Solitary waves and Bohmian mechanics

Abstract

We study a Schrödinger-like equation with a nonlinear term. This nonlinearity has the effect of allowing the existence of highly concentrated stable solitary waves of a topological nature. Such solitary waves tend to move according to Bohmian mechanics. Therefore our model can be considered a nonsingular realization of de Broglie pilot wave theory.

1. Introduction

The Schrödinger equation of one particle in an external field

1.1

is usually interpreted as the law governing the probability density |ψ|² of finding the particle in a certain place. Therefore quantum mechanics does not assign a definite value to positions until a measurement is made. This fact causes many well known logical troubles, and most of all forces the complete description of an experiment to consist of two separate worlds: the small objects, which evolve accordingly to the laws of quantum mechanics, and the large experimental apparatus, which has to be considered classical.

In the early 1950s, David Bohm presented an alternative approach, now known as Bohmian Mechanics (ref. 1 is the original paper, while in ref. 2 one can find some beautiful descriptions of the theory; see ref. 3 for a complete technical presentation). He considered a particle as consisting of both a point mass, the position of which is Q, and an internal field ψ, which evolves according to Eq. 1.1. The only detectable physical object is the point mass, which is guided by the field ψ according the equation

1.2

where S(x, t) is the argument of the complex number ψ(x, t). This approach presents no logical troubles with the theory of measurements, and it can be proved to be experimentally equivalent to usual quantum mechanics, provided one assumes that the initial probability distribution for the point mass is |ψ(x, 0)|² (this assumption can be justified; see for example ref. 4).

Bohmian mechanics is in a certain sense a simplification of de Broglie's pilot wave theory, developed in the 1920s, then abandoned for 30 years by the author until he read about the Bohm approach (see refs. 5 and 6). In de Broglie's pilot wave theory one looks at the singular solutions of Eq. 1.1 and discovers that the singularity evolves according to Eq. 1.2. Hence one is tempted to associate the particle to the singularity of the wave ψ. However, this picture presents some problems: an infinite amount of energy is concentrated in the singularity, and it is not clear how a finite energy wave can guide an infinite energy particle without being affected. Moreover, the singular solutions of the Schrödinger equation are unstable, meaning that any smooth approximation of them will just spread out in finite time. de Broglie was aware of such problems and believed that the true equations should contain some nonlinear part, and that singular solutions had to be replaced by smooth solutions having a highly concentrated peak (solitary waves). The nonlinear term should be so small that it does not affect ψ out from the peak, its effect acting only on the peak by stabilizing it. However, at that time the techniques needed to deal with nonlinear equations were not developed enough, and his program remained unfulfilled.

The aim of this article is to present a model in which the Schrödinger equation (1.1) is perturbed by a suitable small nonlinear term

1.3

and we show that such an equation has stable, highly concentrated solutions. Moreover, we show that as ɛ tends to zero, the peak tends to move according to Eq. 1.2.

2. The Model

The linear Schrödinger equation for a single particle under the action of an external field with potential V : ℝ³ ↦ ℝ is

2.1

Writing the complex valued function ψ as

where both u and S are real-valued, Eq. 2.1 becomes

2.2

2.3

The Lagrangian density for system Eqs. 2.2 and 2.3 is given by

Our model will be obtained by starting from a suitable modification of the Lagrangian density ℒ₀. The choice of such a modification is largely inspired by ref. 7, where a Lorentz invariant system is built, starting from the semilinear wave equation. The first difference is that we deal with a wave function ψ, which takes values in ℂ⁴. In other words,

where u_j and S_j are real-valued. We will often write u = (u₁, … , u₄), S = (S₁, … , S₄). Our new Lagrangian density will be

Here δ = δ(ɛ), σ = σ(ɛ) are positive and infinitesimal for ɛ → 0. p is a constant strictly larger than 3. Let ζ = (1, 0, 0, 0) ∈ ℝ⁴; W is a nonnegative C² real-valued function on ℝ⁴∖{ξ} such that

1. W(0) = 0, W′(0) = 0, W"(0) = 0;

2. there exist c₁, c₂ > 0 such that

   

   The Euler–Lagrange equations given by ℒ are

   2.4

   2.5

   The above system has some first integrals. The first one is the energy, given by the time invariance,

   

   Then there is the momentum of each component, given by the invariance with respect to space translations,

   

   Finally, there is the L² norm of each component, given by the invariance with respect to phase shifts,

   

3. Stationary Solutions of the Free System

In this section we are interested in a case with no external field, that is V ≡ 0, and we want to look for stationary solutions, that is solutions of the form

where the ϕ_j are real-valued and ω₀ is a real constant. Then ϕ = (ϕ₁, … , ϕ₄) solves the system

3.1

The solutions of the above equation are extremal points of the functional

Since W has a singularity in ζ, which prevents the map ϕ to take the value ζ/σ, F will be defined on the Sobolev space of continuous maps (recall that p > 3)

It is easy to see that Λ has a countable number of connected components, indexed by the number of times u wraps around the singularity,

Set Λ* = ∪_q∈ℤ*Λ_q. Clearly Λ* does not contain the map ϕ = 0, and thus any minimum of F in Λ* is a nontrivial solution of Eq. 3.1 with finite energy. It will be called a solitary wave for systems 2.4 and 2.5. The following result is a slight modification of theorem 2.2 of ref. 7 and can be proved basically in the same way.

