice of ρ ε sincethe results of this section are independent of this choice.Next we set χ ε ( t, x ) = p ϕ ( ρ ε ( t, x ))where ϕ ( s ) = s ≤ s − ≤ s ≤
21 if s ≥ χ ε ( t, x ) = 1 where ρ ε ( t, x ) ≥ χ ε ( t, x ) = 0 where ρ ε ( t, x ) ≤ χ ε ( t, x ) as a sort of approximation of thecharacteristic function of the region occupied by the soliton. Finally, we setΨ ε ( t, x ) = ψ ε ( t, x ) χ ε (36) ϕ ε ( t, x ) = ψ ε ( t, x ) [1 − χ ε ] (37)Ψ ε ( t, x ) is the soliton and ϕ ε ( t, x ) is the wave; the regionΣ ε,t = (cid:8) ( t, x ) ∈ R N +1 | < ρ ( t, x ) < (cid:9) (38)= (cid:8) ( t, x ) ∈ R N +1 | < χ ε ( t, x ) < (cid:9) (39)is the region where the soliton and the wave interact with each other; we willrefer to it as the halo of the soliton. Definition 6
We define the following quantities relative to the soliton: • the barycenter : q ε ( t ) = R x | Ψ ε | dx R | Ψ ε | dx • the momentum : p ε ( t ) = Z ∇ S ε | Ψ ε | dx • the mass: m ε ( t ) = m Z | Ψ ε | dx. Remark 7
Notice that the mass of the soliton m ε ( t ) depends on t. The globalmass is constant (namely m ) but it is shared between the soliton and the wavewhose mass is R u ε [1 − χ ε ] dx q ε ( t ) and p ε ( t ) and theirderivatives. Theorem 8
The following equations hold ˙ q ε = p ε m ε + 1 m ε Z Σ ε,t ( x − q ε ) (cid:2) u ε ∇ S ε · ∇ ρ ε − ∇ · J ρ ε (cid:3) dx (40)˙ p ε = − Z ∇ V | Ψ ε | dx − Z Σ ε,t (cid:2) T · ∇ ρ ε + u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1)(cid:3) dx. (41) Remark 9
The term R Σ ε,t T · ∇ ρ ε dx represents the pressure of the wave onthe soliton; if ε → and ∂ Σ ε,t is sufficiently regular then Z Σ ε,t T · ∇ ρ ε dx = Z σ ε T · n dσ where σ ε = { x | ρ ε ( x ) = 1 } and n is its outer normal. Proof of Th. 8.
We calculate the first derivative of the barycenter.˙ q ε ( t ) = ddt R x | Ψ ε | dx R | Ψ ε | dx ! = ddt R x | Ψ ε | dx R | Ψ ε | dx − (cid:0)R x | Ψ ε | dx (cid:1) (cid:0) ddt R | Ψ ε | dx (cid:1)(cid:0)R | Ψ ε | dx (cid:1) = ddt R x | Ψ ε | dx R | Ψ ε | dx − q ε ( t ) ddt R | Ψ ε | dx R | Ψ ε | dx = R ( x − q ε ( t )) ddt ( | Ψ ε | ) dx R | Ψ ε | dx . We have ∇ χ = ∇ ϕ ( ρ ε ( t, x )) = ϕ ′ ( ρ ε ( t, x )) ∇ ρ ε = I Σ ε,t ∇ ρ ε and ddt χ = ddt ϕ ( ρ ε ( t, x )) = ϕ ′ ( ρ ε ( t, x )) ∂ t ρ ε = I Σ ε,t ∂ t ρ ε = − I Σ ε,t ∇ · J ρ ε where I Σ ε,t is the characteristic function of Σ ε,t . So, we have 15 q ε ( t ) = R ( x − q ε ( t )) ddt ( χ u ε ) dx R | Ψ ε | dx = R ( x − q ε ( t )) (cid:0) χ ddt u ε + u ε ddt χ (cid:1) dx R | Ψ ε | dx (42)= R R N ( x − q ε ( t )) χ ddt u ε dx − R Σ ε,t ( x − q ε ( t )) ∇ · J ρ ε dx R R N | Ψ ε | dx . For the first term we use the continuity equation (26). We have Z ( x − q ε ( t )) χ ddt u ε dx = Z ( x − q ε ( t )) ∇ · (cid:18) u ε ∇ S ε m (cid:19) χ dx = 1 m Z (cid:0) u ε ∇ S ε (cid:1) χ dx + 1 m Z ( x − q ε ( t )) u ε ∇ S ε · ∇ χ dx = p ε ( t ) m + 1 m Z Σ ε,t ( x − q ε ( t )) u ε ∇ S ε · ∇ ρ ε dx. Concluding, we get the first equation of motion:˙ q ε ( t ) = p ε ( t ) m