Download Energy-momentum relation for solitary waves of relativistic wave

Russ. J. Math. Physics 9 (2002), no. 2, 153-160
arXiv:math-ph/0508045v1 23 Aug 2005
Energy-momentum relation for solitary waves
of relativistic wave equations
T.V. Dudnikova
1
Mathematics Department
Elektrostal Polytechnic Institute
Elektrostal, 144000 Russia
e-mail: [email protected]
A.I. Komech
2
Mechanics and Mathematics Department
Moscow State University
Moscow, 119899 Russia
e-mail: [email protected]
H. Spohn
Zentrum Mathematik
Technische Universität
München D-80290, Germany
e-mail: [email protected]
Abstract
Solitary waves of relativistic invariant nonlinear wave equation with symmetry
group U (1) are considered. We prove that the energy-momentum relation for spherically symmetric solitary waves coincides with the Einstein energy-momentum relation
for point particles.
1
Supported partly by research grants of DFG (436 RUS 113/615/0-1), INTAS-OPEN-97-10812 and RFBR
(01-01-04002).
2
Supported partly by the Institute of Mathematics of the Vienna University, by Max-Planck Institute for
Mathematics in the Sciences (Leipzig) and by the START project (FWF Y137-TEC) of N.J. Mauser.
1
Introduction
The paper concerns the old problem of mathematically describing elementary particles in
field theory. The problem was raised in classical electrodynamics after the discovery of the
electron by Thomson in 1897. Abraham realized that a point electron would be unstable
due to infinite self-energy, and introduced the model of the “extended” electron with finite
electrostatical energy. However, the electrostatic model is also unstable due to electrostatic
repulsion. The corresponding tension of the extended electron was analyzed by Poincaré in
[13]. To avoid the instability problem for point particles, Einstein suggested that particles
could be described as singularities of solutions to the field equations [5]. To obtain stationary
finite energy solutions, Born and Infeld introduced a “unitary field”, which is a nonlinear
modification of the Maxwell equations [14, 15]. The last approach was not developed further
because the relation to the corresponding quantum version was not clarified.
Rosen [17] was the first who proposed a description of particles for the coupled KleinGordon-Maxwell equations, which are invariant with respect to the Lorentz group and to the
global gauge group U(1) : the particle at rest is described by a finite energy solution that
has “Schrödinger’s” form ψ(x)eiωt (“nonlinear eigenfunctions” or “solitary waves”). The
particle with the nonzero velocity v , |v| < 1 , is obtained by the corresponding Lorentz (or
Poincaré) transformation. Some numerical analysis has been done, showing the existence of
radial solutary waves for a finite range of ω . The particle in the “normal” state is identified
with the solitary wave of minimal energy.
The existence of radial solitary waves (“ground states”) and nonradial solitary waves
(“excited states” with nonzero angular momentum) has been analyzed numerically by many
authors for diverse Lagrangian field theories [8, 9], [17]-[24], [26]. For the complex scalar
field, most general results were achieved by Beresticky and Lions [3]. For the nonlinear Dirac
equation, the existence is proved in [1, 7]. For the Maxwell-Dirac field, the existence was
proved by Esteban, Georgiev and Séré in [6].
The next step after establishing the existence of solitons would be to study their stability
as one of the main features of elementary particles. A nonrigourous analysis in [17] suggests
an instability of solitons with negative total energy treated as a negative mass of the soliton.
The first rigourous proof of instability of solitons with ω = 0 was given by Derrick in [4]. The
proof also concerns negative energy and is based on the virial Derrick-Pokhozhaev identity
[4, 12].
These instability results hindered the development of the approach to elementary particles
as the solitons. On the other hand, the solitons with ω 6= 0 can be stable; this follows from
the general criterion discovered by Grillakis, Shatah and Strauss and other authors (see [11]
and the references therein). Note that in [2] the asymptotic stability of solitons was proved
for the 1D nonlinear Schrödinger equation when ω 6= 0 .