Theorem 3.1.

If ω₀ ≥ 0, F has a minimum inΛ*.

Because the equations are translation-invariant, ϕ(⋅ + q) is a solution whenever ϕ is. We will say that ϕ is centered in q ∈ ℝ³ if ϕ₁ attains its maximum value in q. Let Σ_ω0 be the set of the functions ϕ that minimize F in Λ*. Σ_ω0 is locally compact. Denote by dist the distance in the W^1,2 ∩ W^1,p metric. The minimal solutions given by Theorem 3.1 are orbitally stable as shown by the next result.

Theorem 3.2.

For every λ > 0 there exists η > 0 such that, if dist (u₀, Σ_ω0) < η and∥∇S₀∥_L² < η, the solution (u(t), S(t)) of Eqs. 2.4 and2.5 (still with V ≡ 0), the initial value of which is (u₀, S₀),satisfies dist (u(t), Σ_ω0) < λ for every real t.

Now let ϕ = ϕ^ɛ be a nontrivial solution of Eq. 3.1 centered in q ∈ ℝ³, and let ϕ(x) = (1/σ)Φ(x − q/ɛ). Then Φ solves the system

We make the choice

3.2

such that Φ solves

3.3

Assume that

3.4

Then Eq. 3.3 admits a limiting problem, as the next theorem shows.

Theorem 3.3.

Assume that Φ = Φ^ɛ is one of the nontrivial solutions of Eq. 3.3 given by Theorem 3.1, centered in q ∈ ℝ³. If Eq. 3.4 holds, Φ^ɛ tends toΦ⁰, a unique solution of

in the L² ∩ W^1,ptopology, as ɛ → 0.

In particular, the only nonvanishing component of Φ⁰ is the first one, so that Φ⁰ is actually a scalar function. Moreover, Φ⁰ is radially symmetric. Theorem 3.3 shows that the stationary solutions ϕ^ɛ become more and more concentrated and peaked about their center as ɛ → 0.

4. The Guidance Formula

Now we want to consider the full evolution of Eqs. 2.4 and 2.5 with the initial data given by the superposition of some fixed wave function ψ⁰ = u⁰e^iS0 and of a stationary solution ϕ^ɛ of the free system, with frequency ω₀(ɛ) given by Eq. 3.2. More precisely, we impose the initial conditions

4.1

4.2

where ϕ^ɛ is one of the solutions of Eq. 3.1 given by Theorem 3.1, centered in some q ∈ ℝ³, independent of ɛ. u⁰ ∈ W^1,2 ∩ W^1,p and S⁰ ∈ C¹. We make the following assumption on the evolution problem:

Assumption 4.1.

The system (Eqs. 2.4 and2.5) has a solution (u^ɛ, S^ɛ) with initial condition (Eqs.4.1 and 4.2). Moreover,u^ɛ ∈ C¹(ℝ; W^1,2 ∩ W^1,p).

We start with the particular case V ≡ 0, u⁰ ≡ 0, S⁰(x) = v⋅x. In this case we have an explicit formula for the solution,

where ω(ɛ) = (1/2)|v|² − ω₀(ɛ). Thus the solution is given by a phase-shifted copy of the solitary wave ϕ^ɛ traveling with uniform velocity v = ∇S^ɛ(x, t).

Now we deal with the general case. We claim that is possible to find some positive R such that

for x belonging to some ball of radius ɛR, for every t (the center of such a ball may depend on t). We choose a function α_ɛ ∈ C^∞(ℝ³) such that

Choose a positive number M such that

If (u^ɛ, S^ɛ) is the solution of Eqs. 2.4 and 2.5 with initial conditions given by Eqs. 4.1 and 4.2, we set

It is natural to define the position Q_ɛ(t) of the solitary wave as the point where G_ɛ(⋅, t) assumes its maximum value. We have the following estimate for the Hessian of G_ɛ

For a positive constant m set

Γ_ɛ,m is the set of instants t when G_ɛ assumes a nondegenerate global maximum, and m gives a lower estimate on the smallest eigenvalue of its Hessian in such a point. We regularize ∇S by taking

Lemma 4.1.

Assume that t ∈ Γ_ɛ,m. Then

Here is a sketch of the proof. Because Q_ɛ is a critical point for G_ɛ,

Differentiating the above equation with respect to time and by using Eq. 2.5, we get

Integrating by parts we have

Setting

we get

Multiplying by the vector (dQ_ɛ/dt) − ∇, we have

Because t ∈ Γ_ɛ,m, we finally deduce

Lemma 4.1 allows us to derive the guidance formula, where the solution is sufficiently smooth.

Theorem 4.2.

Assume that ∇S(⋅, t) → ∇S₁(⋅, t) in the Lipschitz norm. If∇S₁ is Lipschitz continuous, then there exists T > 0 such that

where

It seems reasonable to conjecture that the convergence assumed in the hypothesis of the Theorem 4.2 holds. S₁ should be the phase of the solution of the linear Schrödinger equation with initial data u⁰e^iS0, at least when parameters are chosen such that the energy of the solitary wave ϕ^ɛ is negligible with respect to the energy of the initial data u⁰e^iS0.