ε + R Σ ε,t ( x − q ε ( t )) (cid:2) u ε ∇ S ε · ∇ ρ ε − ∇ · J ρ ε (cid:3) dxm ε Next, we will get the second one. We have that˙ p ε = Z (cid:18) ∂∂t u ε ∇ S ε (cid:19) χ dx + Z u ε ∇ S ε ∂∂t χ dx. (43)Now, using (31) we have that Z (cid:18) ∂∂t u ε ∇ S ε (cid:19) χ dx = − Z ∇ V | Ψ ε | dx + Z ∇ · T χ dx = − Z ∇ V | Ψ ε | dx − Z T · ∇ χ dx = − Z ∇ V | Ψ ε | dx − Z Σ ε,t T · ∇ ρ ε dx. The second term of eq. (43) takes the form: Z u ε ∇ S ε ∂∂t χ dx = − Z Σ ε,t u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1) dx. It is possible to give a “pictorial” interpretation to equations (40) and(41). We may assume that u ε represents the density of a fluid; so the solitonis a bump of fluid particles which stick together and the halo Σ ε,t can beregarded as the interface where the soliton and the wave might exchangeparticles, momentum and energy.Hence, • m ε ( t ) is the mass of the soliton • ∇ S ε m is the velocity of the fluid particles and ∇ S ε is their momentumSo each term of the equations (40) and (41) have the following interpre-tation • p ε ( t ) m ε is the average velocity of each particle; in fact p ε ( t ) m ε = R ∇ S ε | Ψ ε | dxm ε = R ∇ S ε m | Ψ ε | dx R | Ψ ε | dx • the “halo term” m ε R Σ ε,t ( x − q ε ) (cid:2) u ε ∇ S ε · ∇ ρ ε − ∇ · J ρ ε (cid:3) dx describes thechange of the average velocity of the soliton due to the exchange of fluidparticles • the term − R ∇ V | Ψ ε | dx describes the volume force acting on the soli-ton • the term − R Σ ε,t T · ∇ ρ ε dx describes the surface force exerted by thewave on the soliton • the term − R Σ ε,t u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1) dx describes the change of the momen-tum of the soliton due to the exchange of fluid particles with the wave. In this section, we analyze the dynamics of the soliton as ε → .1 Analysis of the concentration point of the soliton If ψ ε ( t, x ) is a solution of the problem ( P ), we say that ˆ q ε ( t ) is the concen-tration point of ψ ε ( t, x ) if it minimizes the following quantity f ( q ) = k| ψ ε ( t, x ) | − U ε ( x − q ) k L . (44)It is easy to see that f ( q ) has a minimizer; of course, it might happenthat it is not unique; in this case we denote by ˆ q ε ( t ) one of the minimizers of f at the time t. Basically ˆ q ε ( t ) is a good candidate for the position of our soliton, but itcannot satisfy an equation of type (17) since in general it is not uniquelydefined and a fortiori is not differentiable. ˆ q ε ( t ) could be uniquely defined ifwe make assumptions on the non degeneracy of the ground state, but we donot like to make such assumptions since they are very hard to be verified andin general they do not hold. Actually the position of the soliton is supposedto be q ε ( t ) as in Def. 6. However, as we will see, ˆ q ε is useful to recover someestimates on q ε . So, in this subsection we will analyze some properties ofˆ q ε ( t ) . We start with a variant of a result contained in [6].
