Derrick's theorem

Derrick's paper,{{ cite journal

|author=G. H. Derrick

|title=Comments on nonlinear wave equations as models for elementary particles

|journal=J. Math. Phys.

|volume=5

|issue=9

|pages=1252–1254

|year=1964

|doi=10.1063/1.1704233

|bibcode=1964JMP.....5.1252D |doi-access=free

}}

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

:$\backslash nabla^2\; \backslash theta-\backslash frac\{\backslash partial^2\backslash theta\}\{\backslash partial\; t^2\}=\backslash frac\; 1\; 2\; f\text{'}(\backslash theta),$

\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 $f\text{'}(0)=0$.)

The energy of the time-independent solution $\backslash theta(x)\backslash ,$ 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 $\backslash delta^2\; E\backslash ge\; 0\backslash ,$. Suppose $\backslash theta(x)\backslash ,$ is a localized solution of $\backslash delta\; E=0\backslash ,$. Define $\backslash theta\_\backslash lambda(x)=\backslash theta(\backslash lambda\; x)\backslash ,$ where $\backslash lambda$ is an arbitrary constant, and write $I\_1=\backslash int(\backslash nabla\backslash theta)^2\; d^3\; x$, $I\_2=\backslash int\; f(\backslash theta)\; d^3\; x$. 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_{\lambda=1}=-I_1-3 I_2=0.\,

and since $I\_1>0\backslash ,$,

:$$

\left.\frac{d^2E_\lambda}{d\lambda^2}\right|_{\lambda=1}=2 I_1+12 I_2=-2 I_1\,<0.

That is, for a variation corresponding to

a uniform stretching of the particle.

Hence the solution $\backslash theta(x)\backslash ,$ is unstable.

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

More generally,{{ cite journal

|author=Berestycki, H. and Lions, P.-L.

|title=Nonlinear scalar field equations, I. Existence of a ground state

|journal=Arch. Rational Mech. Anal.

|volume=82

|issue=4

|year=1983

|pages=313–345

|doi=10.1007/BF00250555

|bibcode=1983ArRMA..82..313B

|s2cid=123081616

}}

let $g$ be continuous, with $g(0)=0$.

Denote $G(s)=\backslash int\_0^s\; g(t)\backslash ,dt$.

Let

:$u\backslash in\; L^\backslash infty\_\{\backslash mathrm\{loc\}\}(\backslash R^n),$

\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

:$-\backslash nabla^2\; u=g(u)$,

in the sense of distributions.

Then $u$ satisfies the relation

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

known as Pokhozhaev's identity (sometimes spelled as Pohozaev's identity).{{ cite journal

|author=Pokhozhaev, S. I.

|title=On the eigenfunctions of the equation $\backslash Delta\; u+\backslash lambda\; f(u)=0$

|journal=Dokl. Akad. Nauk SSSR

|volume=165

|pages=36–39

|year=1965

|url=http://mi.mathnet.ru/rus/dan/v165/i1/p36

}}

This result is similar to the virial theorem.

We may write the equation

$\backslash partial\_t^2\; u=\backslash nabla^2\; u-\backslash frac\{1\}\{2\}f\text{'}(u)$

in the Hamiltonian form

$\backslash partial\_t\; u=\backslash delta\_v\; H(u,v)$,

$\backslash partial\_t\; v=-\backslash delta\_u\; H(u,v)$,

where $u,\backslash ,v$ are functions of $x\backslash in\backslash R^n,\backslash ,t\backslash in\backslash R$,

the Hamilton function is given by

:$$

H(u,v)=\int_{\R^n}\left(

\frac{1}{2}|v|^2+\frac{1}{2}|\nabla u|^2+\frac{1}{2}f(u)

\right)\,dx,

and $\backslash delta\_u\; H\backslash ,$, $\backslash delta\_v\; H\backslash ,$

are the

variational derivatives of $H(u,v)\backslash ,$.

Then the stationary solution $u(x,t)=\backslash theta(x)\backslash ,$

has the energy

$H(\backslash theta,0)=\backslash int\_\{\backslash R^n\}\backslash left($

\frac{1}{2}|\nabla\theta|^2+\frac{1}{2}f(\theta)

\right)\,d^n x

and

satisfies the equation

:$$

0=\partial_t \theta(x)=-\partial_u H(\theta,0)=\frac{1}{2}E'(\theta),

with

$E\text{'}\backslash ,$ denoting a variational derivative

of the functional

$E=\backslash int\_\{\backslash R^n\}[\backslash vert\backslash nabla\backslash theta\backslash vert^2+f(\backslash theta)]\backslash ,d^n\; x$.

Although the solution $\backslash theta(x)\backslash ,$

is a critical point of $E\backslash ,$ (since $E\text{'}(\backslash theta)=0\backslash ,$),

Derrick's argument shows that

at $\backslash lambda=1\backslash ,$,

hence

$u(x,t)=\backslash theta(x)\backslash ,$

is not a point of the local minimum of the energy functional $H\backslash ,$.

Therefore, physically, the solution $\backslash theta(x)\backslash ,$ 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 $n\backslash ge\; 2$) was obtained by R. H. Hobart in 1963.{{ cite journal

|author=R. H. Hobart

|title=On the instability of a class of unitary field models

|journal=Proc. Phys. Soc.

|volume=82

|issue=2

|pages=201–203

|year=1963

|doi=10.1088/0370-1328/82/2/306

|bibcode=1963PPS....82..201H

}}

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.{{ cite journal

|author=P. Karageorgis and W. A. Strauss

|title=Instability of steady states for nonlinear wave and heat equations

|journal=J. Differential Equations

|volume=241

|pages=184–205

|year=2007

|issue=1

|doi=10.1016/j.jde.2007.06.006 |arxiv=math/0611559

|bibcode=2007JDE...241..184K

|s2cid=18889076

}}

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{{ cite journal

|author=Вахитов, Н. Г. and Колоколов, А. А.

|title=Стационарные решения волнового уравнения в среде с насыщением нелинейности

|journal=Известия высших учебных заведений. Радиофизика

|volume=16

|year=1973

|pages=1020–1028 }} {{ cite journal

|author=N. G. Vakhitov and A. A. Kolokolov

|title=Stationary solutions of the wave equation in the medium with nonlinearity saturation

|journal=Radiophys. Quantum Electron.

|volume=16

|issue=7

|year=1973

|pages=783–789

|doi=10.1007/BF01031343

|bibcode=1973R&QE...16..783V |s2cid=123386885

}} that a time-periodic solitary wave $u(x,t)=\backslash phi\_\backslash omega(x)e^\{-i\backslash omega\; t\}\backslash ,$ with frequency $\backslash omega\backslash ,$ may be orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied.

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

Category:Stability theory

Category:Solitons