Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation
aa r X i v : . [ m a t h . A P ] D ec NEWTON’S LAW FOR A TRAJECTORY OF CONCENTRATIONOF SOLUTIONS TO NONLINEAR SCHRODINGER EQUATION
ANATOLI BABIN AND ALEXANDER FIGOTIN
Dedicated to the memory of M.I.Vishik
Abstract.
One of important problems in mathematical physics concerns deriva-tion of point dynamics from field equations. The most common approach tothis problem is based on WKB method. Here we describe a different methodbased on the concept of trajectory of concentration. When we applied thismethod to nonlinear Klein-Gordon equation, we derived relativistic Newton’slaw and Einstein’s formula for inertial mass. Here we apply the same approachto nonlinear Schrodinger equation and derive non-relativistic Newton’s law forthe trajectory of concentration. Introduction
A. Einstein remarks in his letter to Ernst Cassirer of March 16, 1937, [50, pp.393-394]: ”One must always bear in mind that up to now we know absolutely noth-ing about the laws of motion of material points from the standpoint of ”classicalfield theory.” For the mastery of this problem, however, no special physical hy-pothesis is needed, but ”only” the solution of certain mathematical problems”. Inthis paper we treat this circle of problems. Namely, we derive here point dynamicsgoverned by Newton’s law from wave dynamics using concepts and methods of theclassical field theory. That continues our study of a neoclassical model of electriccharges which is designed to provide an accurate description of charge interactionwith the electromagnetic field from macroscopic down to atomic scales, [7]-[11].In this paper we derive Newtonian dynamics for localized solutions of NonlinearSchrodinger equations (NLS) in the asymptotic limit when the solutions convergeto delta-functions. The derivation is based on a method of concentrating solutionswhich was developed in our previous paper [11] where we derived the relativisticdynamics and Einstein’s formula for localized solutions of the Nonlinear Klein-Gordon (KG) equation.In many problems of physics and mathematics a common way to establish arelation between wave and point dynamics is by means of the WKB method, seefor instance [41], [44, Sec. 7.1]. We remind that the WKB method is based onthe quasiclassical ansatz for solutions to a hyperbolic partial differential equationand their asymptotic expansion. The leading term of the expansion satisfies theeikonal equation, and wavepackets and their energy propagate along the so-calledcharacteristics of this equations. Consequently, these characteristics represent the
Mathematics Subject Classification.
Key words and phrases.
Nonlinear Schr¨odinger equation, Newton’s law, Lorentz force, tra-jectory of concentration.Supported by AFOSR grant number FA9550-11-1-0163. point dynamics and are determined from the corresponding system of ODEs whichcan be interpreted as a law of motion or a law of propagation. The construction ofthe characteristics involves only local data. Asymptotic derivation of the Newtoniandynamics which does not rely on the fast oscillation of solutions as the WKBmethod does was developed for soliton-like solutions of the Nonlinear Schrodingerand Nonlinear Klein-Gordon equations in a number of papers, see [2], [25], [29],[40]. In the mentioned papers the derivation of the limit Newtonian dynamics of thesoliton center relies on the asymptotic analysis of an ansatz structure of soliton-likesolutions.
The proposed here approach also relates waves governed by certain PDE’sin asymptotic regimes to the point dynamics but it differs fundamentally from thementioned methods . In particular, our approach is not based on any specific ansatzand it is not not entirely local but rather it is semi-local. Our semi-local method isin the spirit of the Ehrenfest theorem and is based on an analysis of a sequence ofconcentrating solutions which are not subjected to structural conditions. Comparedto the mentioned approaches which use the WKB method or soliton-like structureof solutions, our arguments only use very mild assumptions of a localization ofsolutions in a small neighborhood of the trajectory combined with the systematicuse there of integral identities derived from the energy-momentum and densityconservation laws for the NLS. Our method allows to derive Newton’s law underminimal restrictions on the time and spatial dependence of the variable coefficientsand on the nonlinearity and allows to consider arbitrarily long time intervals.The nonlinear Schrodinger (NLS) equation we study in this paper is of the form(1.1) i ˜ ∂ t ψ = χ m h − ˜ ∇ ψ + G ′ a ( ψ ∗ ψ ) ψ i . In the above equation ψ = ψ ( t, x ) is a complex valued wave function, G ′ a is a realvalued nonlinearity, and the covariant differention operators ˜ ∂ t and ˜ ∇ are definedby(1.2) ˜ ∂ t = ∂ t + i qχ ϕ, ˜ ∇ = ∇ − i qχ c A , where ϕ ( t, x ), A ( t, x ) are given twice differentiable functions of time t and spatialvariables x ∈ R interpreted as potentials of external electric and magnetic fields.The quantity q | ψ | is naturally interpreted as the charge density with q being thevalue of the charge.In the case when the potentials ϕ and A are zero, the charge can be considered as”free” and the NLS equation (1.1) has localized solutions corresponding to restingand uniformly moving charges. Newton’s law involves acceleration, and to producean accelerated motion non-zero potentials are required. One may expect point-like dynamics for strongly localized solutions, and the nonlinearity G ′ providesexistence of localized solutions of the NLS equation. The only condition imposedon the nonlinearity (in addition to natural continuity assumptions, see Condition1) is the existence of a positive radial solution ψ = ˚ ψ ( | x | ) of the steady-stateequation for a free charge:(1.3) − ∇ ˚ ψ + G ′ ( (cid:12)(cid:12)(cid:12) ˚ ψ (cid:12)(cid:12)(cid:12) )˚ ψ = 0 . RAJECTORY OF CONCENTRATION 3
We assume that ˚ ψ ( | x | ) is twice continuously differentiable for all x ∈ R and issquare integrable, that is(1.4) Z R (cid:12)(cid:12)(cid:12) ˚ ψ (cid:12)(cid:12)(cid:12) d x = υ < ∞ . Importantly, we assume that the nonlinearity depends explicitly on the size param-eter a > G ′ a ( s ) = a − G ′ (cid:0) a s (cid:1) , s > . Note that the steady-state equation with a general a > ∇ ˚ ψ a + G ′ a ( (cid:12)(cid:12)(cid:12) ˚ ψ a (cid:12)(cid:12)(cid:12) )˚ ψ a = 0and has the following solution:(1.7) ˚ ψ a ( r ) = a − / ˚ ψ (cid:0) a − r (cid:1) , r = | x | ≥ , where ˚ ψ is the solution to equation (1.3) where a = 1. Obviously ˚ ψ a satisfiesthe normalization condition (1.4) with the same υ . According to (1.7) a can beinterpreted as a natural size parameter which describes ˚ ψ a .A typical example of the ground state ˚ ψ a is the Gaussian ˚ ψ a = a − / e − r /a corresponding to a logarithmic nonlinearity G ′ a discussed in Example 2. Evidently, υ − ˚ ψ a converges to Dirac’s delta-function δ ( x ) as a →
0. We study the behaviorof localized solutions of the NLS equation in the asymptotic regime a → ψ = ψ n concentrate at a given trajectory ˆr ( t ) if their charge densities q | ψ | ( x , t ) restricted to R n -neighborhoods Ω n of ˆr ( t )locally converge to q ∞ ( t ) δ ( x − ˆr ( t )) as(1.8) a n → , R n → , a n /R n → , n → ∞ . Now we describe the concept of concentrating solutions in a little more detail.There are two relevant spatial scales for the NLS: the microscopic size parameter a which determines the size of a free charge and the macroscopic length scale R ex of order 1 at which the potentials ϕ and A vary significantly. We introduce thethird intermediate spatial scale R n which can be called the confinement scale and weassume that ψ n asymptotically vanish at the boundary ∂ Ω n = {| x − ˆr ( t ) | = R n } ofthe neighborhood Ω n . According to (1.8) R n /a → ∞ , therefore this assumptionis quite natural for solutions with typical spatial scale a which are localized at ˆr . At the same time, R n /R ex →
0, therefore general potentials ϕ ( t, x ), A ( t, x )can be replaced successfully in Ω n by their linearizations at x = ˆr ( t ). The exactdefinition of the concentrating solutions (or concentrating asymptotic solutions)involves two major assumptions:(1) volume integrals over the balls Ω n of the charge density q | ψ | and themomentum density P should be bounded;(2) surface integrals over the spheres ∂ Ω n of certain quadratic expressionswhich involve ψ and its first order derivatives and originate from the el-ements of the energy-momentum tensor of the NLS equation tend to zeroas R n → a n /R n →
0. Details of these assumptions can be found inDefinitions 4 and 5.
ANATOLI BABIN AND ALEXANDER FIGOTIN
A concise formulation of our main result, Theorem 6, is as follows. We provethat if a sequence of asymptotic solutions of the NLS equation (1.1) concentratesat a trajectory ˆr ( t ) then this trajectory must satisfy the Newton equation (1.9) m∂ t ˆr = f Lor ( t, ˆr ) , where f Lor is the Lorentz force which is defined by the classical formula(1.10) f Lor ( t, ˆr ) = q E ( t, ˆr ) + q c ∂ t ˆr × B ( t, ˆr ) , with the electric and magnetic fields E and B defined in terms of the potentials ϕ, A in (1.2) according to the standard formula(1.11) E = −∇ ϕ − ∂ t A , B = ∇ × A . Notice that that according to Newton’s equation (1.9) the parameter m enteringthe NLS equation (1.1) can be interpreted as the mass of the charge.Now we would like to comment on certain subjects related to our main Theo-rem 6. First of all, the assumptions imposed on the concentrating solutions arenot restrictive. Indeed, the solutions are local and have only to be defined in atubular neighborhood of the trajectory and no initial or boundary conditions areimposed on them. Second of all, only the charge and the momentum conservationlaws are used in the derivation of the theorem. Since only conservation laws areused for the argument, the asymptotic solutions for the NLS equation are intro-duced as functions ψ for which the charge and the momentum conservation lawshold approximately. Namely, certain integrals which involve quantities that enterthe conservation laws vanish in an asymptotic limit, see Definition 5 for details.Remarkably, such mild restrictions still completely determine all possible trajecto-ries of concentration, that is the trajectory ˆr ( t ) is uniquely defined by the initialdata ˆr ( t ) and ∂ t ˆr ( t ) as a solution of (1.9).Equation (1.9) yields a necessary condition for solutions of the NLS equationto concentrate at a trajectory. To obtain this necessary condition we have to de-rive the point dynamics governed by an ordinary differential equation (1.9) fromthe dynamics of waves governed by a partial differential equation. The conceptof ” concentration of functions at a given trajectory ˆr ( t )”, see Definition 4, is thefirst step in relating spatially localized fields ψ to point trajectories. The definitionof concentration of functions has a sufficient flexibility to allow for general regulartrajectories ˆr ( t ) and plenty of functions are localized about the trajectory. But if asequence of functions concentrating at a given trajectory also satisfy or asymptoti-cally satisfy the conservation laws for the NLS equation then, according to Theorems , the trajectory and the limit energy must satisfy Newton’s equation .To derive Newton’s equation (1.9), we consider ψ ( x , t ) restricted to a narrowtubular neighborhood of the trajectory of radius R n and consider adjacent chargecenters(1.12) r n ( t ) = 1¯ ρ n Z | x − ˆr |≤ R n x q | ψ n | d x , ¯ ρ n = Z | x − ˆr |≤ R n q | ψ n | d x , of the concentrating solutions. Then we infer integral equations for the restrictedmomentum and for the adjacent centers from the momentum conservation law andthe continuity equation, and pass to the limit as R n → a n → ρ and a similar momentum density involves integration over a large relative to a RAJECTORY OF CONCENTRATION 5 spatial domain of radius R n , R n >> a . Therefore it is natural to call our method ofdetermination of the point trajectories semi-local . This semi-local feature appliedto the nonlinear Klein-Gordon equation in [10], [11] allowed us to derive Einstein’srelation between mass and energy with the energy being an integral quantity.If the form factor ˚ ψ ( θ ) decays fast enough as θ → ∞ (faster than θ − ), wealso prove the converse statement (Theorem 8) to Theorem 6. Namely, if ˆ r ( t ) is a solution of equation (1.9) then it is a trajectory of concentration for a se-quence of asymptotic solutions to the NLS. The proof of this theorem is basedon an explicit construction of wave-corpuscle solutions of the NLS of the form ψ ( t, x ) = e i S ˚ ψ ( x − ˆr ) for a general enough class of potentials ϕ and A with acertain phase function S . We choose auxiliary potentials ϕ aux and A aux from thisclass to approximate ϕ and A at ˆr ( t ). The wave-corpuscle solutions constructedfor the auxiliary potentials ϕ aux and A aux turn out to form a sequence of concen-trating asymptotic solutions of the NLS. Therefore, for given twice continuouslydifferentiable potentials ϕ and A , a trajectory is a trajectory of concentration ofasymptotic solutions of the NLS if and only if this trajectory satisfies Newton’slaw (1.9). The phase function S of the wave-corpuscle can be interpreted as thephase of de Broglie wave (see [7] for details). Hence, the described relation betweenNewtonian trajectories and concentrating asymptotic solutions can be interpretedas what is known as the wave-particle duality.The paper’s structure is as follows. In the following Subsection 1.1 we define thecharge density, current density and momentum density for a solution of NLS andwrite down corresponding conservation laws. The customary field theory derivationof the conservation laws is given in Appendix 1. In Section 2 we describe the classof trajectories and in Subsection 2.2 we formulate restrictions on the fields ϕ and A and give the definition of solutions concentrating to a trajectory (Definition 4).In Subsections 2.3 and 2.4 we prove our main result on the characterization oftrajectories of concentration of solutions to the NLS equation (Theorem 3).In Section 3 we consider ”wave-corpuscles” that exactly preserve their shape inan accelerated motion and describe a class of potentials ϕ , A which allow for such amotion. In Subsection 3.4 we prove that the wave corpuscles provide an example ofconcentrating solutions to the NLS equation (Theorem 5). In Section 4 we providean extension of results of Section 2 to asymptotic solutions of the NLS equation,including the main Theorem 6. In Section 4.1 we prove that if a trajectory satisfiesNewton’s law then it is a trajectory of concentration of asymptotic solutions of theNLS equation.1.1. Conservation laws for NLS.
