Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation

One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the 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 nothing about the laws of motion of material points from the standpoint of "classical field theory." For the mastery of this problem, however, no special physical hypothesis is needed, but "only" the solution of certain mathematical problems". In this paper we treat this circle of problems. Namely, we derive here point dynamics governed by Newton's law from wave dynamics using concepts and methods of the classical field theory. That continues our study of a neoclassical model of electric charges which is designed to provide an accurate description of charge interaction with the electromagnetic field from macroscopic down to atomic scales, [7]- [11].
In this paper we derive Newtonian dynamics for localized solutions of Nonlinear Schrodinger equations (NLS) in the asymptotic limit when the solutions converge to delta-functions. The derivation is based on a method of concentrating solutions which was developed in our previous paper [11] where we derived the relativistic dynamics 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 a relation between wave and point dynamics is by means of the WKB method, see for instance [41], [44,Sec. 7.1]. We remind that the WKB method is based on the quasiclassical ansatz for solutions to a hyperbolic partial differential equation and their asymptotic expansion. The leading term of the expansion satisfies the eikonal equation, and wavepackets and their energy propagate along the so-called characteristics of this equations. Consequently, these characteristics represent the point dynamics and are determined from the corresponding system of ODEs which can be interpreted as a law of motion or a law of propagation. The construction of the characteristics involves only local data. Asymptotic derivation of the Newtonian dynamics which does not rely on the fast oscillation of solutions as the WKB method does was developed for soliton-like solutions of the Nonlinear Schrodinger and 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 the soliton center relies on the asymptotic analysis of an ansatz structure of soliton-like solutions. The proposed here approach also relates waves governed by certain PDE's in asymptotic regimes to the point dynamics but it differs fundamentally from the mentioned methods. In particular, our approach is not based on any specific ansatz and it is not not entirely local but rather it is semi-local. Our semi-local method is in the spirit of the Ehrenfest theorem and is based on an analysis of a sequence of concentrating solutions which are not subjected to structural conditions. Compared to the mentioned approaches which use the WKB method or soliton-like structure of solutions, our arguments only use very mild assumptions of a localization of solutions in a small neighborhood of the trajectory combined with the systematic use there of integral identities derived from the energy-momentum and density conservation laws for the NLS. Our method allows to derive Newton's law under minimal restrictions on the time and spatial dependence of the variable coefficients and 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 In the above equation ψ = ψ (t, x) is a complex valued wave function, G ′ a is a real valued nonlinearity, and the covariant differention operators∂ t and∇ are defined by where ϕ (t, x), A (t, x) are given twice differentiable functions of time t and spatial variables x ∈ R 3 interpreted as potentials of external electric and magnetic fields. The quantity q |ψ| 2 is naturally interpreted as the charge density with q being the value 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 resting and uniformly moving charges. Newton's law involves acceleration, and to produce an accelerated motion non-zero potentials are required. One may expect pointlike dynamics for strongly localized solutions, and the nonlinearity G ′ provides existence of localized solutions of the NLS equation. The only condition imposed on the nonlinearity (in addition to natural continuity assumptions, see Condition 1) is the existence of a positive radial solution ψ 1 =ψ 1 (|x|) of the steady-state equation for a free charge: We assume thatψ (|x|) is twice continuously differentiable for all x ∈ R 3 and is square integrable, that is Importantly, we assume that the nonlinearity depends explicitly on the size parameter a > 0 as follows: (1.5) G ′ a (s) = a −2 G ′ 1 a 3 s , s > 0. Note that the steady-state equation with a general a > 0 takes the form )ψ a = 0 and has the following solution: (1.7)ψ a (r) = a −3/2ψ 1 a −1 r , r = |x| ≥ 0, whereψ 1 is the solution to equation (1.3) where a = 1. Obviouslyψ a satisfies the normalization condition (1.4) with the same υ 0 . According to (1.7) a can be interpreted as a natural size parameter which describesψ a . A typical example of the ground stateψ a is the Gaussianψ a = a −3/2 e −r 2 /a 2 corresponding to a logarithmic nonlinearity G ′ a discussed in Example 2. Evidently, υ −1 0ψ 2 a converges to Dirac's delta-function δ (x) as a → 0. We study the behavior of localized solutions of the NLS equation in the asymptotic regime a → 0.