The key role of elementary particles suggests that the set of all moving solitons forms
a global attractor for all finite energy solutions to the dynamical field equations. However,
this is still an unproved conjecture.
Finally, it would be of importance to develop a particle-like dynamics for moving solitons.
We make a step in this direction for relativistic-invariant scalar Klein-Gordon equations.
Namely, we prove that, for spherically symmetric solitary waves (constructed in [3]), the
energy-momentum relation coincides with that of a relativistic particle.
1
Furthermore, we suggest that this fact fails for non-spherically symmetric solitary waves
of some angular structure if such solitary waves exist indeed.
A further step would be a justification of effective dynamics for a solitary wave in an
external slowly varying potential. A result of this type was obtained in [16] for the solitons
of the Abraham model of classical electrodynamics and in [25] for localized wave packets of
the Dirac equation. Recently, an effective dynamics was announced for the solitons of the
nonrelativistic nonlinear Hartree equation [10].
2
Standing solitary waves
Consider the relativistic-invariant nonlinear wave equation
ψ̈(x, t) = △ψ(x, t) + f (ψ(x, t)),
x ∈ IRn ,
(2.1)
where n ≥ 1 and ψ ∈ C d , d ≥ 1 . Assume that
f (ψ) = −∇ψ V (ψ),
ψ ∈ C d,
where V is a real potential, and ∇ψ stands for the gradient with respect to u = Re ψ and
v = Im ψ ; in other words, ∇ψ = ∇u + i∇v . Then Eq. (2.1) formally becomes a Hamiltonian
system with the Hamiltonian functional
H(ψ, ψ̇) =
Z "
|ψ̇|2 |∇ψ|2
+
+ V (ψ) dx.
2
2
#
(2.2)
Further, assume that V (ψ) = V(|ψ|) . Then Eq. (2.1) is U(1) -invariant, i.e.,
f (eiθ ψ) = eiθ f (ψ),
θ ∈ IR.
Consider a (standing) solitary wave ψ0 (x, t) = a(x)e−iωt with the energy
H(ψ0 , ψ̇0 ) =
Z
ω 2|a|2 |∇a|2
+
+ V(|a|) dx.
2
2
!
(2.3)
The amplitude a(x) is a solution of the Helmholtz stationary nonlinear equation
−ω 2 a(x) = △a(x) + f (a(x)),
x ∈ IRn .
(2.4)
The existence of nonzero solitary waves was proved in [3] under the following assumptions:
S0
d = 1,
S1
−∞ < lim f (a)/a + ω 2 < 0,
S2
S3
f ∈ C(C),
f (0) = 0,
a→0+
ω 2 a20
< 0,
2
∃α ≥ 0 : f (a) = −αal + o(al ), a → ∞,
∃a0 > 0 : V(a0 ) −
2
l :=
n+2
.
n−2
Further, the solution a(x) to (2.4) is real and satisfies the following properties:
a(x) = R(|x|),
A1
A2
∃C, δ > 0 : |R(k) (r)| ≤ Ce−δr , k = 0, 1, 2.
Property A1 means that the solitary wave thus constructed is a radial (spherically symmetric) “ground state”. Below we discuss the excited states with higher angular momentum for
n = 2.
3
Moving solitary waves
Consider a (moving) solitary wave with velocity v ∈ IRn :
ψv (x, t) = ψ0 (Λv (x, t)).
Here |v| < 1 and Λv is a Lorentz transformation with velocity v :
Λv (x, t) = (γ(xk − vt) + x⊥ , γ(t − vx)),
where xk + x⊥ = x , xk kv, x⊥ ⊥ v, γ = √
1
. In other words,
1 − v2
ψv (x, t) = ψ0 (γ(xk − vt) + x⊥ , γ(t − vx)) = a(γ(xk − vt) + x⊥ )e−iωγ(t−vx) .