The charge density ρ and current density J are defined by the formulas ρ = qψψ ∗ , (1.13) J = − i qχ m (cid:16) ψ ∗ ˜ ∇ ψ − ψ ˜ ∇ ∗ ψ ∗ (cid:17) , (1.14)and for solutions of (1.1) they satisfy the continuity equation :(1.15) ∂ t ρ + ∇ · J = 0 . One can verify the validity of the continuity equation (1.15) directly by simplymultiplying (1.1) by ψ ∗ and taking the imaginary part. An alternative derivationbased on the Lagrangian field formalism is provided in Section 5. The momentum ANATOLI BABIN AND ALEXANDER FIGOTIN density is defined by the formula(1.16) P = i χ h ψ · ˜ ∇ ∗ ψ ∗ − ψ ∗ · ˜ ∇ ψ i , and it is proportional to the current density, namely(1.17) P = mq J . The momentum density satisfies the momentum equation of the form(1.18) ∂ t P + ∂ i T ij = f , where f is the Lorentz force density defined by the formula(1.19) f = ρ E + 1c J × B ,T ij are the entries of the energy-momentum tensor defined by (5.21), (5.22), and theEM fields E , B are defined in terms of the potentials ϕ , A by the standard formulas(1.11). The momentum equation can be derived from the NLS equation usingmultiplication by ˜ ∇ ∗ ψ ∗ and some rather lengthy vector algebra manipulations. Thederivation of the momentum equation based on the standard field theory formalismis given in Section 5.2. Derivation of non-relativistic point dynamics for localizedsolutions of NLS equation
In this section we consider solutions of NLS equation (1.1) that are localizedaround a trajectory in the three dimensional space R and find the necessary con-dition for such a trajectory which coincides with Newton’s law of motion.2.1. Trajectories and their neighborhoods.
The first step in introducing theconcept of concentration to a trajectory is to describe the class of trajectories.
Definition 1 (trajectory) . A trajectory ˆr ( t ) , T − ≤ t ≤ T + , is a twice continuouslydifferentiable function with values in R satisfying (2.1) | ∂ t ˆr ( t ) | ≤ C, (cid:12)(cid:12) ∂ t ˆr ( t ) (cid:12)(cid:12) ≤ C for T − ≤ t ≤ T + . Being given a trajectory ˆr ( t ), we consider a family of neighborhoods contractingto it. Namely, we introduce a ball of radius R centered at x = ˆr ( t ):(2.2) Ω ( ˆr ( t ) , R ) = n x : | x − ˆr ( t ) | ≤ R o ⊂ R , R > . Definition 2 (concentrating neighborhoods) . Concentrating neighborhoods ˆΩ ( ˆr ( t ) , R n ) ⊂ [ T − , T + ] × R of a trajectory ˆr ( t ) are defined as a family of tubular domains (2.3) ˆΩ ( ˆr ( t ) , R n ) = n ( t, x ) : T − ≤ t ≤ T + , | x − ˆr ( t ) | ≤ R n o where R n satisfy the contraction condition: (2.4) R n → as n → ∞ . The cross-section of the tubular domain at a fixed t is given by (2.2) and is denotedby Ω n : (2.5) Ω n = Ω n ( t ) = Ω ( ˆr ( t ) , R n ) ⊂ R . RAJECTORY OF CONCENTRATION 7
Localized NLS equations.
Let us consider the NLS equation (1.1) in aneighborhood of the trajectory ˆr ( t ). We remind that m is the mass parameter, q is the value of the charge, χ is a parameter similar to the Planck constant, cis the speed of light all of which are fixed. The size parameter a and potentials ϕ ( t, x ) , A ( t, x ) form a sequence. We consider the NLS in a shrinking vicinity of thetrajectory and make certain regularity assumptions on behavior of its coefficients.In the definitions below we use the following notations:(2.6) ∂ = c − ∂ t , ∇ x ϕ = ∇ ϕ = ( ∂ ϕ, ∂ ϕ, ∂ ϕ ) , |∇ x ϕ | = |∇ ϕ | = | ∂ ϕ | + | ∂ ϕ | + | ∂ ϕ | , (2.7) ∇ , x ϕ = ( ∂ ϕ, ∂ ϕ, ∂ ϕ, ∂ ϕ ) , |∇ , x ϕ | = | ∂ ϕ | + | ∂ ϕ | + | ∂ ϕ | + | ∂ ϕ | . Now we formulate the continuity assumtions we impose on the nonlinearity G ′ inthe NLS equation (1.1). Condition 1.
The real-valued function G ′ ( s ) is continuous for s > . It coincideswith the derivative of the potential G ( s ) which is differentiable for s > andcontinuous for s ≥ . We assume that the function G ( ψ ∗ ψ ) of the complexvariable ψ ∈ C is differentiable for all ψ and its differential has the form dG ( ψ ∗ ψ ) = g ( ψ ) dψ ∗ + g ∗ ( ψ ) dψ where g ( ψ ) = (cid:26) G ′ ( ψ ∗ ψ ) ψ for ψ ∈ C , ψ = 0 , for ψ = 0 , and g ( ψ ) is continuous for ψ ∈ C . Note that the above condition allows a mild singularity of G ′ at zero, for examplethe logarithmic nonlinearity (3.10) satisfies this condition. Definition 3 (localized NLS equations) . Let ˆr ( t ) be a trajectory with its concen-trating neighborhoods ˆΩ ( ˆr , R n ) , and let a = a n , ϕ = ϕ n , A = A n be a sequenceof parameters and potentials entering the NLS equation (1.1). We say that NLSequations are localized in ˆΩ ( ˆr , R n ) if the following conditions are satisfied. Theparameters R n and a n as n → ∞ become vanishingly small, that is (2.8) a = a n → , R n → and the ratio θ = R n /a grows to infinity: (2.9) θ n = R n a n → ∞ as n → ∞ . We introduce at the trajectory ˆr ( t ) the limit potentials ϕ ∞ ( t, x ) A ∞ ( t, x ) whichare linear in x and are written in the form (2.10) ϕ ∞ ( t, x ) = ϕ ∞ ( t ) + ( x − ˆr ) · ∇ ϕ ∞ ( t ) , (2.11) A ∞ ( t, x ) = A ∞ ( t ) + ( x − ˆr ) · ∇ A ∞ ( t ) , where the coefficients ϕ ∞ , ∇ ϕ ∞ , A ∞ , ∇ A ∞ satisfy the following boundedness con-ditions: (2.12) | ϕ ∞ | ≤ C, |∇ , x ϕ ∞ | ≤ C for T − ≤ t ≤ T + , ANATOLI BABIN AND ALEXANDER FIGOTIN (2.13) | A ∞ | ≤ C, |∇ , x A ∞ | ≤ C for T − ≤ t ≤ T + . We suppose the EM potentials ϕ n ( t, x ) , A n ( t, x ) to be twice continuously differ-entiable in ˆΩ ( ˆr , R n ) . The potentials ϕ n ( t, x ) , A n ( t, x ) locally converge to the limitlinear potentials ϕ ∞ ( t, x ) , A ∞ ( t, x ) , namely they satisfy the following relations: (i) convergence as n → ∞ : (2.14) max T − ≤ t ≤ T + ,x ∈ Ω n ( | ϕ n ( t, x ) − ϕ ∞ ( t, x ) | + |∇ ,x ϕ n ( t, x ) − ∇ ,x ϕ ∞ ( t, x ) | ) → , (2.15) max T − ≤ t ≤ T + ,x ∈ Ω n ( | A n ( t, x ) − A ∞ ( t, x ) | + |∇ ,x A n ( t, x ) − ∇ ,x A ∞ ( t, x ) | ) → uniform in n estimates: (2.16) | ϕ n ( t, x ) | ≤ C , |∇ , x ϕ n ( t, x ) | ≤ C for ( t, x ) ∈ ˆΩ ( ˆr , R n ) , (2.17) | A n ( t, x ) | ≤ C , |∇ , x A n ( t, x ) | ≤ C for ( t, x ) ∈ ˆΩ ( ˆr , R n ) . The limit EM fields E ∞ , B ∞ at the trajectory are defined in terms of thelinear potentials (2.10), (2.11) by (1.11), namely (2.18) E ∞ = −∇ ϕ ∞ ( t, ˆr ) − ∂ t A ∞ ( t, ˆr ) , B ∞ = ∇ × A ∞ ( t, ˆr ) . Note that according to (2.14), (2.15)(2.19) E = E n → E ∞ , B = B n → B ∞ in ˆΩ ( ˆr ( t ) , R n ).Throughout this paper we denote constants which do not depend on n by theletter C with different indices. Sometimes the same letter C with the same indicesmay denote in different formulas different constants. Below we often omit index n in a n , ϕ n etc.The most important case where we apply the above definition is described in thefollowing example. Example 1.
