Derrick's theorem


Derrick's theorem is an argument due to a physicist G.H. Derrick
which shows that stationary localized solutions to a nonlinear wave equation
or nonlinear Klein-Gordon equation
in spatial dimensions three and higher are unstable.

Original argument

Derrick's paper,
which was considered an obstacle to
interpreting soliton-like solutions as particles,
contained the following physical argument
about non-existence of stable localized stationary solutions
to the nonlinear wave equation
now known under the name of Derrick's Theorem.
The energy of the time-independent solution
is given by
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
Whence
and since,
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,
let be continuous, with.
Denote.
Let
be a solution to the equation
in the sense of distributions.
Then satisfies the relation
known as Pokhozhaev's identity.
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
and,
are the
variational derivatives of.
Then the stationary solution
has the energy
and
satisfies the equation
with
denoting a variational derivative
of the functional
Although the solution
is a critical point of ,
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
was obtained by R.H. Hobart in 1963.

Relation to linear instability

A stronger statement, linear instability of localized stationary solutions
to the nonlinear wave equation is proved
by P. Karageorgis and W.A. Strauss in 2007.

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 that a time-periodic solitary wave with frequency may be orbitally stable if the Vakhitov-Kolokolov stability criterion is satisfied.