Note that ψv (x, t) is a solution of (2.1).
4
Energy-momentum relation for radial states
For v ∈ IRn , |v| < 1 , we denote by
Ev := H(ψv , ψ̇v ), Pv := −Re
Z
ψ̇v ∇ψv dx
the energy and momentum of the moving solitary wave, respectively. Let us assume in what
follows that
V(a)+
S4
ω 2 a2
≥ 0, a ≥ 0.
2
In this section, we consider spherically symmetric nonzero solitary waves (i.e. waves with
zero angular momentum). This spherical symmetry is a typical property of a “ground state”
with minimal energy.
Theorem 4.1 Let a(x)e−iωt be a standing solitary wave for (2.1), and let A1, A2 hold.
Then the following “particle-like” energy-momentum relation holds:
Ev = √
Here E0 > 0 for ω 6= 0 .
Ev
E0
√ 0
,
P
=
.
v
1 − v2
1 − v2
3
(4.1)
Remark For a nonzero solitary wave, we have E0 := H(ψ0 , ψ̇0 ) > 0 . In fact, if H(ψ0 , ψ̇0 ) =
0 , then a(x) ≡ const by S4 and (2.3). However, this implies a(x) ≡ 0 by A2.
Proof of Theorem 4.1. Step 1. We choose the coordinates in such a way that v =
(|v|, 0, . . . , 0) . Below we write everywhere v instead |v| to simplify the notations. Let us
write
y1 = γ(x1 − vt), yk = xk ; y = (y1 , ..., yn ).
Then
ψv (x, t)
ψ̇v (x, t)
∇1 ψv (x, t)
∇k ψv (x, t)
a(y)e−iωγ(t−vx1 ) ,
(−γv(∇1 a)(y) − iγωa(y)) e−iωγ(t−vx1 ) ,
(γ(∇1 a)(y) + iγωva(y)) e−iωγ(t−vx1 ) ,
(∇k a)(y)e−iωγ(t−vx1 ) , k = 2, ..., n.
=
=
=
=
Substituting these expressions into Ev , we obtain
Ev =
=
Z "
Z
ψ̇v2 |∇ψv |2
+
+ V (|ψv |) dx
2
2
#
!
"
n
X
1
|∇1 a(y)|2γ 2 (v 2 + 1) + |a(y)|2ω 2 γ 2 (v 2 + 1) +
|∇k a(y)|2 + V (|a(y)|) dx.
2
k=2
#
Then
Ev =
Z
n
i
h
1X
1
γ2
dy.
|∇k a|2 + (v 2 + 1) |∇1 a|2 + |a|2 ω 2 + V (|a|)
2 k=2
2
γ
!
Write
1
2
Z
|a(y)|2 dy,
1
Ik =
2
Z
|∇k a(y)|2 dy,
I0 =
V0 =
Z
V (|a(y)|) dy,
(4.2)
k = 1, ..., n.
(4.3)
Hence, in the notation (4.2) and (4.3), we represent Ev as
Ev
1
=
γ
E0 =
n
X
n
X
!
Ik + γ 2 (v 2 + 1)(I1 + I0 ω 2) + V0 ,
k=2
(4.4)
Ik + I0 ω 2 + V0 .
(4.5)
k=1
Step 2. We now derive (4.1) from this representation for Ev with the help of the following
lemma.
Lemma 4.2 ([3, 4, 12]) The following Derrick-Pokhozhaev identity holds:
1
−(n − 2)
2
Z
2
|∇a(y)| dy = n
1
V (|a(y)|) − ω 2 a2 (y) dy.
2
Z 4
(4.6)
Proof. Let us explain the proof formally. Relation (2.4) implies the variational identity
Z
1Z
1
2
−δ
|∇a| dx = δ
V (|a|) − ω 2 a2 dx.
2
2
Therefore,
Z x 2
1 2 2 x x
d d 1 Z V a
− ω a
dx.