The crucial role in our analysis is played by the key concept of concentrating solutions. Very roughly speaking, solutions ψ = ψ n concentrate at a given trajectorŷ r (t) if their charge densities q |ψ| 2 (x, t) restricted to R n -neighborhoods Ω n ofr (t) locally converge to q ∞ (t) δ (x −r (t)) as (1.8) a n → 0, R n → 0, a n /R n → 0, 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 the third intermediate spatial scale R n which can be called the confinement scale and we assume that ψ n asymptotically vanish at the boundary ∂Ω n = {|x −r (t)| = R n } of the neighborhood Ω n . According to (1.8) R n /a → ∞, therefore this assumption is quite natural for solutions with typical spatial scale a which are localized at r. 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 =r (t). The exact definition 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| 2 and the momentum density P should be bounded; (2) surface integrals over the spheres ∂Ω n of certain quadratic expressions which involve ψ and its first order derivatives and originate from the elements of the energy-momentum tensor of the NLS equation tend to zero as R n → 0, a n /R n → 0. Details of these assumptions can be found in Definitions 4 and 5.
A concise formulation of our main result, Theorem 6, is as follows. We prove that if a sequence of asymptotic solutions of the NLS equation (1.1) concentrates at a trajectoryr (t) then this trajectory must satisfy the Newton equation 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 Notice that that according to Newton's equation (1.9) the parameter m entering the 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 Theorem 6. First of all, the assumptions imposed on the concentrating solutions are not restrictive. Indeed, the solutions are local and have only to be defined in a tubular neighborhood of the trajectory and no initial or boundary conditions are imposed on them. Second of all, only the charge and the momentum conservation laws are used in the derivation of the theorem. Since only conservation laws are used for the argument, the asymptotic solutions for the NLS equation are introduced as functions ψ for which the charge and the momentum conservation laws hold approximately. Namely, certain integrals which involve quantities that enter the conservation laws vanish in an asymptotic limit, see Definition 5 for details. Remarkably, such mild restrictions still completely determine all possible trajectories of concentration, that is the trajectoryr (t) is uniquely defined by the initial datar (t 0 ) and ∂ tr (t 0 ) as a solution of (1.9).
Equation (1.9) yields a necessary condition for solutions of the NLS equation to concentrate at a trajectory. To obtain this necessary condition we have to derive the point dynamics governed by an ordinary differential equation (1.9) from the dynamics of waves governed by a partial differential equation. The concept of "concentration of functions at a given trajectoryr (t)", see Definition 4, is the first step in relating spatially localized fields ψ to point trajectories. The definition of concentration of functions has a sufficient flexibility to allow for general regular trajectoriesr (t) and plenty of functions are localized about the trajectory. But if a sequence of functions concentrating at a given trajectory also satisfy or asymptotically satisfy the conservation laws for the NLS equation then, according to Theorems 3 and 6, 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 narrow tubular neighborhood of the trajectory of radius R n and consider adjacent charge centers (1.12) r n (t) = 1 ρ n |x−r|≤Rn xq |ψ n | 2 dx,ρ n = |x−r|≤Rn q |ψ n | 2 dx, of the concentrating solutions. Then we infer integral equations for the restricted momentum and for the adjacent centers from the momentum conservation law and the continuity equation, and pass to the limit as R n → 0, a n → 0 as is shown in the proofs of Theorems 1-3. Note that the determination of the restricted charge densitȳ ρ and a similar momentum density involves integration over a large relative to a spatial domain of radius R n , R n >> a. Therefore it is natural to call our method of determination of the point trajectories semi-local. This semi-local feature applied to the nonlinear Klein-Gordon equation in [10], [11] allowed us to derive Einstein's relation between mass and energy with the energy being an integral quantity. If the form factorψ 1 (θ) decays fast enough as θ → ∞ (faster than θ −2 ), we also prove the converse statement (Theorem 8) to Theorem 6. Namely, ifr (t) is a solution of equation (1.9) then it is a trajectory of concentration for a sequence of asymptotic solutions to the NLS. The proof of this theorem is based on an explicit construction of wave-corpuscle solutions of the NLS of the form ψ (t, x) = e iSψ (x −r) for a general enough class of potentials ϕ and A with a certain phase function S. We choose auxiliary potentials ϕ aux and A aux from this class to approximate ϕ and A atr (t). The wave-corpuscle solutions constructed for the auxiliary potentials ϕ aux and A aux turn out to form a sequence of concentrating asymptotic solutions of the NLS. Therefore, for given twice continuously differentiable potentials ϕ and A, a trajectory is a trajectory of concentration of asymptotic solutions of the NLS if and only if this trajectory satisfies Newton's law (1.9). The phase function S of the wave-corpuscle can be interpreted as the phase of de Broglie wave (see [7] for details). Hence, the described relation between Newtonian trajectories and concentrating asymptotic solutions can be interpreted as what is known as the wave-particle duality.
The paper's structure is as follows. In the following Subsection 1.1 we define the charge density, current density and momentum density for a solution of NLS and write down corresponding conservation laws. The customary field theory derivation of the conservation laws is given in Appendix 1. In Section 2 we describe the class of trajectories and in Subsection 2. In Section 3 we consider "wave-corpuscles" that exactly preserve their shape in an accelerated motion and describe a class of potentials ϕ,A which allow for such a motion. In Subsection 3.4 we prove that the wave corpuscles provide an example of concentrating solutions to the NLS equation (Theorem 5). In Section 4 we provide an 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 satisfies Newton's law then it is a trajectory of concentration of asymptotic solutions of the NLS equation.