Lemma 10
Given u ∈ H , we define (if it exists) ˆ q ∈ R N to be a minimizerof the function q
7→ || U ( x − q ) − u ( x ) || L . For any η there exists a δ ( η ) such that, if u ∈ J c + δ ( η ) ∩ S (see section 1.1), ˆ q exists and it holds || U ( x − ˆ q ) − u || H ≤ η (45) Z R N r B (ˆ q, ˆ R η ) u dx ≤ η (46) where ˆ R η = − C log( η ) and U ∈ Γ . Proof : The proof of (45) can be found in [6]. If U ∈ Γ, again by [6] weknow that, for R sufficiently large, Z | x | >R U ( x ) dx < Z | x | >R C e − C | x | . Thus Z | x | >R U ( x ) dx = C Z ∞ R ρ N − e − C ρ dρ = C R N e − C R ≤ e − C R C i ’s are suitable positive constants. We remark that R does notdepend on U .Now, it is sufficient to take ˆ R η > − C log( η ) and by (45) we obtain (46). (cid:3) We define the set of admissible initial data as follows: B ε,M = n ψ ( x ) = U ε ( x − q ) e ip · x + ϕ ( x ) : E ε ( ψ ) ≤ c ε + M and k ψ k L = 1 o Lemma 11
For every η > , there exists ε = ε ( η ) > such that Z R N r B (ˆ q ε ,ε ˆ R η ) | ψ ε ( t, x ) | dx < η (47) where ψ ε ( t, x ) is a solution of problem ( P ), with initial data in B ε,M and ˆ q ε is the concentration point of ψ ε . Proof.
By the conservation law, the energy E ε ( ψ ε ( t, x )) is constant withrespect to t . Then we have, by hypothesis on the initial datum E ε ( ψ ε ( t, x )) = E ε ( ψ ε (0 , x )) ≤ c ε + M. Thus J ε ( ψ ε ( t, x )) = E ε ( ψ ( t, x )) − G ( ψ ( t, x ))= E ε ( ψ ε ( t, x )) − Z R N (cid:20) |∇ S ε ( t, x ) | m + V ( x ) (cid:21) u ε ( t, x ) dx ≤ E ε ( ψ ε ( t, x )) ≤ c ε + M (48)because V ≥
0. By rescaling the inequality (48), and setting y = x/ε we get J ( | ε N/ ψ ε ( t, εy ) | ) ≤ c + ε M (49)We choose ε small such that ε M ≤ δ ( η ) . Then ε N/ ψ ε ( t, εy ) ∈ J c + δ ( η ) ∩ S , and so applying Lemma 10. Z R N r B (ˆ q, ˆ R η ) ε N | ψ ε ( t, εy ) | dy < η (50)Now, making the change of variable x = εy, we obtain the desired result. (cid:3) emma 12 If ψ ε ( t, x ) is a solution of problem ( P ), with initial data in B ε,M and ε sufficiently small, then Z R N r B ( ˆ q ε , √ ε ) | ψ ε ( t, x ) | dx = η ( ε ) (51) where η ( ε ) → as ε → Proof.
First we prove that for every η > , there exists ε ( η ) > ψ ε (0 , x ) ∈ B ε ( η ) ,M , we have Z R N r B (cid:16) ˆ q ε , √ ε ( η ) (cid:17) | ψ ε ( t, x ) | dx < η. Arguing as in the proof of Lemma 11, if ε ( η ) ≤ min (cid:20)q δ ( η ) M , R η (cid:21) , we get(47). At this point, since ε ( η ) ≤ R η we have that ε ( η ) ˆ R η ≤ p ε ( η ).Now set ε ( η ) = min η ≤ ζ ε ( ζ ) . Clearly, ε ( η ) is a non-increasing function (which might be discontinuous)and ε ( η ) → η →