∇x a
dx =
− dσ σ=1 2
σ
dσ σ=1
σ
2
σ
Equivalently,
−
Z
d 1
d 1Z
V (a(y)) − ω 2 a2 (y) dy.
σ n−2
|∇y a(y)|2 dy =
σn
dσ σ=1
2
dσ σ=1
2
This gives the Pokhozhaev identity (4.6), or (in the notation of (4.2) and (4.3))
−(n − 2)
n
X
k=1
h
i
Ik = n V0 − ω 2 I0 .
(4.7)
✷
Step 3. To derive (4.1), let us eliminate V0 and ω 2 from (4.4) and (4.7) obtaining
ω 2 I0 =
n−2
(I1 + ... + In ) + V0 .
n
(4.8)
Hence, E0 and Ev (see (4.4) and (4.5)) become
E0 =
n
2n − 2 X
Ik + 2V0 ,
n k=1
(4.9)
n
n
1 X
n−2 X
Ev =
Ik + γ 2 (v 2 + 1) I1 +
Ik + V0 + V0
γ k=2
n k=1
"
#
n
n − 2
n − 2 X
1
2 2
I1 γ (v +1) 1+
+
Ik 1+γ 2(v 2 + 1)
+V0 (γ 2 (v 2 +1)+1) .(4.10)
=
γ
n
n
k=2
"
Note that γ 2 (v 2 + 1) + 1 = 2γ 2 and 1 + γ 2 (v 2 + 1)
by using (4.9), we can represent Ev as
Ev = γ
#
n−2
2n − 2
2v 2
= γ2
− γ2
. Therefore,
n
n
n
n
n
X
2v 2 2n − 2 X
I1 (n − 1) −
Ik + γ2V0
Ik + γ
n k=1
n
k=2
= γE0 + γ
n
X
2v 2 I1 (n − 1) −
Ik .
n
k=2
(4.11)
Using assumption A1, we obtain I1 = I2 = ... = In , which implies identity (4.1) for Ev .
Step 4. We have
Pv = −Re
= −Re
Z
Z
ψ̇v ∇ψv dx
(−γv(∇1 a)(y) − iωγa(y)) γ(∇1 a)(y) + iωγva(y), ∇2 a(y), . . . , ∇n a(y)
5
1
γ
dy.
Since a(x) is a real function, we obtain (in the notation of (4.2) and (4.3))
Pv = γv
Z
(|∇1 a(y)|2 + ω 2 a2 (y)) dy, 0, . . . , 0 = γv2(I1 + ω 2 I0 ), 0, . . . , 0 ,
(4.12)
because ∇1 a(y) is an even function with respect to any variable y2 , ..., yn , while ∇k a(y) is
an odd function of yk , k = 2, ..., n . Using (4.8) and (4.5), we obtain
γv2(I1 + ω 2 I0 ) = γvE0 +
n
X
2γv (n − 1)I1 −
Ik .
n
k=2
(4.13)
Therefore, relations (4.12) and (4.13), together with I1 = I2 = ... = In , imply
Pv = (γvE0 , 0, . . . , 0),
what yields (4.1) for Pv .
✷
Remarks. i) Formulas (4.11) and (4.8) show that identity (4.1) holds for general non-radial
solitary waves if and only if
I2 + ... + In = I1 (n − 1).
(4.14)
ii) Relation (4.1) implies the Einstein identity E0 = m0 with m0 > 0 (see the remark
after Theorem 4.1).
iii) For ω = 0 , condition S4 contradicts S2. Hence, in this case we have E0 ≤ 0 , i.e.,
“the mass” is negative. This corresponds to the instability of solitons with ω = 0 , what was
proved in [4]. This fact forces us to consider solitary waves with ω 6= 0 for U(1) -invariant
equations.