If the potentials ϕ n , A n are the restrictions of fixed twice continuouslydifferentiable potentials ϕ, A to the domain ˆΩ ( ˆr ( t ) , R n ) , then ϕ ∞ , A n is the linearpart of ϕ, A at ˆr and conditions (2.12)-(2.17) are satisfied with ϕ ∞ ( t, x ) = ϕ ( t, ˆr ) + ( x − ˆr ) · ∇ ϕ ( t, ˆr ) , (2.20) A ∞ ( t, x ) = A ( t, ˆr ) + ( x − ˆr ) · ∇ A ( t, ˆr ) , that is the coefficients in (2.10), (2.11) are defined as follows: (2.21) ϕ ∞ ( t ) = ϕ ( t, ˆr ( t )) , ∇ A ∞ ( t ) = ∇ A ( t, ˆr ) , and the EM fields E ∞ , B ∞ in (2.18) are directly expressed in terms of ϕ, A by(1.11), namely (2.22) E ∞ = −∇ ϕ ( t, ˆr ) − ∂ t A ( t, ˆr ) , B ∞ = ∇ × A ( t, ˆr ) . We introduce the local value of the charge restricted to domain Ω ( ˆr ( t ) , R n ) bythe formula(2.23) ¯ ρ n ( t ) = Z Ω( ˆr ( t ) ,R n ) ρ n d x = Z Ω( ˆr ( t ) ,R n ) q | ψ n | d x , RAJECTORY OF CONCENTRATION 9 with ρ n ( t, x ) being the charge density defined by (1.13), and we call ¯ ρ n adjacentcharge value. Definition 4 (concentrating solutions) . Let ˆr ( t ) be a trajectory. We say thatsolutions ψ to the NLS equations concentrate at the trajectory ˆr ( t ) if Condition 1 issatisfied and the following conditions are fulfilled. First, a sequence of concentratingneighborhoods ˆΩ ( ˆr , R n ) , parameters a = a n and potentials ϕ = ϕ n , A = A n are selected as in Definition 3. Second, there exists a sequence of functions ψ = ψ n which are twice continuously differentiable, that is ψ n ∈ C (cid:16) ˆΩ ( ˆr , R n ) (cid:17) , andsuch that the charge density ρ = q | ψ n | and the momentum density P defined by(1.16) for this sequence satisfy the following conditions: (i) every function ψ n is a solution to the NLS equation (1.1) in ˆΩ ( ˆr , R n ) ; (ii) the momentum density P defined by (1.16) for this sequence is such thatthe following integrals are bounded: (2.24) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω n P n ( t ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ;(iii) the local charge value defined by (2.23) is bounded from above and below forsufficiently large n : (2.25) C ≥ ¯ ρ n ( t ) ≥ c > for n ≥ n , T − ≤ t ≤ T + ;(iv) there exists t ∈ [ T − , T + ] such that the sequence of local charge values con-verges: (2.26) lim n →∞ ¯ ρ n ( t ) = ¯ ρ ∞ . (v) We also impose conditions on the following surface integrals over the spheres ∂ Ω n = {| x − ˆr | = R n } : (2.27) Q = Z tt Z ∂ Ω n ¯n i T ij d σ d t ′ , where T ij are given by (5.21), (5.22) and we make summation over repeat-ing indices; (2.28) Q = Z tt Z ∂ Ω n Pˆv · ¯n d σ d t ′ , where P is defined by (1.16); (2.29) Q = − Z ∂ Ω n ( x − r ) ˆv · ¯n ρ d σ, where ρ is defined by (1.13); (2.30) Q = Z ∂ Ω n ( x − r ) n · J d x , where J is given by (1.14); and (2.31) Q = Z tt Z ∂ Ω n ˆv · n ρ d x d t ′ − Z tt Z ∂ Ω n n · J d x d t ′ . The integrals defined above are assumed to satisfy the following limit rela-tions uniformly on the time interval [ T − , T + ] : (2.32) Q → , (2.33) Q → , (2.34) Q → , (2.35) Q → , (2.36) Q → . (vi) We introduce the following integrals over Ω n with vanishing at ˆr weights (2.37) Q = Z Ω n ( E − E ∞ ) ρ d x , (2.38) Q = Z Ω n J × ( B − B ∞ ) d x , and assume that (2.39) Z tt ( Q + Q ) d t ′ → uniformly on the time interval [ T − , T + ] . If all the above conditions arefulfilled, we call ˆr ( t ) a concentration trajectory of the NLS equation (1.1). Obviously conditions (ii)-(iv) and (vi) provide boundedness and convergence ofcertain volume integrals over Ω n , and condition (v) provides asymptotical vanishingof surface integrals over the boundary. Notice that condition (ii) provides for theboundedness of the momentum over domain Ω n , condition, (v) provides for a properconfinement of ψ to Ω n and estimate from below in condition (iii) ensures thatthe sequence is non-trivial. According to (2.25), ¯ ρ n ( t ) is a bounded sequence,consequently it always contains a converging subsequence. Hence condition (iv) isnot really an additional constraint but rather it assumes that such a subsequence isselected. The choice of a particular subsequence limit ¯ ρ ∞ is discussed in Remark 1.This condition describes the amount of charge which concentrates at the trajectoryat the time t . Condition (i) can be relaxed and replaced by the assumption that ψ n is an asymptotic solution, see Definition 5 for details. We could also allowparameters χ, m, q to form sequences and depend on n , but for simplicity in thispaper we assume them fixed. The wave-corpuscles constructed in Section 3 provide a non-trivial example ofsolutions which concentrate at trajectories of accelerating charges .2.3.
Properties of concentrating solutions of NLS.
We define the adjacentcharge center r n by the formula(2.40) r n ( t ) = 1¯ ρ n Z Ω( ˆr ( t ) ,R n ) x ρ n d x . Since ψ a ( t, x ) is a function of class C with respect to ( t, x ), and ˆr ( t ) is differen-tiable, the vector r ( t ) is a differentiable function of time, and we denote by v thevelocity of the adjacent charge center:(2.41) v = v n ( t ) = ∂ t r . RAJECTORY OF CONCENTRATION 11
Below we often make use of the following elementary identity:(2.42) Z Ω n ∂ t f ( t, x ) d x = ∂ t Z Ω n f ( t, x ) d x − Z ∂ Ω n f ( t, x ) ˆv · ¯n d σ, where ¯n is the external normal to ∂ Ω n , ˆv = ∂ t ˆr . Lemma 1.
Let charge densities ρ n satisfy (2.25). Then the adjacent ergocenters r n ( t ) of the solutions converge to ˆr ( t ) uniformly on the time interval [ T − , T + ] .Proof. By (2.40)(2.43) Z Ω n ( x − r ) ρ n d x = 0 , and according to (2.25)(2.44) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω( ˆr ( t ) ,R n ) ( x − ˆr ) ρ n d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ R n Z Ω( ,R n ) ρ n d y → . Therefore ( r − ˆr ) ¯ ρ = Z Ω n ( x − ˆr ) ρ n d x − Z Ω n ( x − r ) ρ n d x → . Using (2.25) we conclude that(2.45) | ˆr − r n | → T − , T + ]. (cid:3) Lemma 2.
Let the current J and the charge density ρ in (1.15) are such that Q defined by (2.31) satisfies (2.36). Then the local charge values converge uniformlyon [ T − , T + ] to a constant: (2.46) ¯ ρ n ( t ) → ¯ ρ ∞ as n → ∞ . Proof.
Integrating the continuity equation we obtain(2.47) ¯ ρ ( t ) − ¯ ρ ( t ) − Z tt Z ∂ Ω n ˆv · n ρ d x d t ′ + Z tt Z ∂ Ω n n · J d x d t ′ = 0 . We use (2.36) and obtain (2.46). (cid:3)
Lemma 3.
Assume that (2.24) holds. Then there is a subsequence of the concen-trating sequence of solutions of the NLS such that (2.48) Z Ω( ˆr ( t ) ,R n ) P n ( t ) d x → p ∞ as n → ∞ . Assume also that boundary integrals (2.27) and (2.28) satisfy assumptions (2.32)and (2.33). Then for this subsequence (2.49) Z Ω n P n ( t ) d x = Z tt Z Ω n f d x d t ′ + p ∞ + Q where (2.50) Q → as n → ∞ uniformly on [ T − , T + ] . Proof.
According to (2.24) we can select a subsequence which has a limit p ∞ and(2.48) holds. We use the momentum equation (1.18)(2.51) ∂ t P + ∂ i T ij = f . Integrating it over Ω ( ˆr ( t ) , R n ) = Ω n and then with respect to time, we obtain theequation(2.52) Z tt Z Ω n ∂ t P ( t ′ ) d x d t ′ − Z tt Z Ω n f d x d t ′ + Q = 0where Q is defined by (2.27). Using (2.42) we rewrite the equation in the form(2.53) Z Ω n P ( t ) d x − Z Ω n P ( t ) d x − Q − Z tt Z Ω n f d x d t ′ + Q = 0 , where Q is defined by (2.28), Ω n = Ω ( ˆr ( t ) , R n ). Using (2.32), (2.33) and (2.48)we obtain (2.49) and (2.50) from (2.53). (cid:3) Lemma 4.
In addition to the assumptions of Lemmas 1, 2, 3 assume that theboundary integrals (2.29) and (2.30) vanish as n → ∞ , namely (2.34) and (2.35)are fulfilled uniformly on the time interval [ T − , T + ] . Then (2.54) Z Ω n J d x = v ¯ ρ ∞ + Q , (2.55) Z Ω n P d x = m v ¯ ρ ∞ q + mq Q , where v = ∂ t r and (2.56) Q → uniformly on the time interval [ T − , T + ] . Proof.
According to the continuity equation, we obtain the identity Z Ω n ( x − r ) ∂ t ρ d x + Z Ω n ( x − r ) ∇ · J d x = 0 . Using the commutation relation(2.57) ∂ j ( x i f ) − x i ∂ j f = δ ij f to transform the second integral, we obtain the following equation(2.58) Z Ω n ∂ t (( x − r ) ρ ) d x + ∂ t r Z ∂ Ω n ρ d x + Z ∂ Ω n ( x − r ) n · J d x = Z Ω n J d x . Using the definition of the charge center r and (2.42) we infer that the first termin the above equation has the following form:(2.59) Z Ω n ∂ t (( x − r ) ρ ) d x = − Z ∂ Ω n ( x − r ) ˆv · ¯n ρ d σ. where ˆv = ∂ t ˆr . We can express then v = ∂ t r from (2.58), and using (2.24), (2.30),(2.45) and (2.25) can estimate the integrals which enter (2.58) concluding that(2.60) | v | ≤ C for − T ≤ t ≤ T. RAJECTORY OF CONCENTRATION 13
We rewrite (2.58) in the form(2.61) Z Ω n J d x = v ¯ ρ ∞ + Q + Q + Q , where ¯ ρ ∞ is the same as in (2.26), Q = v (cid:18)Z Ω n ρ n d x − ¯ ρ ∞ (cid:19) . According to (2.26) and (2.60) Q →
0. Using (2.29) and (2.30) we conclude that(2.54) and (2.56) hold with Q = Q + Q + Q . Using (1.17) we deduce (2.55)from (2.54). (cid:3) Derivation of Newton’s equation for the trajectory of concentration.Theorem 1.
For a concentrating sequence of solutions of the NLS equation theadjacent center velocities v = v n = ∂ t r n satisfy the equation (2.62) m v q ¯ ρ ∞ = Z tt (cid:18) ¯ ρ ∞ E ∞ ( t ′ ) + ¯ ρ ∞ v × B ∞ ( t ′ ) (cid:19) d t ′ + p ∞ + Q where Q → uniformly on [ T − , T + ] , p ∞ is the same as in (2.48), and E ∞ , B ∞ are the same as in (2.18).Proof. We substitute (2.55) into (2.49) and obtain the following equation:(2.63) m v q ¯ ρ ∞ + mq Q = Z tt Z Ω n f d x d t ′ + p ∞ + Q . where the Lorentz force density f is given by (1.19). To evaluate the terms involvingthe Lorentz force density, we use the limit EM fields defined in accordance with(2.10), (2.11) by formula (2.18). We obtain(2.64) Z Ω n f d x = E ∞ Z Ω n ρ d x + Z Ω n J d x × B ∞ + Q where Q = Q + Q is expressed in terms of (2.37), (2.38). Therefore, using(2.54) we obtain from (2.64) that Z Ω n f d x = ¯ ρ ∞ E ∞ + ¯ ρ ∞ v × B ∞ + Q + Q + Q , (2.65) Q = 1c Q × B ∞ , Q = E ∞ (¯ ρ n − ¯ ρ ∞ ) . (2.66)Using (2.39), (2.46), (2.56), (2.12), and (2.13), we obtain (2.62) with Q = Q − mq Q + Z tt ( Q + Q + Q ) d t ′ . (cid:3) The following statement provides an explicit necessary condition for a trajectoryof concentration.