1.1. Conservation laws for NLS. The charge density ρ and current density J are defined by the formulas 14) and for solutions of (1.1) they satisfy the continuity equation: One can verify the validity of the continuity equation (1.15) directly by simply multiplying (1.1) by ψ * and taking the imaginary part. An alternative derivation based on the Lagrangian field formalism is provided in Section 5. The momentum density is defined by the formula and it is proportional to the current density, namely The momentum density satisfies the momentum equation of the form where f is the Lorentz force density defined by the formula T ij are the entries of the energy-momentum tensor defined by (5.21), (5.22), and the EM 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 using multiplication by∇ * ψ * and some rather lengthy vector algebra manipulations. The derivation of the momentum equation based on the standard field theory formalism is given in Section 5.

Derivation of non-relativistic point dynamics for localized solutions of NLS equation
In this section we consider solutions of NLS equation (1.1) that are localized around a trajectory in the three dimensional space R 3 and find the necessary condition for such a trajectory which coincides with Newton's law of motion.

2.1.
Trajectories and their neighborhoods. The first step in introducing the concept of concentration to a trajectory is to describe the class of trajectories.
Definition 1 (trajectory). A trajectoryr (t), T − ≤ t ≤ T + , is a twice continuously differentiable function with values in R 3 satisfying Being given a trajectoryr (t), we consider a family of neighborhoods contracting to it. Namely, we introduce a ball of radius R centered at x =r (t): Definition 2 (concentrating neighborhoods). Concentrating neighborhoodsΩ (r (t) , R n ) ⊂ [T − , T + ] × R 3 of a trajectoryr (t) are defined as a family of tubular domains where R n satisfy the contraction condition: The cross-section of the tubular domain at a fixed t is given by (2.2) and is denoted by Ω n : (2.5) Ω n = Ω n (t) = Ω (r(t), R n ) ⊂ R 3 .

Localized NLS equations.
Let us consider the NLS equation (1.1) in a neighborhood of the trajectoryr(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, c is 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 the trajectory and make certain regularity assumptions on behavior of its coefficients.
In the definitions below we use the following notations: Now we formulate the continuity assumtions we impose on the nonlinearity G ′ 1 in the NLS equation (1.1).

Condition 1.
The real-valued function G ′ 1 (s) is continuous for s > 0. It coincides with the derivative of the potential G 1 (s) which is differentiable for s > 0 and continuous for s ≥ 0. We assume that the function G 1 (ψ * ψ) of the complex variable ψ ∈ C is differentiable for all ψ and its differential has the form and g (ψ) is continuous for ψ ∈ C.
Note that the above condition allows a mild singularity of G ′ 1 at zero, for example the logarithmic nonlinearity (3.10) satisfies this condition.
Definition 3 (localized NLS equations). Letr(t) be a trajectory with its concentrating neighborhoodsΩ (r, R n ), and let a = a n , ϕ = ϕ n , A = A n be a sequence of parameters and potentials entering the NLS equation (1.1). We say that NLS equations are localized inΩ (r, R n ) if the following conditions are satisfied. The parameters R n and a n as n → ∞ become vanishingly small, that is a = a n → 0, R n → 0 and the ratio θ = R n /a grows to infinity: We introduce at the trajectoryr(t) the limit potentials ϕ ∞ (t, x) A ∞ (t, x) which are linear in x and are written in the form where the coefficients ϕ ∞ , ∇ϕ ∞ , A ∞ , ∇A ∞ satisfy the following boundedness conditions: We suppose the EM potentials ϕ n (t, x) , A n (t, x) to be twice continuously differentiable inΩ (r, R n ). The potentials ϕ n (t, x) , A n (t, x) locally converge to the limit linear potentials ϕ ∞ (t, x), A ∞ (t, x), namely they satisfy the following relations: (i) convergence as n → ∞: (ii) uniform in n estimates: The limit EM fields E ∞ ,B ∞ at the trajectory are defined in terms of the linear potentials (2.10), (2.11) by (1.11), namely Note that according to (2.14), (2.15) Throughout this paper we denote constants which do not depend on n by the letter C with different indices. Sometimes the same letter C with the same indices may 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 the following example. Example 1. If the potentials ϕ n , A n are the restrictions of fixed twice continuously differentiable potentials ϕ, A to the domainΩ (r(t), R n ), then ϕ ∞, A n is the linear part of ϕ, A atr and conditions (2.12)-(2.17) are satisfied with that is the coefficients in (2.10), (2.11) are defined as follows: and the EM fields E ∞ , B ∞ in (2.18) are directly expressed in terms of ϕ, A by (1.11), namely We introduce the local value of the charge restricted to domain Ω (r(t), R n ) by the formula with ρ n (t, x) being the charge density defined by (1.13), and we callρ n adjacent charge value.
