Derrick's theorem explained

Derrick's theorem is an argument by physicist G. H. Derrickwhich shows that stationary localized solutions to a nonlinear wave equationor nonlinear Klein - Gordon equationin spatial dimensions three and higher are unstable.

Original argument

Derrick's paper,^[1] which was considered an obstacle tointerpreting soliton-like solutions as particles,contained the following physical argumentabout non-existence of stable localized stationary solutionsto the nonlinear wave equation

\nabla ^{2} \theta - (\partial ^{2\theta})/(= \partial t ^{2}) (1)/(2) f'(\theta), \theta(x,t)\in\R, 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 byE = \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 _{λ(x)=\theta(λ} x) where λ is an arbitrary constant, and write I ^{2} _{1= \int(\nabla\theta)} d ^{3} x , I _{2=\int} f(\theta) d ^{3} x . Then E _{λ =\int\left[(\nabla\theta} ^{2+ f(\theta} ^{λ )\right]} d ^{3} x = I _{1/λ} + I ^{3 .} _{2 / λ} Whence dE _{λ/dλ\vert} _{λ=1} =- I _{1-3} I _{2=0.} and since I _{1>0} ,\left. (d ^{2 E} _{λ})/(d λ ^{2}) \right| _{λ=1} =2 I _{1+12} I _{2=-2} I _{1<0.} That is, \delta ^{2} E< 0 for a variation corresponding toa 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,^[2] let

g be continuous, with g( 0 ) =0 .Denote G(s) = \int ^{s} _{0} g(t)dt .Letu\in L ^{inf ty} _{loc} (\R ^{n), \nabla} u\in L ^{2(\R} ^{n), G(u( ⋅ ))\in} L ^{1(\R} ^{n), 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 _{\R ^{n}} |\nabla u(x)| ^{2 dx = n\int} _{\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

\partial ^{2} _{t} u = \nabla ^{2} u - (1)/(2) f'(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 byH(u,v) = \int _{\R ^{n}} \left( (1)/(2) |v| ^{2+ (1)/(2)} |\nabla u| ^{2+ (1)/(2)} f(u) \right)dx, and \delta _{u} H , \delta _{v} H are thevariational derivatives of H(u,v) .

Then the stationary solution

u(x,t) = \theta(x) has the energyH(\theta, 0 ) = \int _{\R ^{n}} \left( (1)/(2) |\nabla\theta| ^{2+ (1)/(2)} f(\theta) \right)d ^{n} x and satisfies the equation0= \partial _{t} \theta(x) =- \partial _{u} H(\theta, 0 ) = (1)/(2) E'(\theta), withE' denoting a variational derivativeof the functionalE = \int _{\R ^{n}} [\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(d ^{2})/(d λ ^{2}) E(\theta( λ x))< 0 at λ=1 ,henceu(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.^[4]

Relation to linear instability

A stronger statement, linear (or exponential) instability of localized stationary solutionsto the nonlinear wave equation (in any spatial dimension) is provedby P. Karageorgis and W. A. Strauss in 2007.^[5]

Stability of localized time-periodic solutions

u(x,t) = \phi ^{- i\omega t} _{\omega(x)e} 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

1. G. H. Derrick. Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys.. 5. 9. 1252 - 1254. 1964. 10.1063/1.1704233. 1964JMP.....5.1252D . free.
2. Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations, I. Existence of a ground state. Arch. Rational Mech. Anal.. 82. 4. 1983. 313 - 345. 10.1007/BF00250555. 1983ArRMA..82..313B. 123081616.
3. Pokhozhaev, S. I.. On the eigenfunctions of the equation \Delta u +λ f(u) =0 . Dokl. Akad. Nauk SSSR. 165. 36 - 39. 1965.
4. R. H. Hobart. On the instability of a class of unitary field models. Proc. Phys. Soc.. 82. 2. 201 - 203. 1963. 10.1088/0370-1328/82/2/306. 1963PPS....82..201H.
5. P. Karageorgis and W. A. Strauss. Instability of steady states for nonlinear wave and heat equations. J. Differential Equations. 241. 184 - 205. 2007. 1. 10.1016/j.jde.2007.06.006 . math/0611559. 2007JDE...241..184K. 18889076.
6. Вахитов, Н. Г. and Колоколов, А. А.. Стационарные решения волнового уравнения в среде с насыщением нелинейности. Известия высших учебных заведений. Радиофизика. 16. 1973. 1020 - 1028 . N. G. Vakhitov and A. A. Kolokolov. Stationary solutions of the wave equation in the medium with nonlinearity saturation. Radiophys. Quantum Electron.. 16. 7. 1973. 783 - 789. 10.1007/BF01031343. 1973R&QE...16..783V . 123386885.