Derrick's theorem

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.

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
:$\nabla^2 \theta-\frac=\frac 1 2 f'(\theta), \qquad \theta(x,t)\in\R,\quad x\in\R^3,$
now known under the name of Derrick's Theorem. (Above, $f(s)$ is a differentiable function with $f'(0)=0$.)

The energy of the time-independent solution $\theta(x)\,$ is given by

:
E=\int\left \nabla\theta)^2+f(\theta)\right\, d^3 x.

A necessary condition for the solution to be stable is $\delta^2 E\ge 0\,$. Suppose $\theta(x)\,$ is a localized solution of $\delta E=0\,$. Define $\theta_\lambda(x)=\theta(\lambda x)\,$ where $\lambda$ is an arbitrary constant, and write $I_1=\int(\nabla\theta)^2 d^3 x$, $I_2=\int f(\theta) d^3 x$. Then
:
E_\lambda
=\int\left \nabla\theta_\lambda)^2+f(\theta_\lambda)\right\, d^3 x
=I_1/\lambda +I_2/\lambda^3.

Whence $dE_\lambda/d\lambda\vert_=-I_1-3 I_2=0.\,$
and since $I_1>0\,$,
:$\left.\frac\_=2 I_1+12 I_2=-2 I_1\,<0.$
That is, $\delta^2 E<0\,$ for a variation corresponding to
a uniform stretching of the ''particle''.
Hence the solution $\theta(x)\,$ is unstable.

Derrick's argument works for $x\in\R^n$, $n\ge 3\,$.

Pokhozhaev's identity

More generally,
let $g$ be continuous, with $g(0)=0$.
Denote $G(s)=\int_0^s g(t)\,dt$.
Let
:$u\in L^\infty_(\R^n), \qquad \nabla u\in L^2(\R^n), \qquad G(u(\cdot))\in L^1(\R^n), \qquad n\in\N,$
be a solution to the equation
:$-\nabla^2 u=g(u)$,
in the sense of distributions.
Then $u$ satisfies the relation
:$(n-2)\int_, \nabla u(x), ^2\,dx=n\int_G(u(x))\,dx,$
known as Pokhozhaev's identity (sometimes spelled as ''Pohozaev's identity'').
This result is similar to the virial theorem.

Interpretation in the Hamiltonian form

We may write the equation
$\partial_t^2 u=\nabla^2 u-\fracf'(u)$
in the Hamiltonian form
$\partial_t u=\delta_v H(u,v)$,
$\partial_t v=-\delta_u H(u,v)$,
where $u,\,v$ are functions of $x\in\R^n,\,t\in\R$,
the Hamilton function is given by
:$H(u,v)=\int_\left( \frac, v, ^2+\frac, \nabla u, ^2+\fracf(u) \right)\,dx,$
and $\delta_u H\,$, $\delta_v H\,$
are the
variational derivatives of $H(u,v)\,$.

Then the stationary solution $u(x,t)=\theta(x)\,$
has the energy
$H(\theta,0)=\int_\left( \frac, \nabla\theta, ^2+\fracf(\theta) \right)\,d^n x$
and
satisfies the equation
:$0=\partial_t \theta(x)=-\partial_u H(\theta,0)=\fracE'(\theta),$
with
$E'\,$ denoting a variational derivative
of the functional
E=\int_ vert\nabla\theta\vert^2+f(\theta),d^n x.
Although the solution $\theta(x)\,$
is a critical point of $E\,$ (since $E'(\theta)=0\,$),
Derrick's argument shows that
$\fracE(\theta(\lambda x))<0$
at $\lambda=1\,$,
hence
$u(x,t)=\theta(x)\,$
is not a point of the local minimum of the energy functional $H\,$.
Therefore, physically, the solution $\theta(x)\,$ is expected to be unstable.
A related result, showing non-minimization of the energy of localized stationary states
(with the argument also written for $n=3$, although the derivation being valid in dimensions $n\ge 2$) was obtained by R. H. Hobart in 1963.

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.

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 $u(x,t)=\phi_\omega(x)e^{-i\omega t}\,$ with frequency $\omega\,$ may be orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied.

See also

* Orbital stability
* Pokhozhaev's identity
* Vakhitov–Kolokolov stability criterion
* Virial theorem

References

Stability theory
Solitons