Theorem 2 (Trajectory of concentration criterion) . Let solutions ψ of the NLSequation (1.1) concentrate at ˆr ( t ) . Then the trajectory ˆr ( t ) satisfies the equation (2.67) ∂ t ˆr = f ∞ with the Lorentz force f ∞ ( t ) expressed in terms of the potentials (2.10), (2.11) bythe formula (2.68) f ∞ ( t ) = − q E ∞ + q c ∂ t ˆr × B ∞ . Proof.
The function v ( t ) can be considered as a solution of the integral equation(2.62) on the interval [ T − , T + ], and this equation is evidently linear. Since the term Q → v n ( t ) of its solutions converges uniformly to thesolution of the equation(2.69) m v ∞ q ¯ ρ ∞ = ¯ ρ ∞ Z tt (cid:18) E ∞ ( t ′ ) + v ∞ × B ∞ ( t ′ ) (cid:19) d t ′ + p ∞ . Now we want to prove that v ∞ = ˆv = ∂ t ˆr . Note that according to Lemma 1 Z tt v n d t ′ = r n ( t ) − r n ( t ) → ˆr ( t ) − ˆr ( t ) , and, hence, Z tt v ∞ d t ′ = ˆr ( t ) − ˆr ( t ) . The above identity implies ∂ r ˆr = ˆv ( t ) = v ∞ ( t ) and consequently ˆr ( t ) satisfies(2.69):(2.70) m∂ t ˆr q ¯ ρ ∞ = ¯ ρ ∞ Z tt (cid:18) E ∞ ( t ′ ) + ∂ r ˆr × B ∞ ( t ′ ) (cid:19) d t ′ + p ∞ . The derivative of the above equation yields (2.67). (cid:3)
As a corollary of Theorem 2 we obtain the following theorem describing thewhole class of trajectories of concentration of NLS equation (1.1).
Theorem 3 (Non-relativistic Newton’s law) . Assume that (i) potentials ϕ ( t, x ) , A ( t, x ) are defined and regular in a domain D ⊂ R × R ; (ii) the trajectory ( t, ˆr ( t )) lies in this domain and the limit potentials ϕ ∞ , A ∞ are the restriction of thepotentials ϕ and A as in (2.20). Let EM fields E ( t, x ) , B ( t, x ) be defined bythe potentials as in the formula (1.11). Let solutions ψ of the NLS equation (1.1)concentrate at ˆr ( t ) . Then equation (2.67) for the trajectory ˆr takes the form ofNewton’s law of motion with the Lorentz force corresponding to the external EMfields E ( t, x ) and B ( t, x ) , that is (2.71) m∂ t ˆr = q E ( t, ˆr ) + q c ∂ t ˆr × B ( t, ˆr ) . Therefore, for the NLS equation with given potentials ϕ , A any concentrationtrajectory must coincide with the solution of the equation (2.71). For given po-tentials ϕ , A the concentration trajectory through a point ˆr ( t ) is uniquelydetermined by the velocity ∂ t ˆr ( t ). In particular, it does not depend on the pa-rameter χ which enters the NLS equation. All possible concentration trajectoriesare solutions of the equation (2.71). Remark 1.
Consider the situation of Example 1 and Theorem 3. The sequences a n , R n , θ n , ϕ n , A n , ψ n enter the definition of a concentrating solution. If we taketwo different sequences which fit the definition, we obtain the same equation (2.71).More than that, the trajectory ˆr ( t ) is uniquely defined by the initial data ˆr ( t ) , ∂ t ˆr ( t ) and consequently it does not depend on the particular sequence. Remarkably, RAJECTORY OF CONCENTRATION 15 the equation for the trajectory is independent of the value of ¯ ρ ∞ , and this propertydoes not hold in the relativistic case, see [11] . Remark 2.
We can modify the definition of concentration at a trajectory by allow-ing parameter χ to be not fixed but form a sequence. The statements of Theorem2 and Theorem 3 continue to hold in this case. Wave-corpuscles in accelerated motion
An example of concentrating solutions is provided by what we call wave-corpuscles.We define the wave-corpuscle ψ by the formula ψ ( t, x ) = e i S ˚ ψ ( | y | ) , (3.1) S = S ( y , t ) , y = x − r ( t ) , (3.2)where the form factor ˚ ψ satisfies the steady-state equation (1.6) and the normal-ization condition (1.4).To give non-trivial examples of localized form factors, it is convenient to startwith the form factor ˚ ψ ( r ) and to describe the nonlinearity which produces theform factor as a solution of (1.6). To define the nonlinearity, we impose first ourrequirements on the ground state ˚ ψ ( r ) of the charge distribution. A ground stateis a positive function ˚ ψ ( r ), r = | x | , which is twice differentiable, satisfies the chargenormalization condition (1.4) and is monotonically decreasing:(3.3) ∂ r ˚ ψ ( r ) < r > . If ˚ ψ ( r ) is a ground state, we can determine the nonlinearity G ′ from the followingequation obtained from (1.6):(3.4) ∇ ˚ ψ = G ′ ( | ˚ ψ | )˚ ψ. For a radial ˚ ψ we obtain then an expression for the nonlinearity G ′ :(3.5) G ′ (cid:16) ˚ ψ ( r ) (cid:17) = G ′ (cid:16) ˚ ψ ( r ) (cid:17) = ( ∇ ˚ ψ ) ( r )˚ ψ ( r ) . Since ˚ ψ ( r ) is a monotonic function, we can find its inverse r = r (cid:0) ψ (cid:1) , yielding(3.6) G ′ ( s ) = ∇ ˚ ψ ( r ( s ))˚ ψ ( r ( s )) , ψ ( ∞ ) ≤ s ≤ ˚ ψ (0) . Since ˚ ψ ( r ) is smooth and ∂ r ˚ ψ < G ′ ( | ψ | ) is smooth for 0 < | ψ | < ˚ ψ (0); weextend G ′ ( s ) for s ≥ ˚ ψ (0) as a smooth function for all s >
0. To describe thelocalization of the ground state ˚ ψ , we introduce an explicit dependence on the sizeparameter a > ψ ( r ) = ˚ ψ a ( r ) = a − / ˚ ψ (cid:0) a − r (cid:1) , r = | x | ≥ . This corresponds to the dependence of the nonlinearity on the size parameter de-scribed by (1.5). Note that the antiderivative G ( s ) is defined for s ≥ G ( s ) = Z s G ′ ( s ′ ) d s ′ Example 2.
We define a
Gaussian form factor by the formula (3.9) ˚ ψ ( r ) = C g e − r / where C g is a normalization factor, C g = π − / if υ = 1 in (1.4). Such a groundstate is called gausson in [17] , [18] . Elementary computation shows that ∇ ˚ ψ ( r )˚ ψ ( r ) = r − − ln (cid:16) ˚ ψ ( r ) /C g (cid:17) − . Hence, we define the nonlinearity corresponding to the Gaussian by the formula (3.10) G ′ (cid:0) | ψ | (cid:1) = − ln (cid:0) | ψ | /C g (cid:1) − , and refer to it as the logarithmic nonlinearity. The nonlinear potential function hasthe form (3.11) G ( s ) = Z s G ′ ( s ′ ) d s ′ = − s ln s + s (cid:18) ln 1 π / − (cid:19) . Dependence on the size parameter a > is given by the formula (3.12) G ′ a (cid:0) | ψ | (cid:1) = − a − ln (cid:0) a | ψ | /C g (cid:1) − a − . Note that according to Gross inequality the Gaussian provides the minimum ofenergy E = Z u d x subjected to the normalization condition (1.4), where the energy density u is definedby (5.19). More examples of nonlinearities can be found in [7]–[11], and many facts fromthe theory of the NLS equations can be found in [20], [51]. The NLS equations withlogarithmic nonlinearity are studied in [21], [19], [20].Below we find conditions on the external fields ϕ and A which allow the wave-corpuscle of the form (3.1) to preserve exactly its shape | ψ | in accelerated motiongoverned by the NLS equation along a trajectory r ( t ). In particular, Newton’s lawof motion emerges as the necessary condition for such a motion.3.1. Criterion for shape preservation.
It is convenient to rewrite the NLSequation (1.1) in the moving frame using a change of variables x − r ( t ) = y .In y -coordinates the NLS equation (1.1) takes the form(3.13) χ i ∂ t ψ − χ i v · ∇ ψ − qϕψ = χ m " − (cid:18) ∇ − i qχc A (cid:19) ψ + G ′ ( ψ ∗ ψ ) ψ , where v is the center velocity:(3.14) v ( t ) = ∂ t r ( t ) . We substitute (3.1) in (3.13), and, canceling the factor e i S , we arrive to the followingequivalent equation for the phase S : − χ ˚ ψ∂ t S − χ i v · ∇ ˚ ψ + χ v · ∇ S ˚ ψ − qϕ ˚ ψ =(3.15) = χ m " − (cid:18) ∇ − i qχc A + i ∇ S (cid:19) ˚ ψ + G ′ (cid:16) ˚ ψ (cid:17) ˚ ψ . RAJECTORY OF CONCENTRATION 17
Expanding (cid:16) ∇ − i q A χc + i ∇ S (cid:17) ˚ ψ and using (1.6) to exclude the nonlinearity G ′ , werewrite (3.15) in the form − ˚ ψχ∂ t S − χ i v · ∇ ˚ ψ + χ v · ∇ S ˚ ψ − qϕ ˚ ψ =(3.16)= χ m " (cid:18) i qχc A − i ∇ S (cid:19) ∇ ˚ ψ − ˚ ψ ∇ · (cid:18) i ∇ S − i qχc A (cid:19) + (cid:18) ∇ S − qχc A (cid:19) ˚ ψ . Every term in the above equation has either the factor ˚ ψ or ∇ ˚ ψ . Collecting termsat ∇ ˚ ψ and ˚ ψ respectively, we obtain two equations:(3.17) (cid:16) χ ∇ S − qc A − m v (cid:17) · ∇ ˚ ψ = 0 , (3.18) − χ∂ t S + χ v · ∇ S − qϕ = − i χ m ∇ · (cid:16) χ ∇ S − qc A (cid:17) + 12 m (cid:16) χ ∇ S − qc A (cid:17) . Now we would like to find conditions on the potentials ϕ, A under which thereexists the phase S which satisfies the above system of equations (3.17), (3.18). Sucha phase provides for the wave-corpuscle solution to the NLS equation (3.13). Webegin with equation (3.17) first. Since ∇ ˚ ψ = ˚ ψ ′ ( | y | ) y | y | , we conclude that (3.17) is equivalent to the following orthogonality condition:(3.19) (cid:16) qc A − χ ∇ S + m v (cid:17) · y = . To treat the above equation we use the concept of a sphere-tangent field . We call avector field ˘V ( y ) sphere-tangent if it satisfies the orthogonality condition(3.20) ˘V ( y ) · y = 0 for all y . Obviously, a sphere-tangent vector field is tangent to spheres centered at the origin.Any vector field V ( y ) of class C (cid:0) R (cid:1) can be uniquely splitted into a potentialfield ∇ P and a sphere-tangent field ˘V which satisfies the orthogonality condition(3.20), namely(3.21) V = ∇ P + ˘V (see Section 6 for details and explicit formulas). We split the vector potential A into its sphere-tangent and gradient parts, namely(3.22) A ( t, y ) = ˘A ( t, y ) + A ∇ ( t, y )with ˘A ( t, y ) · y = 0 , (3.23) A ∇ ( t, y ) = ∇ P ( t, y ) . (3.24)Equation (3.19) implies that the field qc A − ∇ S + m v is purely sphere-tangent,therefore its gradient part is zero and we conclude that the following equation isequivalent to (3.19):(3.25) χ ∇ S = m v + qc A ∇ . Now let us consider equation (3.18). Expressing ∇ S from (3.25) we write (3.18) inthe form(3.26) − χ∂ t S + v · (cid:16) m v + qc A ∇ (cid:17) − qϕ = i χ m qc ∇ · ˘A + 12 m (cid:16) m v − qc ˘A (cid:17) . Taking the imaginary part of (3.26), we see that the sphere-tangent part of thevector potential must satisfy the following zero divergency condition:(3.27) ∇ · ˘A = 0 . The real part of (3.26) yields equation(3.28) χ∂ t S = m v + qc v · A ∇ − m (cid:16) m v − qc ˘A (cid:17) − qϕ. Hence, (3.28) and (3.25) take the form χ∂ t S = 12 m v + qc v · A − m q c ˘A − qϕ, (3.29) χ ∇ S = m v + qc A ∇ . (3.30)The above relations give an expression for the 4-gradient of χS . The right-handsides of (3.29), (3.30) can be considered as the coefficients of a differential 1-form.Hence, if the phase S which solves (3.16) exists, the form must be exact, andconsequently it must be closed. Conversely, according to Poincare’s lemma, theform on R is exact if it is closed on R , and then the phase S can be found bythe integration of the differential form. Since the right-hand side of (3.30) is thegradient, the form is closed if the following condition is satisfied:(3.31) ∇ (cid:18) m v + qc v · A − m q c ˘A − qϕ (cid:19) = ∂ t (cid:16) m v + qc A ∇ (cid:17) . This equation can be interpreted as a balance of forces which allows to exactlypreserve the shape of the wave-corpuscle.