Definition 4 (concentrating solutions). Letr(t) be a trajectory. We say that solutions ψ to the NLS equations concentrate at the trajectoryr(t) if Condition 1 is satisfied and the following conditions are fulfilled. First, a sequence of concentrating neighborhoodsΩ (r, 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 2 Ω (r, R n ) , and such that the charge density ρ = q |ψ n | 2 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) inΩ (r, R n ); (ii) the momentum density P defined by (1.16) for this sequence is such that the following integrals are bounded: (iii) the local charge value defined by (2.23) is bounded from above and below for sufficiently large n: such that the sequence of local charge values converges: (v) We also impose conditions on the following surface integrals over the spheres ∂Ω n = {|x −r| = R n }: where T ij are given by (5.21), (5.22) and we make summation over repeating indices; (2.28) where P is defined by (1.16); where ρ is defined by (1.13); (2.30) where J is given by (1.14); and The integrals defined above are assumed to satisfy the following limit relations uniformly on the time interval [T − , T + ]: (vi) We introduce the following integrals over Ω n with vanishing atr weights and assume that Obviously conditions (ii)-(iv) and (vi) provide boundedness and convergence of certain volume integrals over Ω n , and condition (v) provides asymptotical vanishing of surface integrals over the boundary. Notice that condition (ii) provides for the boundedness of the momentum over domain Ω n , condition, (v) provides for a proper confinement of ψ to Ω n and estimate from below in condition (iii) ensures that the sequence is non-trivial. According to (2.25),ρ n (t 0 ) is a bounded sequence, consequently it always contains a converging subsequence. Hence condition (iv) is not really an additional constraint but rather it assumes that such a subsequence is selected. The choice of a particular subsequence limitρ ∞ is discussed in Remark 1. This condition describes the amount of charge which concentrates at the trajectory at the time t 0 . 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 allow parameters χ, m, q to form sequences and depend on n, but for simplicity in this paper we assume them fixed.
The wave-corpuscles constructed in Section 3 provide a non-trivial example of solutions which concentrate at trajectories of accelerating charges.

2.3.
Properties of concentrating solutions of NLS. We define the adjacent charge center r n by the formula Since ψ a (t, x) is a function of class C 1 with respect to (t, x), andr(t) is differentiable, the vector r (t) is a differentiable function of time, and we denote by v the velocity of the adjacent charge center: Below we often make use of the following elementary identity: wheren is the external normal to ∂Ω n ,v = ∂ tr .
Proof. Integrating the continuity equation we obtain We use (2.36) and obtain (2.46).
Lemma 3. Assume that (2.24) holds. Then there is a subsequence of the concentrating sequence of solutions of the NLS such that Assume also that boundary integrals (2.27) and (2.28) satisfy assumptions (2.32) and (2.33). Then for this subsequence 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) Integrating it over Ω (r(t), R n ) = Ω n and then with respect to time, we obtain the equation where Q 0 is defined by (2.27). Using (2.42) we rewrite the equation in the form where Q 01 is defined by (2.28), Ω n = Ω (r(t), R n ).
where v = ∂ t r and Proof. According to the continuity equation, we obtain the identity Using the commutation relation to transform the second integral, we obtain the following equation Using the definition of the charge center r and (2.42) we infer that the first term in the above equation has the following form: wherev = ∂ tr . We can express then v = ∂ t r from (2.58), and using ( We rewrite (2.58) in the form whereρ ∞ is the same as in (2.26),  (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: where the Lorentz force density f is given by (1.19). To evaluate the terms involving the Lorentz force density, we use the limit EM fields defined in accordance with (2.10), (2.11) by formula (2.18). We obtain where Q 3 = Q 30 + Q 31 is expressed in terms of (2.37), (2.38). Therefore, using (2.54) we obtain from (2.64) that The following statement provides an explicit necessary condition for a trajectory of concentration.
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 6 → 0 uniformly, the sequence v n (t) of its solutions converges uniformly to the solution of the equation Now we want to prove that v ∞ =v = ∂ tr . Note that according to Lemma 1 The above identity implies ∂ rr =v (t) = v ∞ (t) and consequentlyr (t) satisfies (2.69): The derivative of the above equation yields (2.67).