0. Then it has a “pseudoinverse” function η ( ε ) namelya function which is the inverse in the monotonicity points, which is discon-tinuous where ε ( η ) is constant and constant where ε ( η ) is discontinuous.Moreover η ( ε ) as ε → (cid:3) ρ ε First of all we notice that, in Lemma 12, it is not restrictive to assume that η = η ( ε ) ≥ ε. (52)Now we set ρ ε ( t, x ) = a ( x ) ∗ u ( t, x ) where, a ε ( s ) ∈ C ∞ , a ε ( s ) = | s | ≤ η (cid:16) − η (cid:17) | s | ≥ η (cid:16) η (cid:17) and |∇ a ε ( s ) | ≤ η − . (53)20 emma 13 Take ψ ε a solution of ( P ) with initial data in B ε,M .If | x − ˆ q ε ( t ) | ≤ η (cid:16) − η (cid:17) then ρ ε ( t, x ) ≥ − η ) if | x − ˆ q ε ( t ) | ≥ η (cid:16) η (cid:17) then ρ ε ( t, x ) ≤ η where η = η ( ε ) as in Lemma 12. In particular we have that Σ ε,t ⊂ B (cid:16) ˆ q ε ( t ) , η (cid:16) η (cid:17)(cid:17) \ B (cid:16) ˆ q ε ( t ) , η (cid:16) − η (cid:17)(cid:17) (54) where Σ ε,t is defined by (38). Proof. If | x − ˆ q ε | ≤ η (cid:16) − η (cid:17) , then | x − ˆ q ε | + √ ε ≤ η (cid:16) − η (cid:17) + √ η ≤ η (cid:16) − η (cid:17) and hence B (ˆ q ε , √ ε ) ⊂ B (cid:16) x, η (cid:16) − η (cid:17)(cid:17) . Then, by using Lemma 12, ρ ε ( t, x ) = Z a ε ( y − x ) u ε ( t, y ) dy ≥ Z B ( x, η / − η / ) u ε ( t, y ) dy ≥ Z B (ˆ q ε , ε / ) u ε ( t, y ) dy ≥ − η ) . If | x − ˆ q ε ( t ) | ≥ η (cid:16) η (cid:17) , | x − ˆ q ε | − √ ε ≥ η (cid:16) η (cid:17) − √ η ≥ η (cid:16) η (cid:17) and so B (cid:16) x, η (cid:16) η (cid:17)(cid:17) ⊂ R N r B (ˆ q ε , √ ε ) . Then, using again Lemma 12, ρ ε ( t, x ) = Z a ε ( y − x ) u ε ( t, y ) dy ≤ Z B (cid:16) x, η (cid:16) η (cid:17)(cid:17) u ε ( t, y ) dy ≤ Z R N r B (ˆ q ε , √ ε ) u ε ( t, y ) dy ≤ η (cid:3) ρ ε = a ε ∗ u ε ∈ C ( R N +1 ) and, by (26), it satisfies the continuityequation (35) with J ρ ε = a ε ∗ (cid:0) u ε ∇ S ε (cid:1) . (55)Therefore, the results of Section 2 hold. In particular, we have that thesupport of Ψ ε ( t, x ) is contained in B (cid:16) ˆ q ε , η (cid:16) η (cid:17)(cid:17) when η is suffi-ciently small (namely η < / ε → Theorem 14
The following equations hold ˙ q ε ( t ) = p ε ( t ) m ε ( t ) + K ε ( t ) (56)˙ p ε = −∇ V ( q ε ( t )) + F ε ( q ε ) + H ε ( t ) (57) where sup t ∈ R ( | H ε ( t ) | + | K ε ( t ) | ) → as ε → and F ε ( q ε ) = − Z Σ ε,t T · ∇ ρ ε dx. (59) Moreover we have that ∀ τ , τ , (cid:12)(cid:12)(cid:12)(cid:12)Z τ τ F ε ( q ε ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( ε ) (1 + | τ − τ | ) (60) where c ( ε ) → as ε → . Proof.
We set K ε ( t ) = 1 m ε Z Σ ε,t ( x − q ε ) (cid:2) u ε ∇ S ε · ∇ ρ ε − ∇ · J ρ ε (cid:3) dx,H ,ε ( t ) = Z Σ ε,t u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1) dx,H ,ε ( t ) = ∇ V ( q ε ( t )) − Z ∇ V ( x ) | Ψ ε | dx,H ε ( t ) = H ,ε ( t ) + H ,ε ( t ) , K ε . We have thatsup x ∈ Σ ε,t | x − q ε | ≤ (cid:0) η / + 2 η / (cid:1) ≤ η / (61)since q ε ( t ) , x ∈ B (ˆ q ε , η / + 2 η / ) . Also, by (53) and well known properties on convolutions,sup x ∈ Σ ε,t |∇ ρ ε | ≤ sup x ∈ R N (cid:12)(cid:12) ∇ a ε ( x ) ∗ u ε ( t, x ) (cid:12)(cid:12) (62) ≤ k∇ a ε k L ∞ · k u ε k L ≤ η / If ψ ε (0 , x ) ∈ B ε,M , by (30), we have G ( ψ ) = E ε ( ψ ) − J ε ( ψ ) ≤ c ε + M − c ε = M ; (63)so, by Lemma 12, Z R N r B (ˆ q ε , √ ε ) u |∇ S ε | ≤ (cid:18)Z R N r B (ˆ q ε , √ ε ) u ε (cid:19) (cid:18)Z R N r B (ˆ q ε , √ ε ) u ε |∇ S ε | (cid:19) ≤ η [2 mG ( ψ )] ≤ const.