'''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,<ref>{{ 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 }}</ref> 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 :<math>\nabla^2 \theta-\frac{\partial^2\theta}{\partial t^2}=\frac 1 2 f'(\theta), \qquad \theta(x,t)\in\R,\quad x\in\R^3, </math> now known under the name of Derrick's Theorem. (Above, <math>f(s)</math> is a differentiable function with <math>f'(0)=0</math>.)
The energy of the time-independent solution <math>\theta(x)\,</math> is given by
:<math> E=\int\left[(\nabla\theta)^2+f(\theta)\right] \, d^3 x. </math>
A necessary condition for the solution to be stable is <math>\delta^2 E\ge 0\,</math>. Suppose <math>\theta(x)\,</math> is a localized solution of <math>\delta E=0\,</math>. Define <math>\theta_\lambda(x)=\theta(\lambda x)\,</math> where <math>\lambda</math> is an arbitrary constant, and write <math>I_1=\int(\nabla\theta)^2 d^3 x</math>, <math>I_2=\int f(\theta) d^3 x</math>. Then :<math> E_\lambda =\int\left[(\nabla\theta_\lambda)^2+f(\theta_\lambda)\right] \, d^3 x =I_1/\lambda +I_2/\lambda^3. </math> Whence <math> dE_\lambda/d\lambda\vert_{\lambda=1}=-I_1-3 I_2=0.\,</math> and since <math>I_1>0\,</math>, :<math> \left.\frac{d^2E_\lambda}{d\lambda^2}\right|_{\lambda=1}=2 I_1+12 I_2=-2 I_1\,<0. </math> That is, <math>\delta^2 E<0\,</math> for a variation corresponding to a uniform stretching of the ''particle''. Hence the solution <math>\theta(x)\,</math> is unstable.
Derrick's argument works for <math>x\in\R^n</math>, <math>n\ge 3\,</math>.
==Pokhozhaev's identity==
More generally,<ref>{{ 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 }}</ref> let <math>g</math> be continuous, with <math>g(0)=0</math>. Denote <math>G(s)=\int_0^s g(t)\,dt</math>. Let :<math>u\in L^\infty_{\mathrm{loc}}(\R^n), \qquad \nabla u\in L^2(\R^n), \qquad G(u(\cdot))\in L^1(\R^n), \qquad n\in\N, </math> be a solution to the equation :<math>-\nabla^2 u=g(u)</math>, in the sense of distributions. Then <math>u</math> satisfies the relation :<math>(n-2)\int_{\R^n}|\nabla u(x)|^2\,dx=n\int_{\R^n}G(u(x))\,dx,</math> known as Pokhozhaev's identity (sometimes spelled as ''Pohozaev's identity'').<ref>{{ cite journal |author=Pokhozhaev, S. I. |title=On the eigenfunctions of the equation <math>\Delta u+\lambda f(u)=0</math> |journal=Dokl. Akad. Nauk SSSR |volume=165 |pages=36–39 |year=1965 |url=http://mi.mathnet.ru/rus/dan/v165/i1/p36 }}</ref> This result is similar to the virial theorem.
==Interpretation in the Hamiltonian form==
We may write the equation <math>\partial_t^2 u=\nabla^2 u-\frac{1}{2}f'(u)</math> in the Hamiltonian form <math>\partial_t u=\delta_v H(u,v)</math>, <math>\partial_t v=-\delta_u H(u,v)</math>, where <math>u,\,v</math> are functions of <math>x\in\R^n,\,t\in\R</math>, the Hamilton function is given by :<math> 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, </math> and <math>\delta_u H\,</math>, <math>\delta_v H\,</math> are the variational derivatives of <math>H(u,v)\,</math>.
Then the stationary solution <math>u(x,t)=\theta(x)\,</math> has the energy <math>H(\theta,0)=\int_{\R^n}\left( \frac{1}{2}|\nabla\theta|^2+\frac{1}{2}f(\theta) \right)\,d^n x</math> and satisfies the equation :<math> 0=\partial_t \theta(x)=-\partial_u H(\theta,0)=\frac{1}{2}E'(\theta), </math> with <math>E'\,</math> denoting a variational derivative of the functional <math>E=\int_{\R^n}[\vert\nabla\theta\vert^2+f(\theta)]\,d^n x</math>. Although the solution <math>\theta(x)\,</math> is a critical point of <math>E\,</math> (since <math>E'(\theta)=0\,</math>), Derrick's argument shows that <math>\frac{d^2}{d\lambda\,^2}E(\theta(\lambda x))<0</math> at <math>\lambda=1\,</math>, hence <math>u(x,t)=\theta(x)\,</math> is not a point of the local minimum of the energy functional <math>H\,</math>. Therefore, physically, the solution <math>\theta(x)\,</math> is expected to be unstable. A related result, showing non-minimization of the energy of localized stationary states (with the argument also written for <math>n=3</math>, although the derivation being valid in dimensions <math>n\ge 2</math>) was obtained by R. H. Hobart in 1963.<ref>{{ 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 }}</ref>
==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.<ref>{{ 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 }}</ref>
==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<ref>{{ 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 }}</ref> that a time-periodic solitary wave <math>u(x,t)=\phi_\omega(x)e^{-i\omega t}\,</math> with frequency <math>\omega\,</math> 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== <references />
Category:Stability theory Category:Solitons