As a corollary of Theorem 2 we obtain the following theorem describing the whole 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 3 ; (ii) the trajectory (t,r (t)) lies in this domain and the limit potentials ϕ ∞ , A ∞ are the restriction of the potentials ϕ and A as in (2.20). Let EM fields E (t, x), B (t, x) be defined by the potentials as in the formula (1.11). Let solutions ψ of the NLS equation (1.1) concentrate atr(t). Then equation (2.67) for the trajectoryr takes the form of Newton's law of motion with the Lorentz force corresponding to the external EM fields E (t, x) and B (t, x), that is Therefore, for the NLS equation with given potentials ϕ, A any concentration trajectory must coincide with the solution of the equation (2.71). For given potentials ϕ, A the concentration trajectory through a pointr (t 0 ) is uniquely determined by the velocity ∂ tr (t 0 ). In particular, it does not depend on the parameter χ which enters the NLS equation. All possible concentration trajectories are 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 take two different sequences which fit the definition, we obtain the same equation (2.71). More than that, the trajectoryr (t) is uniquely defined by the initial datar (t 0 ), ∂ tr (t 0 ) and consequently it does not depend on the particular sequence. Remarkably, the equation for the trajectory is independent of the value ofρ ∞ , and this property does not hold in the relativistic case, see [11].
Remark 2. We can modify the definition of concentration at a trajectory by allowing parameter χ to be not fixed but form a sequence. The statements of Theorem 2 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 where the form factorψ satisfies the steady-state equation (1.6) and the normalization condition (1.4).
To give non-trivial examples of localized form factors, it is convenient to start with the form factorψ (r) and to describe the nonlinearity which produces the form factor as a solution of (1.6). To define the nonlinearity, we impose first our requirements on the ground stateψ (r) of the charge distribution. A ground state is a positive functionψ (r), r = |x|, which is twice differentiable, satisfies the charge normalization condition (1.4) and is monotonically decreasing: Ifψ (r) is a ground state, we can determine the nonlinearity G ′ from the following equation obtained from (1.6): For a radialψ we obtain then an expression for the nonlinearity G ′ : Sinceψ 2 (r) is a monotonic function, we can find its inverse r = r ψ 2 , yielding Sinceψ (r) is smooth and ∂ rψ < 0, G ′ (|ψ| 2 ) is smooth for 0 < |ψ| 2 <ψ 2 (0); we extend G ′ (s) for s ≥ψ 2 (0) as a smooth function for all s > 0. To describe the localization of the ground stateψ, we introduce an explicit dependence on the size parameter a > 0 as in (1.7): (3.7)ψ (r) =ψ a (r) = a −3/2ψ 1 a −1 r , r = |x| ≥ 0. This corresponds to the dependence of the nonlinearity on the size parameter described by (1.5). Note that the antiderivative G (s) is defined for s ≥ 0 according to Condition 1 and is given by the formula. where C g is a normalization factor, C g = π −3/4 if υ 0 = 1 in (1.4). Such a ground state is called gausson in [17], [18]. Elementary computation shows that Hence, we define the nonlinearity corresponding to the Gaussian by the formula (3.10) G ′ |ψ| 2 = − ln |ψ| 2 /C 2 g − 3, and refer to it as the logarithmic nonlinearity. The nonlinear potential function has the form Dependence on the size parameter a > 0 is given by the formula Note that according to Gross inequality the Gaussian provides the minimum of energy E = u dx subjected to the normalization condition (1.4), where the energy density u is defined by (5.19).
More examples of nonlinearities can be found in [7]- [11], and many facts from the theory of the NLS equations can be found in [20], [51]. The NLS equations with logarithmic nonlinearity are studied in [21], [19], [20].
Below we find conditions on the external fields ϕ and A which allow the wavecorpuscle of the form (3.1) to preserve exactly its shape |ψ| in accelerated motion governed by the NLS equation along a trajectory r (t). In particular, Newton's law of motion emerges as the necessary condition for such a motion.
3.1. Criterion for shape preservation. It is convenient to rewrite the NLS equation (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 where v is the center velocity: We substitute (3.1) in (3.13), and, canceling the factor e iS , we arrive to the following equivalent equation for the phase S: Expanding ∇ − iqA χc + i∇S 2ψ and using (1.6) to exclude the nonlinearity G ′ , we rewrite (3.15) in the form Every term in the above equation has either the factorψ or ∇ψ. Collecting terms at ∇ψ andψ respectively, we obtain two equations: Now we would like to find conditions on the potentials ϕ, A under which there exists the phase S which satisfies the above system of equations (3.17), (3.18). Such a phase provides for the wave-corpuscle solution to the NLS equation (3.13). We begin with equation (3.17) first. Since we conclude that (3.17) is equivalent to the following orthogonality condition: To treat the above equation we use the concept of a sphere-tangent field. We call a vector fieldV (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 1 R 3 can be uniquely splitted into a potential field ∇P and a sphere-tangent fieldV 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

23)
A ∇ (t, y) = ∇P (t, y) . Equation (3.19) implies that the field q c A − ∇S + mv is purely sphere-tangent, therefore its gradient part is zero and we conclude that the following equation is equivalent to (3.19): Taking the imaginary part of (3.26), we see that the sphere-tangent part of the vector potential must satisfy the following zero divergency condition: The real part of (3.26) yields equation Hence, (3.28) and (3.25) take the form The above relations give an expression for the 4-gradient of χS. The right-hand sides 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, and consequently it must be closed. Conversely, according to Poincare's lemma, the form on R 4 is exact if it is closed on R 4 , and then the phase S can be found by the integration of the differential form. Since the right-hand side of (3.30) is the gradient, the form is closed if the following condition is satisfied: This equation can be interpreted as a balance of forces which allows to exactly preserve the shape of the wave-corpuscle. Equation (3.31) together with condition (3.27) constitutes the criterion for the preservation of the shape |ψ|.