Equation (3.31) together with condition(3.27) constitutes the criterion for the preservation of the shape | ψ | . Trajectory and phase of a wave-corpuscle.
If equation (3.31) holds, thephase function S can be found by integrating the exact 1-form:(3.32) S ( t, y ) = 1 χ Z Γ (cid:18) m v + qc v · A − m q c ˘A − qϕ (cid:19) d t + (cid:16) m v + qc A ∇ (cid:17) · d y where Γ is a curve in time-space connecting (0 , ) with ( t, y ). Since equation (3.31)is fulfilled, the integral does not depend on the curve. In particular, we take asΓ = Γ a curve formed by two straight-line segments: the first segment from (0 , )to ( t, ) and the second from ( t, ) to ( t, y ). That yields S ( t, y ) = 1 χ Z Γ (cid:18) m v + qc v · A − m q c ˘A − qϕ (cid:19) d t + (cid:16) m v + qc A ∇ (cid:17) · d y (3.33) = m v · y + qc y · A ( t, ) + s p ( t ) + s p2 ( t, y ) , RAJECTORY OF CONCENTRATION 19 where s p ( t ) = 1 χ Z t (cid:18) m v ( t ) + q c v · A ( t, − qϕ ( t, (cid:19) d t, (3.34) s p2 ( t, y ) = qχ c Z y · ( A ( t, s y ) − A ( t, )) d s, (3.35)and s p2 ( t, y ) is at least quadratic in y . In the above derivation we used that ˘A ( t, ) = 0, y · ˘A ( t, y ) = 0, and A ∇ ( t, ) = A ( t, ).Singling out the linear part of the phase we can write above formulas in the form(3.36) S ( t, y ) = 1 χ m ˜v · y + s p ( t ) + s p2 ( t, y ) , where(3.37) ˜v = v + qmc A ( t, ) , (3.38) s p ( t ) = 1 χ Z t (cid:18) m ˜v ( t ) − q mc ( A ( t, − qϕ ( t, (cid:19) d t. The integrability condition (3.31) can be written now in the form(3.39) m∂ t r = qc ∇ ( v · A ) − q c m ∇ ˘A − q ∇ ϕ − qc ∂ t A ∇ . The above equation together with (3.27) is a system of equations which involvesthe EM potentials ϕ, A and the corpuscle center trajectory r ( t ). Its fulfillmentguarantees that the wave-corpuscle preserves its shape in the dynamics describedby the NLS equation. We refer to equations (3.39), (3.27) non-relativistic wave-corpuscle dynamic balance conditions . Obviously the conditions do not depend onthe nonlinearity G .By setting y = 0 in (3.39) and taking into account that ˘A ex ( t,
0) = 0 we obtainthe point balance condition: (3.40) m∂ t r = qc ∇ ( v · A ) ( t, − q ∇ ϕ ( t, − qc ∂ t A ( t, . To interpret this condition, we note that y = 0 in the moving frame in the aboveequation corresponds to x = r in the resting frame and(3.41) ∂ t A ( t, x ) x = r = ∂ t A ( t, y ) y = − v · ∇ A ( t, y ) y = , hence the point balance condition (3.40) takes in the resting frame the form(3.42) m∂ t r = qc ∇ ( v · A ) ( t, r ) − q ∇ ϕ ( t, r ) − qc ∂ t A ( t, r ) − qc v · ∇ A ( t, r ) . Recall that the expression for the Lorentz force in terms of the EM potentials isgiven by the formula f Lor = − q ∇ ϕ − qc ∂ t A + qc ∇ ( v · A ) − qc ( v · ∇ ) A , therefore the right-hand side of (3.42) coincides with the Lorentz force (1.10).Hence, the point balance condition (3.40) in x -coordinates has the form m∂ t r = f Lor ( t, r ) , (3.43) f Lor ( t, r ) = q E ( t, r ) + qc v × B ( t, r )(3.44) where EM fields E and B are given by (1.11). Hence the point balance conditioncoincides with Newton’s law of motion (1.9) of a charged point subjected to theLorentz force f Lor .We always assume that the trajectory r ( t ) satisfies the point balance condition(3.43), therefore the integrability condition (3.39) after taking into account (3.40)can be written in the form qc ( ∇ ( v · A ( t, y ) − ∇ ( v · A ) ( t, − q c m ∇ ˘A − q ( ∇ ϕ ( t, y ) − ∇ ϕ ( t, − qc ∂ t ( A ∇ ( t, y ) − A ∇ ( t, )) = where the left hand side explicitly vanishes for spatially constant fields A andconstant ∇ ϕ .Now let us discuss the universality of the non-relativistic dynamic balance condi-tions. The conditions (3.27), (3.45) of preservation of the shape | ψ | are universal ,namely they do not depend on the nonlinearity G or on the form factor ˚ ψ . Recallthat when deriving the conditions we set coefficients at ∇ ˚ ψ and ˚ ψ in (3.16) to bezero. We would like to show that this is a necessary requirement for the conditionsto be universal.Equation (3.16) has the form Q ˚ ψ ( r ) + ˚ ψ ′ ( r ) Q · y /r = 0 , where ˚ ψ ( r ) is a form factor, and coefficients Q , Q obviously do not depend on ˚ ψ .If Q · y is not identically zero, we can separate the variables: Q rQ · y = − ∂ r ln ˚ ψ ( r ) , where the left-hand side is ˚ ψ -independent and the right-hand side depends on ˚ ψ .It is impossible, and hence Q · y must be zero as well Q , leading to equations(3.17), (3.18).3.3. Wave-corpuscles in the EM field.
The simplest example of fields ϕ, A forwhich dynamic balance conditions are fulfilled are spatially constant ϕ ( t ) , A ( t );equations (3.27), (3.45) are obviously fulfilled. Now we construct a more generalexample of the fulfillment of the condition (3.45). In this example we prescribearbitrary EM potentials which are linear in x , take a trajectory which satisfiesNewton’s law (3.43), and assume that second and higher order components of theEM potentials expansion at r ( t ) must satisfy certain restrictions. Namely, weassume that A involves a linear in y = x − r part which can be given by anarbitrary 3 × A ( t ) with time-dependent elements:(3.46) A ( t, y ) = A ( t ) y = y · ∇ A ( t, ) . We assume that quadratic and higher order components of the sphere-tangent partof A are set to zero, but allow an arbitrary potential part(3.47) A ( t, y ) = A ( t, ) + A ( t, y ) + A ∇ ( t, y ) , where A ∇ has at least the second order zero at the origin and is potential: A ∇ ( t, y ) = ∇ P with an arbitrary P . For such a field, its potential and sphere-tangent parts aregiven respectively by formulas A ∇ ( t, y ) = ∇ P + ∇ P + ∇ P , (3.48) ˘A ( t, y ) = 12 A ( t ) y − A T1 ( t ) y , (3.49)where(3.50) P = y · A ( t, ) , P ( t, y ) = 12 ( y · A ( t ) y ) , A T1 stands for A transposed, and P is a function with zero of at least the thirddegree at the origin. Note that an action of an anti-symmetric matrix can be writtenusing the cross product, therefore (3.49) can be written as follows:(3.51) ˘A ( t, y ) = 12 ˘B ( t ) × y , where ˘B = ∇ × ˘A . The electric potential ϕ has the form(3.52) ϕ = ϕ ( t, ) + y · ∇ ϕ ( t, ) + ϕ ( t, y ) , where ϕ ( t, ) and ∇ ϕ ( t, ) are given continuously differentiable functions of t, and ϕ ( t, y ) has order two or higher in y and is subject to the condition (3.54) weformulate below.Let us verify the conditions which guarantee that the wave-corpuscle is an exactsolution. Notice that condition (3.27) for ˘A defined by (3.49) is fulfilled. Hence, itis sufficient to satisfy (3.45), which takes the form(3.53) qc ∇ ( v · ∇ P ) − q c m ∇ ˘A − q ∇ ϕ ( t, y ) − qc ∂ t ( ∇ P ( t, y ) + ∇ P ( t, y )) = . To satisfy (3.53), we set(3.54) ϕ ( t, y ) = − qc m ˘A − c ∂ t ( P ( t, y ) + P ( t, y )) + 1 c v · ∇ P . We want now to determine the phase S of the wave-corpuscle using (3.34), (3.35).If P is homogenious of third degree, we obtain(3.55) s p ( t ) = 1 χ Z t (cid:18) m v ( t ) + qc v · A ( t, − qϕ ( t, (cid:19) d t,s p2 ( t, y ) = qc χ Z y · A ( t, y ) sds + qc χ Z y · A ∇ ( t, y ) s d s (3.56) = q c χ y · A ( t, y ) + q c χ y · A ∇ ( t, y ) . In this case the phase function of the wave-corpuscle involves a term s p2 ( t, y ).Formula (3.1) takes the form(3.57) ψ ( t, x ) = e i S ˚ ψ ( | x − r ( t ) | ) , (3.58) S ( t, y ) = m χ v · y + qc χ y · A ( t, ) + s p ( t ) + s p2 ( t, y ) . where y = x − r ( t ) , v = ∂ t r . If the external magnetic field satisfies the following anti-symmetry conditions y · A ( t, y ) = 0 , (3.59) y · A ∇ ( t, y ) = 0then s p2 = 0, and the phase function is linear in y .The above calculations can be summarized in the following statement Theorem 4.
Let the potentials ϕ, A have the form (3.52), (3.46), (3.47) where ϕ ( t, ) and ∇ ϕ ( t, ) are given continuously differentiable functions of t . Supposealso that A ( t, ) is a given continuously differentiable function of t , A ( t ) is anarbitrary × matrix which is continuously differentiably depends on t , ∇ P isa continuously differentiable function of t and y . Let the quadratic part ϕ of thepotential ϕ satisfy (3.54). Let also trajectory r ( t ) satisfies Newton’s equation (3.43)-(3.44). Then the wave-corpuscle defined by formula (3.57)-(3.58) is a solution toNLS equation (1.1). Remark 3.
The above construction does not depend on the nonlinearity G ′ = G ′ a aslong as (1.6) is satisfied. It is also uniform with respect to a > , and the dependenceon a in (3.57) is only through ˚ ψ ( | x − r | ) = a − / ˚ ψ (cid:0) a − | x − r | (cid:1) . Obviously, if ψ ( t, x ) is defined by (3.1) then | ψ ( t, x ) | → δ ( x − r ) as a → . Remark 4.
The form (3.1) of exact solutions is the same as the WKB ansatz in thequasi-classical approach, [41] . The trajectories of the charge center coincide withtrajectories that can be found by applying well-known quasiclassical asymptotics tosolutions of (1.1) if one neglects the nonlinearity. Note though that there are twoimportant effects of the nonlinearity not presented in the standard quasiclassicalapproach. First of all, due to the nonlinearity the charge preserves its shape inthe course of evolution on unbounded time intervals whereas in the linear modelany wavepacket disperses over time. Second of all, the quasiclassical asymptoticexpansions produce infinite asymptotic series which provide for a formal solution,whereas the properly introduced nonlinearity as in (1.6), (3.6) allows one to obtainan exact solution. For a treatment of mathematical aspects of the approach tononlinear wave mechanics based on the WKB asymptotic expansions we refer thereader to [41] , [35] and references therein. Remark 5.
