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.
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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. (Above,
The energy of the time-independent solution
A necessary condition for the solution to be stable is
Whence
That is,
The above argument also works for
Pohozaev's identity
More generally, let
be a solution to the equation
in the sense of distributions. Then
known as Pohozaev's identity. It result is similar to the Virial theorem.
Interpretation in the Hamiltonian form
We may write the equation
and
Then the stationary solution
with
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