3.2.
Trajectory and phase of a wave-corpuscle. If equation (3.31) holds, the phase function S can be found by integrating the exact 1-form: where Γ is a curve in time-space connecting (0, 0) with (t, y). Since equation (3.31) is fulfilled, the integral does not depend on the curve. In particular, we take as Γ = Γ 0 a curve formed by two straight-line segments: the first segment from (0, 0) to (t, 0) and the second from (t, 0) to (t, y). That yields and s p2 (t, y) is at least quadratic in y. In the above derivation we used that A (t, 0) = 0, y ·Ȃ (t, y) = 0, and A ∇ (t, 0) = A (t, 0).
Singling out the linear part of the phase we can write above formulas in the form The integrability condition (3.31) can be written now in the form The above equation together with (3.27) is a system of equations which involves the EM potentials ϕ, A and the corpuscle center trajectory r (t). Its fulfillment guarantees that the wave-corpuscle preserves its shape in the dynamics described by the NLS equation. We refer to equations (3.39), (3.27) non-relativistic wavecorpuscle dynamic balance conditions. Obviously the conditions do not depend on the nonlinearity G. By setting y = 0 in (3.39) and taking into account thatȂ ex (t, 0) = 0 we obtain the point balance condition: To interpret this condition, we note that y = 0 in the moving frame in the above equation corresponds to x = r in the resting frame and hence the point balance condition (3.40) takes in the resting frame the form Recall that the expression for the Lorentz force in terms of the EM potentials is given by the formula 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 where EM fields E and B are given by (1.11). Hence the point balance condition coincides with Newton's law of motion (1.9) of a charged point subjected to the Lorentz 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 where the left hand side explicitly vanishes for spatially constant fields A and constant ∇ϕ. Now let us discuss the universality of the non-relativistic dynamic balance conditions. 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 factorψ. Recall that when deriving the conditions we set coefficients at ∇ψ andψ in (3.16) to be zero. We would like to show that this is a necessary requirement for the conditions to be universal.
Equation (3.16) has the form Q 0ψ (r) +ψ ′ (r) Q 1 · y/r = 0, whereψ (r) is a form factor, and coefficients Q 0 , Q 1 obviously do not depend onψ. If Q 1 · y is not identically zero, we can separate the variables: where the left-hand side isψ-independent and the right-hand side depends onψ. It is impossible, and hence Q 1 · y must be zero as well Q 0 , leading to equations (3.17), (3.18).

3.3.
Wave-corpuscles in the EM field. The simplest example of fields ϕ, A for which 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 general example of the fulfillment of the condition (3.45). In this example we prescribe arbitrary EM potentials which are linear in x, take a trajectory which satisfies Newton's law (3.43), and assume that second and higher order components of the EM potentials expansion at r (t) must satisfy certain restrictions. Namely, we assume that A involves a linear in y = x − r part which can be given by an arbitrary 3 × 3 matrix A 1 (t) with time-dependent elements: We assume that quadratic and higher order components of the sphere-tangent part of A are set to zero, but allow an arbitrary potential part where A ∇2 has at least the second order zero at the origin and is potential: with an arbitrary P 3 . For such a field, its potential and sphere-tangent parts are given respectively by formulas A T 1 stands for A 1 transposed, and P 3 is a function with zero of at least the third degree at the origin. Note that an action of an anti-symmetric matrix can be written using the cross product, therefore (3.49) can be written as follows: The electric potential ϕ has the form where ϕ (t, 0) and ∇ϕ (t, 0) are given continuously differentiable functions of t, and ϕ 2 (t, y) has order two or higher in y and is subject to the condition (3.54) we formulate below. Let us verify the conditions which guarantee that the wave-corpuscle is an exact solution. Notice that condition (3.27) forȂ defined by (3.49) is fulfilled. Hence, it is sufficient to satisfy (3.45), which takes the form To satisfy (3.53), we set We want now to determine the phase S of the wave-corpuscle using (3.34), (3.35). If P 3 is homogenious of third degree, we obtain In this case the phase function of the wave-corpuscle involves a term s p2 (t, y). Formula (3.1) takes the form s p2 (t, y) .
