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 equation in spatial dimensions three and higher are unstable.
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 solutions to the nonlinear wave equation now known under the name of Derrick's Theorem.
is a differentiable function with
The energy of the time-independent solution
is given by A necessary condition for the solution to be stable is
( x ) = θ ( λ x )
is an arbitrary constant, and write
for a variation corresponding to a uniform stretching of the particle.
Derrick's argument works for
Let be a solution to the equation in the sense of distributions.
satisfies the relation known as Pokhozhaev's identity (sometimes spelled as Pohozaev's identity).
[3] This result is similar to the virial theorem.
We may write the equation
in the Hamiltonian form
, the Hamilton function is given by and
are the variational derivatives of
and satisfies the equation with
denoting a variational derivative of the functional
is a critical point of
), Derrick's argument shows that
( θ ( λ x ) ) < 0
is not a point of the local minimum of the energy functional
Therefore, physically, the solution
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] 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] 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
may be orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied.