Derrick's theorem is an argument by physicist G. H. Derrick
which shows that stationary localized solutions to a nonlinear wave equation
or nonlinear Klein–Gordon equationin spatial dimensions three and higher are unstable.
Original argument
Derrick's paper,[1]which was considered an obstacle to
interpreting soliton-like solutions as particles,
contained the following physical argument
about non-existence of stable localized stationary solutionsto the nonlinear wave equation
![{\displaystyle \nabla ^{2}\theta -{\frac {\partial ^{2}\theta }{\partial t^{2}}}={\frac {1}{2}}f'(\theta ),\qquad \theta (x,t)\in \mathbb {R} ,\quad x\in \mathbb {R} ^{3},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
now known under the name of Derrick's Theorem. (Above,
is a differentiable function with
.)
The energy of the time-independent solution
is given by
![{\displaystyle E=\int \left[(\nabla \theta )^{2}+f(\theta )\right]\,d^{3}x.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A necessary condition for the solution to be stable is
. Suppose
is a localized solution of
. Define
where
is an arbitrary constant, and write
,
. Then
![{\displaystyle E_{\lambda }=\int \left[(\nabla \theta _{\lambda })^{2}+f(\theta _{\lambda })\right]\,d^{3}x=I_{1}/\lambda +I_{2}/\lambda ^{3}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Whence
and since
,
![{\displaystyle \left.{\frac {d^{2}E_{\lambda }}{d\lambda ^{2}}}\right|_{\lambda =1}=2I_{1}+12I_{2}=-2I_{1}\,<0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
That is,
for a variation corresponding to
a uniform stretching of the particle.
Hence the solution
is unstable.
Derrick's argument works for
,
.
Pokhozhaev's identity
More generally,[2]let
be continuous, with
.
Denote
.
Let
![{\displaystyle u\in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n}),\qquad \nabla u\in L^{2}(\mathbb {R} ^{n}),\qquad G(u(\cdot ))\in L^{1}(\mathbb {R} ^{n}),\qquad n\in \mathbb {N} ,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
be a solution to the equation
,
in the sense of distributions.
Then
satisfies the relation
![{\displaystyle (n-2)\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G(u(x))\,dx,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
known as Pokhozhaev's identity (sometimes spelled as Pohozaev's identity).[3]This result is similar to the virial theorem.
Interpretation in the Hamiltonian form
We may write the equation
in the Hamiltonian form
,
,
where
are functions of
,
the Hamilton function is given by
![{\displaystyle H(u,v)=\int _{\mathbb {R} ^{n}}\left({\frac {1}{2}}|v|^{2}+{\frac {1}{2}}|\nabla u|^{2}+{\frac {1}{2}}f(u)\right)\,dx,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
,
are thevariational derivatives of
.
Then the stationary solution
has the energy
and
satisfies the equation
![{\displaystyle 0=\partial _{t}\theta (x)=-\partial _{u}H(\theta ,0)={\frac {1}{2}}E'(\theta ),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
denoting a variational derivative
of the functional
.
Although the solution
is a critical point of
(since
),
Derrick's argument shows that
at
,
hence
is not a point of the local minimum of the energy functional
.
Therefore, physically, the solution
is expected to be unstable.
A related result, showing non-minimization of the energy of localized stationary states
(with the argument also written for
, although the derivation being valid in dimensions
) was obtained by R. H. Hobart in 1963.[4]
Relation to linear instability
A stronger statement, linear (or exponential) instability of localized stationary solutions
to the nonlinear wave equation (in any spatial dimension) is proved
by P. Karageorgis and W. A. Strauss in 2007.[5]
Stability of localized time-periodic solutions
Derrick describes some possible ways out of this difficulty, including the conjecture that Elementary particles might correspond to stable, localized solutions which are periodic in time, rather than time-independent.Indeed, it was later shown[6] that a time-periodic solitary wave
with frequency
may be orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied.
See also
References
- ^ G. H. Derrick (1964). "Comments on nonlinear wave equations as models for elementary particles". J. Math. Phys. 5 (9): 1252–1254. Bibcode:1964JMP.....5.1252D. doi:10.1063/1.1704233.
- ^ Berestycki, H. and Lions, P.-L. (1983). "Nonlinear scalar field equations, I. Existence of a ground state". Arch. Rational Mech. Anal. 82 (4): 313–345. Bibcode:1983ArRMA..82..313B. doi:10.1007/BF00250555. S2CID 123081616.
{{cite journal}}
: CS1 maint: multiple names: authors list (link) - ^ Pokhozhaev, S. I. (1965). "On the eigenfunctions of the equation Δ u + λ f ( u ) = 0 {\displaystyle \Delta u+\lambda f(u)=0}
". Dokl. Akad. Nauk SSSR. 165: 36–39.
- ^ R. H. Hobart (1963). "On the instability of a class of unitary field models". Proc. Phys. Soc. 82 (2): 201–203. Bibcode:1963PPS....82..201H. doi:10.1088/0370-1328/82/2/306.
- ^ P. Karageorgis and W. A. Strauss (2007). "Instability of steady states for nonlinear wave and heat equations". J. Differential Equations. 241 (1): 184–205. arXiv:math/0611559. Bibcode:2007JDE...241..184K. doi:10.1016/j.jde.2007.06.006. S2CID 18889076.
- ^ Вахитов, Н. Г. and Колоколов, А. А. (1973). "Стационарные решения волнового уравнения в среде с насыщением нелинейности". Известия высших учебных заведений. Радиофизика. 16: 1020–1028.
{{cite journal}}
: CS1 maint: multiple names: authors list (link) N. G. Vakhitov and A. A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16 (7): 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343. S2CID 123386885.