If the external magnetic field satisfies the following anti-symmetry conditions y · A 1 (t, y) = 0, (3.59) y · A ∇2 (t, y) = 0 then s p2 = 0, and the phase function is linear in y.
The above calculations can be summarized in the following statement Remark 3. The above construction does not depend on the nonlinearity G ′ = G ′ a as long as (1.6) is satisfied. It is also uniform with respect to a > 0, and the dependence on a in (3.57) is only throughψ (|x − r|) = a −3/2ψ Remark 4. The form (3.1) of exact solutions is the same as the WKB ansatz in the quasi-classical approach, [41]. The trajectories of the charge center coincide with trajectories that can be found by applying well-known quasiclassical asymptotics to solutions of (1.1) if one neglects the nonlinearity. Note though that there are two important effects of the nonlinearity not presented in the standard quasiclassical approach. First of all, due to the nonlinearity the charge preserves its shape in the course of evolution on unbounded time intervals whereas in the linear model any wavepacket disperses over time. Second of all, the quasiclassical asymptotic expansions produce infinite asymptotic series which provide for a formal solution, whereas the properly introduced nonlinearity as in (1.6), (3.6) allows one to obtain an exact solution. For a treatment of mathematical aspects of the approach to nonlinear wave mechanics based on the WKB asymptotic expansions we refer the reader to [41], [35] and references therein.
Remark 5. We can use in the definition of a wave-corpuscle (3.1) a radial form factorψ which instead of (1.6) satisfies the eigenvalue problem 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 NLS equation (1.1) with the nonlinearity G ′ if and only if the function e −i χ 2m λt ψ is a solution of the NLS equation with the nonlinearity G ′ − λ. Therefore the wavecorpuscle solutions constructed in Theorem 4 based onψ with the nonlinearity G ′ −λ provide solutions to the NLS equation with the original nonlinearity, one has only to add to the phase function S an additional term χ 2m λt. Existence of many solutions of 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 a given NLS equation with the same trajectory of motion r (t) and the same EM fields which correspond to different form factors.

3.4.
Wave-corpuscles as concentrating solutions. The wave-corpuscles constructed in Theorem 4 provide an example of concentrating solutions. In order to see that we have to verify all the requirements of Definition 4. It is important to note that the dependence on a in (3.1) is only throughψ (|x − r|) = a −3/2ψ 1 a −1 |x − r| whereψ 1 is a given smooth function. The verification of conditions imposed in the Definition 4 is straightforward. For instance, 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 momentum density (1.16), (1.2) and for the wave-corpuscles 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  (2.19). Using continuity of E (t, y) and B (t, y) we conclude that |E n − E ∞ | → 0 and |B n − B| → 0 in Ω 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 parts given by (5.21) and (5.22): According to (5.22) to estimate |Q 0, = | it is sufficient to prove that Using this estimate and (5.22) we obtain To estimate Q 0,= we use (5.21) is a radial function andn i = y i / |y| is odd with respect to y i -reflection, we see that in the above integral We also note that where ∂ t S and ϕ are bounded functions in Ω n . Under the assumption (3.66) straightforward estimates produce that where R n /a = θ → ∞, R n → 0. We assume that the form factorψ 1 (r) and its derivativeψ ′ 1 (r) satisfy the following decay conditions: (3.75) θ 3 ψ ′ 1 (θ) ≤ C as θ → ∞, and we take such a n , R n that (3.76) a n R −4 n → 0, a n → 0, R n → 0.
Under these assumptions we conclude that (3.66) is fulfilled. Summing up the above arguments we obtain the following statement: Theorem 5. Letψ 1 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. Then the sequence ψ n concentrates at the trajectoryr(t) = r(t).

Concentration of asymptotic solutions
From the proofs of Section 2 one can see that in the derivation of the Newtonian dynamics we use only the conservation laws (1.15) and (1.18). Since we use only asymptotic properties, it is natural to consider fields which satisfy NLS equations and the conservation laws not exactly, but approximately. Namely, we assume now that the conservation laws (1.15) and (1.18) are replaced by 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′ , and quantities f ′ are perturbation terms which vanish as n → ∞. We show now how to modify the proofs of statements in Section 2 on concentrating solutions to obtain similar statements on concentrating asymptotic solutions.