We can use in the definition of a wave-corpuscle (3.1) a radial formfactor ˚ ψ which instead of (1.6) satisfies the eigenvalue problem (3.60) − ∇ ˚ ψ + G ′ ( (cid:12)(cid:12)(cid:12) ˚ ψ (cid:12)(cid:12)(cid:12) )˚ ψ = λ ˚ ψ. Obviously the above equation (3.60) can be considered as the steady state equation(1.6) with a modified nonlinearity G ′ − λ . Note that ψ is a solution of the NLSequation (1.1) with the nonlinearity G ′ if and only if the function e − i χ m λt ψ isa solution of the NLS equation with the nonlinearity G ′ − λ . Therefore the wave-corpuscle solutions constructed in Theorem 4 based on ˚ ψ with the nonlinearity G ′ − λ provide solutions to the NLS equation with the original nonlinearity, one has only toadd to the phase function S an additional term χ m λt . Existence of many solutionsof the nonlinear eigenvalue problems of the form (3.60) was proved in many papers,see [14] , [15] , [18] , [30] . Therefore there are many wave-corpuscle solutions of agiven NLS equation with the same trajectory of motion r ( t ) and the same EMfields which correspond to different form factors. RAJECTORY OF CONCENTRATION 23
Wave-corpuscles as concentrating solutions.
The wave-corpuscles con-structed in Theorem 4 provide an example of concentrating solutions. In order tosee that we have to verify all the requirements of Definition 4. It is important to notethat the dependence on a in (3.1) is only through ˚ ψ ( | x − r | ) = a − / ˚ ψ (cid:0) a − | x − r | (cid:1) where ˚ ψ is a given smooth function. The verification of conditions imposed in theDefinition 4 is straightforward. For instance,¯ ρ n = q Z Ω( ˆr ( t ) ,R n ) a − (cid:12)(cid:12)(cid:12) ˚ ψ (cid:0) a − | x − r | (cid:1)(cid:12)(cid:12)(cid:12) d x (3.61) = q Z Ω( ,R n /a ) ˚ ψ ( | y | ) d y → q Z R ˚ ψ ( | y | ) d y , where R n /a → ∞ according to (2.9) and using that we obtain (2.25) and (2.26).To estimate integral (2.24) we note that according to the definition of momentumdensity (1.16), (1.2)(3.62) P = i χ (cid:20) ∇ ψ ∗ ψ ∗ − ∇ ψψ + 2 i qχ c A (cid:21) ψ ∗ ψ, and for the wave-corpuscles(3.63) P = χ (cid:20) ∇ S − qχ c A (cid:21) ˚ ψ . Since the phase S in (3.58) is a smooth function which does not depend on a , and A satisfies (2.17), the estimate (2.24) can be obtained using (3.61). Using (3.63)and (3.61) we obtain (2.33):(3.64) | Q | ≤ | t − t | max T − ≤ s ≤ T + | ∂ t ˆr | max T − ≤ s ≤ T + Z ∂ Ω n | P | d σ → . Similarly we obtain (2.34), (2.35) and (2.36). Note that | E n (0) − E ∞ (0) | = | E (0) − E ∞ (0) | = 0 and | B n (0) − B ∞ (0) | = | B (0) − B ∞ (0) | = 0 according to(2.19). Using continuity of E ( t, y ) and B ( t, y ) we conclude that | E n − E ∞ | → | B n − B | → n since R n →
0. Therefore filfillment of (2.39) follows from(3.61) and (3.63).To obtain (2.32) we split the tensor T ij into diagonal and non-diagonal partsgiven by (5.21) and (5.22):(3.65) T ij = T iji = j + T iji = j . According to (2.27) Q = Q , = + Q , = where Q , = = Z tt Z ∂ Ω n ¯n i T iji = j d σ d t ′ , Q , = = Z tt Z ∂ Ω n ¯n i T iji = j d σ d t ′ . According to (5.22) to estimate | Q , = | it is sufficient to prove that(3.66) max T − ≤ t ≤ T + Z ∂ Ω n ( χ m |∇ ψ ( t, x ) | + | ψ ( t, x ) | ) d σ → . Using this estimate and (5.22) we obtain(3.67) | Q , = | ≤ | t − t | max T − ≤ s ≤ T + Z ∂ Ω n (cid:12)(cid:12)(cid:12) T iji = j (cid:12)(cid:12)(cid:12) d σ → . To estimate Q , = we use (5.21) Z ∂ Ω n ¯n i T ii d σ = − Z ∂ Ω n χ m ¯n i ˜ ∂ i ψ ˜ ∂ ∗ i ψ ∗ d σ (3.68) + Z ∂ Ω n (cid:20) χ m (cid:18) G (cid:16) | ψ | (cid:17) + (cid:12)(cid:12)(cid:12) ˜ ∇ ψ (cid:12)(cid:12)(cid:12) (cid:19) + i χ (cid:16) ψ ˜ ∂ ∗ t ψ ∗ − ψ ∗ ˜ ∂ t ψ (cid:17)(cid:21) X i ¯n i d σ Since | ψ | = (cid:12)(cid:12)(cid:12) ˚ ψ (cid:12)(cid:12)(cid:12) is a radial function and ¯n i = y i / | y | is odd with respect to y i –reflection, we see that in the above integral(3.69) Z ∂ Ω n G (cid:16) | ψ | (cid:17) ¯n i d σ = 0 . We also note thati χ (cid:16) ψ ˜ ∂ ∗ t ψ ∗ − ψ ∗ ˜ ∂ t ψ (cid:17) = i χ ψ∂ t ψ ∗ − ψ ∗ ∂ t ψ ) + qϕ | ψ | (3.70) = χ | ψ | ∂ t S + qϕ | ψ | where ∂ t S and ϕ are bounded functions in Ω n . Under the assumption (3.66)straightforward estimates produce thati χ Z ∂ Ω n (cid:16) ψ ˜ ∂ ∗ t ψ ∗ − ψ ∗ ˜ ∂ t ψ (cid:17) d σ → , Z ∂ Ω n χ m ¯n i ˜ ∂ i ψ ˜ ∂ ∗ i ψ ∗ d σ → , Z ∂ Ω n χ m (cid:12)(cid:12)(cid:12) ˜ ∇ ψ (cid:12)(cid:12)(cid:12) d σ → T − , T + ]. Hence, if (3.66) holds, we obtain that(3.71) Q , = → , and taking into account (3.67) we conclude that (2.32) holds.Now let us prove that (3.66) holds under certain decay conditions. Since thephase S in (3.58) is bounded and have bounded derivatives in Ω ( ˆr ( t ) , R n ), to obtain(3.66) it is sufficient to estimate surface integrals of (cid:12)(cid:12)(cid:12) ∇ ˚ ψ (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12) ˚ ψ (cid:12)(cid:12)(cid:12) . Obviously, Z ∂ Ω( ˆr ( t ) ,R n ) a − (cid:12)(cid:12)(cid:12) ∇ ˚ ψ (cid:0) a − | x − r | (cid:1)(cid:12)(cid:12)(cid:12) d σ (3.72) = 4 πR n a − (cid:12)(cid:12)(cid:12) ˚ ψ ′ ( R n /a ) (cid:12)(cid:12)(cid:12) = 4 πaR − n θ (cid:12)(cid:12)(cid:12) ˚ ψ ′ ( θ ) (cid:12)(cid:12)(cid:12) , Z ∂ Ω( ˆr ( t ) ,R n ) a − (cid:12)(cid:12)(cid:12) ˚ ψ (cid:0) a − | x − r | (cid:1)(cid:12)(cid:12)(cid:12) d σ =(3.73) = 4 πa − R n (cid:12)(cid:12)(cid:12) ˚ ψ ( R n /a ) (cid:12)(cid:12)(cid:12) = 4 πaR − n θ (cid:12)(cid:12)(cid:12) ˚ ψ ( θ ) (cid:12)(cid:12)(cid:12) , where R n /a = θ → ∞ , R n →
0. We assume that the form factor ˚ ψ ( r ) and itsderivative ˚ ψ ′ ( r ) satisfy the following decay conditions:(3.74) θ (cid:12)(cid:12)(cid:12) ˚ ψ ( θ ) (cid:12)(cid:12)(cid:12) ≤ C as θ → ∞ , RAJECTORY OF CONCENTRATION 25 (3.75) θ (cid:12)(cid:12)(cid:12) ˚ ψ ′ ( θ ) (cid:12)(cid:12)(cid:12) ≤ C as θ → ∞ , and we take such a n , R n that(3.76) a n R − n → , a n → , R n → . Under these assumptions we conclude that (3.66) is fulfilled.Summing up the above arguments we obtain the following statement:
Theorem 5.
Let ˚ ψ satisfy (3.74), (3.75) and a n , R n satisfy (3.76). Let ψ = ψ n be wave-corpuscle solutions of the NLS equation constructed in Theorem 4. Thenthe sequence ψ n concentrates at the trajectory ˆr ( t ) = r ( t ) . Concentration of asymptotic solutions
From the proofs of Section 2 one can see that in the derivation of the Newtoniandynamics we use only the conservation laws (1.15) and (1.18). Since we use onlyasymptotic properties, it is natural to consider fields which satisfy NLS equationsand the conservation laws not exactly, but approximately. Namely, we assume nowthat the conservation laws (1.15) and (1.18) are replaced by(4.1) ∂ t ρ + ∂ t ρ ′ + ∇ · J + ∇ · J ′ = 0 , (4.2) ∂ t P + ∂ t P ′ + ∂ i T ij + ∂ i T ij ′ = f + f ′ where charge density ρ , current density J and tensor elements T ij are defined by(1.13),(1.14) and (5.22) in terms of given functions ψ, ϕ, A and ρ ′ , J ′ P ′ T ij ′ , andquantities f ′ are perturbation terms which vanish as n → ∞ . We show now how tomodify the proofs of statements in Section 2 on concentrating solutions to obtainsimilar statements on concentrating asymptotic solutions.In the proof of Lemma 3 equation (2.51) is replaced by (4.2). This leads toreplacement of the equation (2.53) by the equation Z Ω( ˆr ( t ) ,R n ) P ( t ) d x − Z Ω( ˆr ( t ) ,R n ) P ( t ) d x − Q (4.3) − Z tt Z Ω n f d x d t ′ + Q + Q ′ = 0 , with Q ′ = Z Ω( ˆr ( t ) ,R n ) P ′ ( t ) d x − Z Ω( ˆr ( t ) ,R n ) P ′ ( t ) d x (4.4) − Z tt Z ∂ Ω n P ′ ˆv · ¯n d σ d t ′ + Z tt Z Ω n ∂ i T ij ′ d x d t ′ − Z tt Z Ω n f ′ d x d t ′ . If(4.5) Q ′ → Q = Q + Q ′ − Q , and we obtain thefollowing lemma: Lemma 5.
Let the conservation law (1.18) be replaced by (4.2) with condition (4.5)satisfied. Then the statement of Lemma 3 remains true: (4.6) Z Ω n P n ( t ) d x = Z tt Z Ω n f d x d t ′ + p ∞ + Q
006 ANATOLI BABIN AND ALEXANDER FIGOTIN where (4.7) Q → as n → ∞ uniformly on [ T − , T + ] . Sufficient conditions for fulfillment of (4.5) are the following limit relations(4.8) Z Ω( ˆr ( t ) ,R n ) P ′ ( t ) d x → , (4.9) Z tt Z ∂ Ω n P ′ ˆv · ¯n d σ d t ′ → , (4.10) Z tt Z ∂ Ω n ¯n i T ij ′ d σ d t ′ → , (4.11) Z tt Z Ω n f ′ d x d t ′ → T − , T + ] . Let us take a look at the proof of Lemma 4. Since we replace the continuityequation (1.15) by (4.1), the equation (2.58) involves now additional terms:(4.12) Z Ω n ∂ t (( x − r ) ρ ) d x + ∂ t r Z Ω n ρ d x + Z ∂ Ω n ( x − r ) n · J d x + Q ′ = Z Ω n J d x , with(4.13) Q ′ = Z Ω n ∂ t (( x − r ) ρ ′ ) d x + ∂ t r Z Ω n ρ ′ d x + Z ∂ Ω n ( x − r ) n · J ′ d x − Z Ω n J ′ d x . We assume that(4.14) Q ′ → Lemma 6.