η (64)and in particular Z Σ ε,t u |∇ S ε | dx ≤ η [2 mG ( ψ )] ≤ const.η . (65)Finally, by (55)sup x ∈ Σ ε,t (cid:12)(cid:12) ∇ · J ρ ε (cid:12)(cid:12) ≤ sup x ∈ R N | ( ∇ · a ε ) ∗ (cid:0) u ε ∇ S ε (cid:1) |≤ k∇ · a ε k L ∞ · Z R N u ε |∇ S ε |≤ η / (cid:18)Z R N u ε (cid:19) · (cid:18)Z R N u ε |∇ S ε | (cid:19) ≤ [2 mG ( ψ )] η / = const.η − . (66)By (54), 23 Σ ε,t | ≤ (cid:12)(cid:12)(cid:12) B (cid:16) ˆ q ε ( t ) , η (cid:16) η (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) B (cid:16) ˆ q ε ( t ) , η (cid:16) − η (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) = ω N h η (cid:16) η (cid:17)i N − ω N h η (cid:16) − η (cid:17)i N = ω N η N (cid:20)(cid:16) η (cid:17) N − (cid:16) − η (cid:17) N (cid:21) ≤ ω N η N · N η ≤ const.η N +18 . (67)So, by (61),....(67) | K ε ( t ) | ≤ Z Σ ε,t (cid:12)(cid:12) ( x − q ε ) (cid:2) u ε ∇ S ε · ∇ ρ ε − ∇ · J ρ ε (cid:3)(cid:12)(cid:12) dx ≤ sup x ∈ Σ ε,t | x − q ε | · "Z Σ ε,t (cid:12)(cid:12) u ε ∇ S ε · ∇ ρ ε (cid:12)(cid:12) dx + Z Σ ε,t (cid:12)(cid:12)(cid:0) ∇ · J ρ ε (cid:1)(cid:12)(cid:12) dx ≤ sup x ∈ Σ ε,t | x − q ε | · " sup x ∈ Σ ε,t |∇ ρ ε | · Z Σ ε,t (cid:12)(cid:12) u ε ∇ S ε (cid:12)(cid:12) + sup x ∈ Σ ε,t (cid:12)(cid:12) ∇ · J ρ ε (cid:12)(cid:12) · Z Σ ε,t dx ≤ η / (cid:2) const.η − / · η / + const.η − / · | Σ ε,t | (cid:3) ≤ const.η / h η − / · η / + η − / · η N +18 i ≤ const. η / . Then, by Lemma 12, | K ε ( t ) | → t .Now, let us estimate | H ,ε ( t ) | ; by (66) and (65) we have | H ,ε ( t ) | ≤ Z Σ ε,t (cid:12)(cid:12) u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1)(cid:12)(cid:12) dx ≤ sup x ∈ Σ ε,t (cid:12)(cid:12) ∇ · J ρ ε (cid:12)(cid:12) · Z Σ ε,t (cid:12)(cid:12) u ε ∇ S ε (cid:12)(cid:12) ≤ const. η / · η = const.η / By the above estimate, | H ,ε ( t ) | → . (69)We recall that Z | Ψ ε | = 1 − o (1)24hen ε →
0, and that suppΨ ε ⊂ B (ˆ q ε , η / + 2 η / ). We have ∇ V ( q ε ( t )) = (1 + o (1)) Z ∇ V ( q ε ( t )) | Ψ ε | and so | H ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ∇ V ( x ) | Ψ ε | dx − ∇ V ( q ε ( t )) (cid:12)(cid:12)(cid:12)(cid:12) = Z |∇ V ( x ) − ∇ V ( q ε ) | | Ψ ε | dx + o (1) Z ∇ V ( q ε ( t )) | Ψ ε | ≤ || V ′′ || C ( R N ) Z | x − q ε | | Ψ ε | dx + o (1) Z ∇ V ( q ε ( t )) | Ψ ε | ≤ o (1) (cid:0) || V ′′ || C ( R N ) + ||∇ V || C ( R N ) (cid:1) = o (1)for all t . By the above equation, (68) and (69), the (58) follows.Let P = P ( ψ ε ) be defined by (34). By the definitions of p ε , and (64), forevery t ∈ R , we have that | p ε ( t ) − P ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ∇ S (cid:0) | Ψ ε | − u ε (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R N r B (ˆ q ε , √ ε ) |∇ S | u ε dx = c ( ε ) (70)where c ( ε ) → ε → . By (31) ˙P = Z (cid:0) − u ε ∇ V + ∇ · T (cid:1) dx and since T ∈ L ( R N ) , ˙P = − R u ε ∇ V dx.