In the proof of Lemma 3 equation (2.51) is replaced by (4.2). This leads to replacement of the equation (2.53) by the equation

If (4.5)
Q ′ 0 → 0 the proof of Lemma 3 remains valid with Q 00 = Q 0 + Q ′ 0 − Q 01 , and we obtain the following lemma:  Sufficient conditions for fulfillment of (4.5) are the following limit relations We assume that (4.14) Q ′ 2 → 0 and conclude that the statement of of Lemma 4 holds: 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) are as follows: Now we consider the proof of Lemma 2. Equation (2.47) is replaced by The proof of Lemma 2 is preserved if we assume that Now we collect all the assumptions required for exact solutions in Section 2 that remain valid for asymptotic solutions in the following definition: Definition 5 (Concentrating asymptotic solutions). We assume all conditions of Definition 4 except condition (i), which is replaced by the following weaker condition: 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) concentrate atr(t). We callr(t) a concentration trajectory of the NLS equation in asymptotic sense if there exists a sequence of asymptotic solutions which concentrates atr(t).
Using Lemmas 5 and 6 instead of Lemmas 3 and 4 we obtain statements of Theorems 2 and 3 under the assumption that asymptotic solutions ψ of the NLS equation (1.1) concentrate atr(t). In particular, we obtain Theorem 6. Assume that potentials ϕ (t, x), A (t, x) are defined and twice continuously differentiable in a domain D ⊂ R × R 3 , the trajectory (t,r (t)) lies in this domain and the limit potentials ϕ ∞ , A ∞ are the restriction of fixed potentials ϕ,A as in (2.20). Let EM fields E (t,  44), and assume 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) As a second step we construct the auxiliary potentials where ϕ 2 is determined by (3.54) and ϕ 2 (t, x − r) is quadratic with respect to (x − r).
The wave-corpuscle ψ described in Theorem 4 is a solution of the NLS equation with the potentials (ϕ aux , A aux ), and hence it exactly satisfies the corresponding 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) The expression for ρ does not depend on the potentials, therefore (4.27) ρ ′ = 0.
From (1.16), (3.63) we see that and according to (1.17) From the expression for the Lorentz density (1. 19) we obtain Using (3.63) and (1.17) we rewrite expression for f ′ in the form To estimate the difference of tensor elements (4.26) we use (5.22) and (5.21). Note that 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: and they are vanishingly small in Ω (r(t), R n ). Now we estimate terms which enter (4.4), (4.13), (4.19). Proof. The proof is based on inequalities (4.32) and (4.31). To obtain (4.8) we use (4.28) and (3.61): and we obtain (4.8) for R n → 0. To obtain (4.9) we observe that according to (4.28) ∂Ω(r(t),Rn) P ′v ·ndσ (4.33) n a −3ψ 2 a (R n /a) = 4πCR n θ 3ψ 2 a (θ) → 0 Similarly we obtain and (4.29). A straightforward estimate of (4.30) yields (4.11). Note that according to (5.21) and (5.19) the terms in T ii involving G do not depend on (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 of terms, 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 2 n . All the terms are easily estimated, for example an elementary inequality  As a corollary we obtain the following theorem. Since asymptotic solutions in the sense of Definition 5 in Theorem 7 are constructed as wave-corpuscles, we obtain the following corollary. Corollary 1. If a sequence of concentrating asymptotic solutions concentrates at a trajectory r(t), then there exists a sequence of concentrating wave-corpuscle asypmptotic solutions which concentrate at the same trajectory.
The EM 4-potential is given by the formula where ϕ (t, x) , A (t, x) are given potentials, and the EM power tensor is defined as follows: The covariant derivatives which involve the EM fields are defined by (1.2), and corresponding 4-differential operators have the form (5.5)∂ µ = 1 c∂ t ,∇ ,∂ µ = 1 c∂ t , −∇ .
Proposition 1. Let EnMT T µν be defined by formula (5.14) for a solution of the NLS equation (1.1). Then T µν satisfies the following EnMT conservation law Proof. According to (5.7) and (5.9 the definition (5.14) can be written in the form where J ν is defined by (5.9). We differentiate the above expression and use the continuity equation (5.13): The NLS equation (1.1) can be written in the form where the Lagrangian L is considered as a function of ψ, ψ * , ψ * 6. Appendix 2: Splitting of a field into potential and sphere-tangent parts Here we describe splitting of a general vector field into sphere-tangent and gradient components which was used in derivation of the properties of wave-corpuscles.
Lemma 9. Any continuously differentiable vector field V (y) can be uniquely splitted into a gradient field ∇P and a sphere-tangent fieldV where P is continuously differentiable,V is continuous and satisfies the orthogonality condition (3.20), namelyV · y = 0.
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 = Ωr where |Ω| = 1, r = |y| we rewrite the above equation in the form Ω · V = ∂ r P (Ωr) .
Note that P is defined from the above equation up to a function C (Ω); since the only 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: If the potential P is defined by (6.2) then y ·V = y ·V − y ·∇P satisfies (3.20).
Now we provide some explicit formulas for polynomial fields V.