Let the continuity equation (1.15) be replaced by (4.1) with condition(4.14) fulfilled. Then the statement of Lemma 4 is true.
Note that according to (2.59) sufficient conditions for fulfillment of (4.14) areas follows:(4.15) Z Ω n ρ ′ d x → , (4.16) Z ∂ Ω n ( x − r ) n · J ′ d x → , (4.17) Z ∂ Ω n ( x − r ) ˆv · ¯n ρ ′ d σ → . Now we consider the proof of Lemma 2. Equation (2.47) is replaced by(4.18) ¯ ρ ( t ) − ¯ ρ ( t ) − Z tt Z ∂ Ω n ˆv · n ρ d x d t ′ + Z tt Z ∂ Ω n n · J d x d t ′ + Q ′ = 0 , RAJECTORY OF CONCENTRATION 27 where(4.19) Q ′ = ¯ ρ ′ ( t ) − ¯ ρ ′ ( t ) − Z ∂ Ω n ˆv · n ρ ′ d x + Z ∂ Ω n n · J ′ d x . The proof of Lemma 2 is preserved if we assume that(4.20) Q ′ → . Lemma 7.
Let the continuity equation (1.15) be replaced by (4.1) with condition(4.20) fulfilled. Then the statement of Lemma 2 is true.
Now we collect all the assumptions required for exact solutions in Section 2 thatremain valid for asymptotic solutions in the following definition:
Definition 5 (Concentrating asymptotic solutions) . We assume all conditions ofDefinition 4 except condition (i), which is replaced by the following weaker condi-tion: conservation laws (4.2) and (4.1) are fulfilled and conditions (4.14), (4.5),(4.20) hold. Then we say that asymptotic solutions of the NLS equation (1.1) con-centrate at ˆr ( t ) . We call ˆr ( t ) a concentration trajectory of the NLS equation inasymptotic sense if there exists a sequence of asymptotic solutions which concen-trates at ˆr ( t ) . Using Lemmas 5 and 6 instead of Lemmas 3 and 4 we obtain statements ofTheorems 2 and 3 under the assumption that asymptotic solutions ψ of the NLSequation (1.1) concentrate at ˆr ( t ). In particular, we obtain Theorem 6.
Assume that potentials ϕ ( t, x ) , A ( t, x ) are defined and twice contin-uously differentiable in a domain D ⊂ R × R , the trajectory ( t, ˆr ( t )) lies in thisdomain and the limit potentials ϕ ∞ , A ∞ are the restriction of fixed potentials ϕ , A as in (2.20). Let EM fields E ( t, x ) , B ( t, x ) be determined in terms of thepotentials by formula (1.11). Let asymptotic solutions ψ of the NLS equation (1.1)asymptotically concentrate at ˆr ( t ) . Then the trajectory ˆr satisfies Newton’s law ofmotion (2.71). Point trajectories as trajectories of asymptotic concentration .
Letus consider NLS quation (1.1) in a domain D ⊂ R × R and assume that potentials ϕ ( t, x ) and A ( t, x ) are defined and twice continuously differentiable in domain D .Let us consider equations (3.43)-(3.44) describing Newtonian dynamics of a pointcharge in EM field. Let us consider a solution r ( t ) of equations (3.43)-(3.44), andassume that the trajectory ( t, r ( t )) lies in D on the time interval T − ≤ t ≤ T + .Now we construct asymptotic solutions of the NLS which concentrate at r ( t ).As a first step we find the linear part of ϕ ( t, x ), A ( t, x ) at r ( t ) as in (2.20) ϕ ∞ ( t, x ) = ϕ ( t, r ) + ( x − r ) ∇ ϕ ( t, r ) , (4.21) A ∞ ( t, x ) = A ( t, r ) + ( x − r ) ∇ A ( t, r ) , As a second step we construct the auxiliary potentials ϕ aux ( t, x ) = ϕ ( t, r ) + ( x − r ) ∇ ϕ ( t, r ) + ϕ ( t, x − r ) , (4.22) A aux ( t, x ) = A ( t, r ) + ( x − r ) ∇ A ( t, r ) , where ϕ is determined by (3.54) and ϕ ( t, x − r ) is quadratic with respect to( x − r ). The wave-corpuscle ψ described in Theorem 4 is a solution of the NLS equa-tion with the potentials ( ϕ aux , A aux ), and hence it exactly satisfies the correspond-ing conservation laws. From fulfillment of (1.15) and (1.18) for P ( ϕ aux , A aux ) ,J ( ϕ aux , A aux ) , T ij ( ϕ aux , A aux ), f ( ϕ aux , A aux ) we obtain fulfillment of (4.1), (4.2)with(4.23) P ′ = P ( ϕ, A ) − P ( ϕ aux , A aux ) , (4.24) J ′ = J ( ϕ, A ) − J ( ϕ aux , A aux ) , (4.25) f ′ = f ( ϕ, A ) − f ( ϕ aux , A aux ) , (4.26) T ij ′ = T ij ( ϕ, A ) − T ij ( ϕ aux , A aux ) . The expression for ρ does not depend on the potentials, therefore(4.27) ρ ′ = 0 . Now we need to verify that conditions (4.14), (4.5), (4.20) hold for the wave-corpuscle ψ .From (1.16), (3.63) we see that(4.28) P ( ϕ, A ) − P ( ϕ aux , A aux ) = − q c ( A − A aux ) ψ ∗ ψ, and according to (1.17)(4.29) J ( ϕ, A ) − J ( ϕ aux , A aux ) = − q m c ( A − A aux ) ψ ∗ ψ. From the expression for the Lorentz density (1.19) we obtain f ( ϕ, A ) − f ( ϕ aux , A aux ) = ρ ( E − E aux ) + 1c ( J × B − J aux × B aux ) == ρ ( E − E aux ) + 1c (( J − J aux ) × B + J aux × ( B − B aux )) . Using (3.63) and (1.17) we rewrite expression for f ′ in the form f ′ = ρ ( E − E aux )(4.30)+ 1c (cid:18) − q m c˚ ψ ( A − A aux ) × B + χ qm ˚ ψ (cid:20) ∇ S − qχ c A aux (cid:21) × ( B − B aux ) (cid:19) . To estimate the difference of tensor elements (4.26) we use (5.22) and (5.21). Notethat according to the construction of ϕ aux and A aux the differences ϕ − ϕ aux and A − A aux have the second order zero at r ( t ). Hence the diffferences E − E aux and B − B aux have a zero of the first order at x = r :(4.31) | A − A aux | ≤ C | x − r | , | ϕ − ϕ aux | ≤ C | x − r | , (4.32) | E − E aux | ≤ C | x − r | , | B − B aux | ≤ C | x − r | , and they are vanishingly small in Ω ( r ( t ) , R n ). Now we estimate terms which enter(4.4), (4.13), (4.19). Lemma 8.
Let P ′ , J ′ , f ′ , T ij ′ be defined by (4.23)-(4.26), conditions (3.74), (3.75)and (3.76) fulfilled. Then (4.8)-(4.11) and (4.15)-(4.17) and (4.20) hold. RAJECTORY OF CONCENTRATION 29
Proof.
The proof is based on inequalities (4.32) and (4.31). To obtain (4.8) we use(4.28) and (3.61): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω( r ( t ) ,R n ) P ′ ( t ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z Ω( r ( t ) ,R n ) | x − r | ˚ ψ a ( | x − r | ) d x ≤ CR n Z Ω( ,R n /a ) ˚ ψ a ( | y | ) d y ≤ C R n , and we obtain (4.8) for R n →
0. To obtain (4.9) we observe that according to(4.28) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω( r ( t ) ,R n ) P ′ ˆv · ¯n d σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.33) ≤ C Z ∂ Ω( r ( t ) ,R n ) | x − r | ˚ ψ a ( | x − r | ) d σ = 4 πCR n a − ˚ ψ a ( R n /a )= 4 πCR n θ ˚ ψ a ( θ ) → T ii involving G do not dependon ( ϕ, A ), and hence G does not enter T ij ′ . According to (5.22), (5.21), (3.70)and (1.2) T ij ( ϕ, A ) is a quadratic function of potentials and T ij ′ equals sum ofterms, every of which involves factors A − A aux or ϕ − ϕ aux which satisfy (4.31)and also factors ψ ∇ ψ ∗ or ∇ ψψ ∗ or ψψ ∗ . The factors A − A aux or ϕ − ϕ aux in Ω ( r ( t ) , R n ) produce coefficient CR n . All the terms are easily estimated, forexample an elementary inequality Z ∂ Ω( r ( t ) ,R n ) (cid:12)(cid:12)(cid:12) ˚ ψ a (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ ˚ ψ a (cid:12)(cid:12)(cid:12) d σ ≤ Z ∂ Ω( r ( t ) ,R n ) (cid:12)(cid:12)(cid:12) ˚ ψ a (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∇ ˚ ψ a (cid:12)(cid:12)(cid:12) d σ allows to use (3.72)-(3.73), and we obtain that Z ∂ Ω( r ( t ) ,R n ) (cid:12)(cid:12)(cid:12) ˚ ψ a (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ ˚ ψ a (cid:12)(cid:12)(cid:12) d σ → . Therefore (4.10) is fulfilled. Since ρ ′ = 0 (4.15) and (4.17) are fulfilled, (4.16)follows from 4.8) and (1.17). To check condition (4.20) we note that ρ ′ = 0 andafter using (1.17) we estimate the surface integral in (4.19) involving J ′ similarlyto (4.33). (cid:3) Theorem 7.
Let potentials ϕ ( t, x ) , A ( t, x ) be twice continuously differentiable ina domain D . Let form factor ˚ ψ satisfy conditions (3.74), (3.75). Let sequences a n , R n satisfy (3.76). Let r ( t ) be a solution of equations (3.43)-(3.44) with tra-jectory in D . Then wave-corpuscles constructed in Theorem 5 based on r ( t ) andpotentials ϕ aux , A aux given by (4.22) provide asymptotic solutions in the sense ofDefinition 5 that concentrate at r ( t ) . As a corollary we obtain the following theorem.
Theorem 8.
Let ϕ ( t, x ) , A ( t, x ) are defined and twice continuously differentiablein a domain D . Then a trajectory r ( t ) which lies in D is a concentration trajectoryof asymptotic solutions of NLS equation (1.1) if and only if it satisfies Newton’slaw (3.43) with the Lorentz force defined by (3.44). Since asymptotic solutions in the sense of Definition 5 in Theorem 7 are con-structed as wave-corpuscles, we obtain the following corollary.
Corollary 1.