So, by (41) and (59)˙ p ε − ˙P = Z ∇ V (cid:0) u ε − | Ψ ε | (cid:1) dx − Z Σ ε,t (cid:2) T · ∇ ρ ε + u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1)(cid:3) dx = Z ∇ V (cid:0) u ε − | Ψ ε | (cid:1) dx + F ε ( q ε ) − Z Σ ε,t u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1) dx. Then, by (69) and Lemma 12, (cid:12)(cid:12)(cid:12) F ε ( q ε ) − (cid:16) ˙ p ε − ˙P (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Σ ε,t u ε ∇ S ε (cid:0) ∇ · J ρ ε (cid:1) dx − Z ∇ V (cid:0) u ε − | Ψ ε | (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ o (1) + k∇ V k L ∞ Z (cid:12)(cid:12) u ε − | Ψ ε | (cid:12)(cid:12) ≤ o (1) + k∇ V k L ∞ Z R N r B (ˆ q ε , √ ε ) u ε dx = c ( ε )25here c ( ε ) → ε → . Finally by (70), ∀ τ , τ , (cid:12)(cid:12)(cid:12)(cid:12)Z τ τ F ε ( q ε ) dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z τ τ (cid:16) ˙ p ε − ˙P (cid:17) dt (cid:12)(cid:12)(cid:12)(cid:12) + c ( ε ) ( τ − τ ) ≤ | p ε ( τ ) − P ( τ ) | + | p ε ( τ ) − P ( τ ) | + c ( ε ) ( τ − τ ) ≤ c ( ε ) + c ( ε ) ( τ − τ ) ≤ c ( ε ) (1 + | τ − τ | )with a suitable choice of c ( ε ) . (cid:3) Collecting the previous results, we get our main theorem and Cor. 2:
Proof of Th. 1
By the def. (36),(37), Lemma 12 and Th. 14, Theorem1 holds with R ε = η (cid:16) η (cid:17) . (cid:3) Proof of Cor. 2.
We rewrite (17), (19) and (20) in integral form andwe get ( q ε ( t ) = q ε (0) + R t p ε ( s ) m ε ( s ) ds + R t K ε ( s ) dsp ε ( t ) = p ε (0) − R t ∇ V ( q ε ( s )) ds + R t [ F ε ( q ε ) + H ε ( s )] ds (71) (cid:26) q ( t ) = q ε (0) + R t p ( s ) m ds p ( t ) = p ε (0) − R t ∇ V ( q ( s )) ds (72)and hence, for any | t | ≤ T | q ε ( t ) − q ( t ) | ≤ Z t (cid:12)(cid:12)(cid:12)(cid:12) p ε ( s ) m ε ( s ) − p ( s ) m (cid:12)(cid:12)(cid:12)(cid:12) ds + Z t | K ε ( s ) | ds ≤ L Z t | p ε ( s ) − p ( s ) | ds + α ( ε )where, by (58), α ( ε ) → ε → | p ε ( t ) − p ( t ) | ≤ Z t |∇ V ( q ε ( s )) − ∇ V ( q ( s )) | ds + (cid:12)(cid:12)(cid:12)(cid:12)Z t F ε ( q ε ) ds (cid:12)(cid:12)(cid:12)(cid:12) + Z t | H ε ( s ) | ds ≤ L Z t | q ε ( s ) − q ( s ) | ds + α ( ε )where, by (58), α ( ε ) → ε → . Then, setting z ε ( t ) = | q ε ( t ) − q ( t ) | + | p ε ( t ) − p ( t ) | , we have z ε ( t ) ≤ L Z t z ε ( s ) ds + α ( ε )26ith a suitable choice of L and α ( ε ) . Now, by the Gronwall inequality, wehave z ε ( t ) ≤ α ( ε ) e Lt and from here, we get the conclusion. (cid:3) References [1] W.K. Abou Salem,
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