5
Nonradial excited states
In the previous section we have considered a spherically symmetric solitary wave (“ground
state”) with zero angular momentum. For n = 1 , the condition (4.14) obviously holds (and
condition (4.1) as well) for all solitary waves. Let us prove that (4.1) holds for n = 2 for
non-radial solitary waves that describe the excited states with nonzero angular momentum.
Introduce polar coordinates by setting x1 = r cos ϕ and x2 = r sin ϕ .
Lemma 5.1 Let ψ0 (x, t) = a(x)e−iωt be a solitary wave for (2.1) with a(x) := R(r)eikϕ ,
where k is an integer, R(r) is real valued, R(0) = 0 , and A2 holds. Then relation (4.1) is
satisfied.
Proof. In the polar coordinates
∇1 a = R′ (r) cos kϕ − ik
R(r)
R(r)
sin kϕ, ∇2 a = R′ (r) sin kϕ + ik
cos kϕ.
r
r
Therefore,
I1
1
≡
2
Z
1
|∇1 a| dy1dy2 =
2
2
Z∞
0
rdr
Z2π
0
h
dϕ |R′ (r)|2 cos2 kϕ + k 2
6
i
|R(r)|2
2
sin
kϕ
r2
I2
π
=
2
Z∞ h
1
≡
2
Z
0
|R(r)|2 i
r dr,
r2
Z∞
1
|∇2 a| dy1dy2 =
2
2
Z∞ h
π
2
=
|R′ (r)|2 + k 2
|R′ (r)|2 + k 2
0
Z2π
rdr
0
0
|R(r)|
r2
2i
h
dϕ |R′ (r)|3 sin2 kϕ + k 2
i
|R(r)|2
2
cos
kϕ
r2
r dr.
Hence, I1 = I2 , i.e., (4.14) holds, which implies (4.1) for Ev .
✷
Now let us prove identity (4.1) for Pv :
Pv : = −Re
= −Re
Z
ψ̇v ∇ψv dx
Z
(−γv(∇1 a)(y) − iωγa(y)) γ(∇1 a)(y) + iωγva(y), ∇2 a(y)
IR
IR
2
2
1
γ
dy. (5.1)
First,
Z
Re
(∇1 a)(y)∇2 a(y) dy
2
= Re
IR
+∞
Z
r dr
0
+∞
Z "
= Re
Z2π
0
R′ (r) cos ϕ − ik
′2
R (r) − k
0
2R
2
#
R(r)
R(r)
sin kϕ R′ (r) sin kϕ − ik
cos kϕ dϕ
r
r
(r)
r dr
r2
Z2π
sin 2kϕ
dϕ = 0.
2
0
(5.2)
Similarly,
Re
= Re
Z
a(y)∇2a(y) dy = Re
IR2
+∞
Z "
+∞
Z
r dr
0
R(r)R′ (r) + k
0
R2 (r)
r dr
r
#
Z2π
0
Z2π
0
R(r)eikϕ R′ (r) sin kϕ − ik
R(r)
cos kϕ dϕ
r
sin 2kϕ
dϕ = 0.
2
(5.3)
Hence, relations (5.1)-(5.3) imply
Pv : = −Re
=
γv
Z
ZIR
IR
2
(−γv(∇1 a)(y) − iωγa(y)) γ(∇1 a)(y) + iωγva(y), 0
(|∇1 a(y)|2 + ω 2 a2 (y)) dy, 0 = γv2(I1 + ω 2 I0 ), 0 .
2
1
γ
dy
(5.4)
as in (4.12). Further, relation (4.13) and the equality I1 = I2 , which is already proved,
imply the validity of (4.1) for Pv .
✷
7
Remark. The existence of solitary waves ψ0 (x, t) = R(r)eikφ e−iωt with k 6= 0 was not
proved in [3]. However, in this case, equation (2.4) reduces to an ordinary differential equation (as for radial solutions). Therefore, it is natural to think that existence can be proved
by modifing the method in [3].
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