Derrick's theorem

Derrick's theorem is an argument by physicist G.H. Derrick which shows that stationary localized solutions to a or nonlinear Klein–Gordon equation in 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 solutions to the nonlinear wave equation

${\displaystyle \nabla ^{2}\theta -{\frac {\partial ^{2}\theta }{\partial t^{2}}}={\frac {1}{2}}f'(\theta ),\qquad \theta (x,t)\in \mathbb {R} ,\quad x\in \mathbb {R} ^{3},}$

now known under the name of Derrick's Theorem. (Above, ${\displaystyle f(s)}$ is a differentiable function with ${\displaystyle f'(0)=0}$.)

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

${\displaystyle E=\int \left[(\nabla \theta )^{2}+f(\theta )\right]\,d^{3}x.}$

A necessary condition for the solution to be stable is ${\displaystyle \delta ^{2}E\geq 0\,}$. Suppose ${\displaystyle \theta (x)\,}$ is a localized solution of ${\displaystyle \delta E=0\,}$. Define ${\displaystyle \theta _{\lambda }(x)=\theta (\lambda x)\,}$ where ${\displaystyle \lambda }$ is an arbitrary constant, and write ${\displaystyle I_{1}=\int (\nabla \theta )^{2}d^{3}x}$, ${\displaystyle I_{2}=\int f(\theta )d^{3}x}$. Then

${\displaystyle E_{\lambda }=\int \left[(\nabla \theta _{\lambda })^{2}+f(\theta _{\lambda })\right]\,d^{3}x=I_{1}/\lambda +I_{2}/\lambda ^{3}.}$

Whence ${\displaystyle dE_{\lambda }/d\lambda \vert _{\lambda =1}=-I_{1}-3I_{2}=0.\,}$ and since ${\displaystyle I_{1}>0\,}$,

${\displaystyle \left.{\frac {d^{2}E_{\lambda }}{d\lambda ^{2}}}\right|_{\lambda =1}=2I_{1}+12I_{2}=-2I_{1}\,<0.}$

That is, ${\displaystyle \delta ^{2}E<0\,}$ for a variation corresponding to a uniform stretching of the particle. Hence the solution ${\displaystyle \theta (x)\,}$ is unstable.

Derrick's argument works for ${\displaystyle x\in \mathbb {R} ^{n}}$, ${\displaystyle n\geq 3\,}$.

Pokhozhaev's identity[]

More generally,^[2] let ${\displaystyle g}$ be continuous, with ${\displaystyle g(0)=0}$. Denote ${\displaystyle G(s)=\int _{0}^{s}g(t)\,dt}$. Let

${\displaystyle u\in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n}),\qquad \nabla u\in L^{2}(\mathbb {R} ^{n}),\qquad G(u(\cdot ))\in L^{1}(\mathbb {R} ^{n}),\qquad n\in \mathbb {N} ,}$

be a solution to the equation

${\displaystyle -\nabla ^{2}u=g(u)}$,

in the sense of distributions. Then ${\displaystyle u}$ satisfies the relation

${\displaystyle (n-2)\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G(u(x))\,dx,}$

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 ${\displaystyle \partial _{t}^{2}u=\nabla ^{2}u-{\frac {1}{2}}f'(u)}$ in the Hamiltonian form ${\displaystyle \partial _{t}u=\delta _{v}H(u,v)}$, ${\displaystyle \partial _{t}v=-\delta _{u}H(u,v)}$, where ${\displaystyle u,\,v}$ are functions of ${\displaystyle x\in \mathbb {R} ^{n},\,t\in \mathbb {R} }$, the Hamilton function is given by

${\displaystyle H(u,v)=\int _{\mathbb {R} ^{n}}\left({\frac {1}{2}}|v|^{2}+{\frac {1}{2}}|\nabla u|^{2}+{\frac {1}{2}}f(u)\right)\,dx,}$

and ${\displaystyle \delta _{u}H\,}$, ${\displaystyle \delta _{v}H\,}$ are the variational derivatives of ${\displaystyle H(u,v)\,}$.

Then the stationary solution ${\displaystyle u(x,t)=\theta (x)\,}$ has the energy ${\displaystyle H(\theta ,0)=\int _{\mathbb {R} ^{n}}\left({\frac {1}{2}}|\nabla \theta |^{2}+{\frac {1}{2}}f(\theta )\right)\,d^{n}x}$ and satisfies the equation

${\displaystyle 0=\partial _{t}\theta (x)=-\partial _{u}H(\theta ,0)={\frac {1}{2}}E'(\theta ),}$

with ${\displaystyle E'\,}$ denoting a variational derivative of the functional ${\displaystyle E=\int _{\mathbb {R} ^{n}}[\vert \nabla \theta \vert ^{2}+f(\theta )]\,d^{n}x}$. Although the solution ${\displaystyle \theta (x)\,}$ is a critical point of ${\displaystyle E\,}$ (since ${\displaystyle E'(\theta )=0\,}$), Derrick's argument shows that ${\displaystyle {\frac {d^{2}}{d\lambda \,^{2}}}E(\theta (\lambda x))<0}$ at ${\displaystyle \lambda =1\,}$, hence ${\displaystyle u(x,t)=\theta (x)\,}$ is not a point of the local minimum of the energy functional ${\displaystyle H\,}$. Therefore, physically, the solution ${\displaystyle \theta (x)\,}$ is expected to be unstable. A related result, showing non-minimization of the energy of localized stationary states (with the argument also written for ${\displaystyle n=3}$, although the derivation being valid in dimensions ${\displaystyle n\geq 2}$) 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 solitary wave ${\displaystyle u(x,t)=\phi _{\omega }(x)e^{-i\omega t}\,}$ with frequency ${\displaystyle \omega \,}$ 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[]