If a sequence of concentrating asymptotic solutions concentrates at atrajectory r ( t ) , then there exists a sequence of concentrating wave-corpuscle asypmp-totic solutions which concentrate at the same trajectory. Appendix 1: Lagrangian field formalism for nonlinearSchr¨odinger Equation
To derive the conservation laws for the NLS equation it is convenient to userelativistic notation. We introduce the following 4-differential operators(5.1) ∂ µ = (cid:18) ∂ t , ∇ (cid:19) , ∂ µ = (cid:18) ∂ t , −∇ (cid:19) , where the indices µ take four values µ = 0 , , , ∂ = ∂ t . We also denote(5.2) ψ ,µ = ∂ µ ψ, ψ ,µ = ∂ µ ψ. The EM 4-potential is given by the formula(5.3) A = A µ = ( ϕ, A ) , where ϕ ( t, x ) , A ( t, x ) are given potentials, and the EM power tensor is defined asfollows:(5.4) F µν = ∂ µ A ν − ∂ ν A µ . The covariant derivatives which involve the EM fields are defined by (1.2), andcorresponding 4-differential operators have the form(5.5) ˜ ∂ µ = (cid:18)
1c ˜ ∂ t , ˜ ∇ (cid:19) , ˜ ∂ µ = (cid:18)
1c ˜ ∂ t , − ˜ ∇ (cid:19) . We also use the following notation for covariant derivatives(5.6) ψ ; µ = ˜ ∂ µ ψ, ψ ; µ ∗ = ˜ ∂ µ ∗ ψ ∗ , ψ ; µ = ˜ ∂ µ ψ, ψ ∗ ℓ ; µ = ˜ ∂ ∗ µ ψ ∗ . Obviously,(5.7) ψ ; µ = ψ ,µ + i qχ c A µ ψ. The NLS equation (1.1) is the Euler-Lagrange field equation for the following La-grangian density:(5.8) L = i χ h ψ ∗ ˜ ∂ t ψ − ψ ˜ ∂ ∗ t ψ ∗ i − χ m h ˜ ∇ ψ ˜ ∇ ∗ ψ ∗ + G ( ψ ∗ ψ ) i where G ( s ) is given by (3.8). The 4-current J ν for the Lagrangian is defined bythe formula(5.9) J ν = − i qχ (cid:18) ∂L∂ψ ; ν ψ − ∂L∂ψ ∗ ; ν ψ ∗ (cid:19) , it can be written in the form(5.10) J = J ν = (c ρ, J ) RAJECTORY OF CONCENTRATION 31 where ρ, J are given by (1.13),(1.14). The Lagrangian (5.8) and the NLS equation(1.1) are gauge invariant, that is invariant with respect to the multiplication of ψ by e i γ with real γ :(5.11) L (cid:0) e i γ ψ, e i γ ψ ; µ , e − i γ ψ ∗ , e − i γ ψ ∗ ℓ ; µ (cid:1) = L (cid:0) ψ, ψ ; µ , ψ ∗ , ψ ∗ ; µ (cid:1) . If we take derivative of the above conditon (5.11) with respect to γ at γ = 0 , weobtain the following structural restriction on the Lagrangian L :(5.12) ∂L∂ψ ; µ ψ ; µ − ∂L∂ψ ∗ ; µ ψ ∗ ; µ + ∂L∂ψ ψ ℓ − ∂L∂ψ ∗ ψ ∗ ℓ = 0 . Direct verification shows that if a Lagrangian L satisfies the above structural con-dition, then the current defined by (5.9) for a solution of the Euler-Lagrange fieldequation (NLS in our case) satisfies the continuity equation(5.13) ∂ ν J ν = 0which can be written in the form (1.15).We introduce the energy-momentum tensor (EnMT) (see [13] for the generaltheory) for the NLS by the following formula:(5.14) T µν = ∂L∂ψ ; µ ψ ; ν + ∂L∂ψ ∗ ; µ ψ ; ν ∗ − g µν L, where g µν is the Minkowski metric tensor, that is(5.15) g µν = 0 for µ = ν, g = 1 , and g jj = − j = 1 , , . Proposition 1.
Let EnMT T µν be defined by formula (5.14) for a solution of theNLS equation (1.1). Then T µν satisfies the following EnMT conservation law (5.16) ∂ µ T µν = f ν where f ν is the Lorentz force density defined by the formula (5.17) f ν = 1c J µ F νµ with J µ and F νµ defined by respectively by (5.9) and (5.4).Proof. According to (5.7) and (5.9 the definition (5.14) can be written in the form T µν = ∂L∂ψ ; µ ψ ,ν + ∂L∂ψ ; µ i qχ c A ν ψ + ∂L∂ψ ∗ ; µ ψ ,ν ∗ − ∂L∂ψ ∗ ; µ i qχ c A ν ψ ∗ − g µν L = ∂L∂ψ ,µ ψ ,ν + ∂L∂ψ ∗ ,µ ψ ,ν ∗ − J µ A ν − g µν L. where J ν is defined by (5.9). We differentiate the above expression and use thecontinuity equation (5.13): ∂ µ T µν = ∂ µ (cid:18) ∂L∂ψ ; µ ψ ,ν (cid:19) + ∂ µ (cid:18) ∂L∂ψ ∗ ; µ ψ ,ν ∗ (cid:19) − ∂ µ ( J µ A ν ) − ∂ µ ( g µν L )= ∂ µ ∂L∂ψ ; µ ψ ,ν + ∂L∂ψ ; µ ∂ µ ψ ,ν + ∂ µ ∂L∂ψ ∗ ; µ ψ ,ν ∗ + ∂L∂ψ ∗ ; µ ∂ µ ψ ,ν ∗ − J µ ∂ µ A ν − g µν ∂ µ ( L ) . The NLS equation (1.1) can be written in the form(5.18) ∂L∂ψ ∗ − ∂ µ ∂L∂ψ ∗ ,µ = 0 , where the Lagrangian L is considered as a function of ψ, ψ ∗ , ψ ∗ ,µ ψ ∗ ,µ , and of thespatial and time variables which enter through A ν . Using (5.18) together with itsconjugate we obtain that ∂ µ T µν = − J µ ∂ µ A ν + (cid:20) ∂L∂ψ ∂ ν ψ + ∂L∂ψ ∗ ∂ ν ψ ∗ + ∂L∂ψ ∗ ,µ ∂ ν ψ ∗ µ + ∂L∂ψ ,µ ∂ ν ψ µ − ∂ ν ( L ) (cid:21) Note that the expression in brackets equals the partial derivative ∂ ν L (we denote ∂ ν ( L ) the complete derivative of L and ∂ ν L the partial one). Therefore ∂ µ T µν = − J µ ∂ µ A ν + ∂ ν L, Note that the partial derivative ∂ ν L can be evaluated as follows: ∂ ν L = i qχ c (cid:18) ∂L∂ψ ; µ ψ − ∂L∂ψ ∗ ; µ ψ ∗ (cid:19) ∂ ν A µ = − J µ ∂ ν A µ = − J µ ∂ ν A µ . Hence, ∂ µ T µν = − J µ ∂ µ A ν + 1c J µ ∂ ν A µ , yielding the EnMT conservation law (5.16). (cid:3) The entries of the EnMT can be interpreted as energy and momentum densities u, p j respectively, namely u = T = χ m (cid:20)(cid:12)(cid:12)(cid:12) ˜ ∇ ψ (cid:12)(cid:12)(cid:12) + G (cid:16) | ψ | (cid:17)(cid:21) , (5.19) p j = T j = i χ (cid:16) ψ ˜ ∂ ∗ j ψ ∗ − ψ ∗ ˜ ∂ j ψ (cid:17) . (5.20)The formula for the momentum density can be written in the form (1.16). Theproportionality (1.17) is a specific property of the NLS Lagrangian and does nothold in a general case. Remaining entries of the EnMT take the form T j = − χ m (cid:16) ˜ ∂ t ψ ˜ ∂ ∗ j ψ ∗ + ˜ ∂ ∗ t ψ ∗ ˜ ∂ j ψ (cid:17) , j = 1 , , , (5.21) T ii = u − χ m ˜ ∂ i ψ ˜ ∂ ∗ i ψ ∗ + i χ (cid:16) ψ ˜ ∂ ∗ t ψ ∗ − ψ ∗ ˜ ∂ t ψ (cid:17) , and for i = j, i, j = 1 , , . (5.22) T ij = − χ m (cid:16) ˜ ∂ i ψ ˜ ∂ ∗ j ψ ∗ + ˜ ∂ j ψ ˜ ∂ ∗ i ψ ∗ (cid:17) . Note that the Lorentz force density f ν in (5.16) can be written in the form(5.23) f ν = 1c J µ F νµ = (cid:0) f , f (cid:1) = (cid:18) J · E , ρ E + 1c J × B (cid:19) where, in particular, the momentum equation has the form (1.18). Remark 6.
Note that the derivation of the conservation law (5.16) is valid for anylocal solution ψ of the NLS equation which has continuous second derivatives, G is continuously differentiable on the range of | ψ | and the nonlinear term G ( ψψ ∗ ) which enters (5.8) has continuous first derivatives. Note that the derivatives G ( ψψ ∗ ) RAJECTORY OF CONCENTRATION 33 Appendix 2: Splitting of a field into potential andsphere-tangent parts
Here we describe splitting of a general vector field into sphere-tangent and gradi-ent components which was used in derivation of the properties of wave-corpuscles.
Lemma 9.
Any continuously differentiable vector field V ( y ) can be uniquely split-ted into a gradient field ∇ P and a sphere-tangent field ˘V (6.1) V = ∇ P + ˘V . where P is continuously differentiable, ˘V is continuous and satisfies the orthogo-nality condition (3.20), namely ˘V · y = .Proof. To determine P we multiply (6.1) by y and obtain y · V = y · ∇ P + y · ˘V = y · ∇ P The directional derivative y · ∇ can be written as r∂ r and using notation y = Ω r where | Ω | = 1, r = | y | we rewrite the above equation in the form Ω · V = ∂ r P ( Ω r ) . Note that P is defined from the above equation up to a function C ( Ω ); since theonly function of this form which is continuous at the origin must be a constant, P is defined uniquely modulo constants. We can find P by integration:(6.2) P ( y ) = Z | y | Ω · V ( Ω r ) d r = Z | y | | y | y · V (cid:18) | y | y r (cid:19) d r. If the potential P is defined by (6.2) then y · ˘V = y · V − y ·∇ P satisfies (3.20). (cid:3) Now we provide some explicit formulas for polynomial fields V .To obtain an explicit expression for P , we assume that V and P are expandedinto series of homogenious expressions:(6.3) V ( y ) = X j V j ( y ) , P = X j P j ( y ) , ˘V ( y ) = X j ˘V j ( y ) , where V j ( ζ y ) = ζ j A j ( y ) , P j ( ζ y ) = ζ j ϕ j ( y ) . For a j -homogenious V j we have P j +1 ( y ) = Z | y | | y | y · V j (cid:18) | y | y r (cid:19) d r = Z | y | | y | | y | j y · V j ( y ) r j d r, implying(6.4) P j +1 ( y ) = 1( j + 1) | y | j +1 y · V j ( y ) | y | j +1 = 1 j + 1 y · V j ( y ) . In particular, the zero order term V corresponds to(6.5) P ( y ) = V (0) · y , ˘V ( y ) = 0 . The first order one corresponds to(6.6) P ( y ) = 12 V ( y ) · y and(6.7) ∇ P ( y ) = 12 V ( y ) + 12 V T1 ( y ) . Obviously, ∇ P ( y ) coincides with the symmetric part of the linear transformation V ( y ), and(6.8) ˘V ( y ) = 12 V ( y ) − V T1 ( y )coincides with the anti-symmetric part. For higher values of j we have ∇ P j +1 ( y ) = 1 j + 1 ∇ ( y · V j ( y )) . Using vector calculus we obtain ∇ ( y · V j ( y )) = ( y · ∇ ) V j + ( V j · ∇ ) y + y × ( ∇ × V j ) + V j × ( ∇ × y )= ( y · ∇ ) V j + V j + y × ( ∇ × V j )where, by Euler’s identity for homogenious functions,( y · ∇ ) V j ( y ) = j V j ( y ) . Hence ∇ P j +1 ( y ) = 1 j + 1 ∇ ( y · V j ( y )) = 1 j + 1 (( y · ∇ ) V j + V j + y × ( ∇ × V j )) =1 j + 1 ( j V j + V j + y × ( ∇ × V j )) = V j + 1 j + 1 y × ( ∇ × V j ) , (6.9) ˘V j ( y ) = V j ( y ) − ∇ P j +1 ( y ) = − j + 1 y × ( ∇ × V j ) . Acknowledgment.
The research was supported through Dr. A. Nachman ofthe U.S. Air Force Office of Scientific Research (AFOSR), under grant numberFA9550-11-1-0163.
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Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, U.S.A.
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