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

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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

 

now known under the name of Derrick's Theorem. (Above,   is a differentiable function with  .)

The energy of the time-independent solution   is given by

 

A necessary condition for the solution to be stable is  . Suppose   is a localized solution of  . Define   where   is an arbitrary constant, and write  ,  . Then

 

Whence   and since  ,

 

That is,   for a variation corresponding to a uniform stretching of the particle. Hence the solution   is unstable.

Derrick's argument works for  ,  .

Pokhozhaev's identity

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More generally,[2] let   be continuous, with  . Denote  . Let

 

be a solution to the equation

 ,

in the sense of distributions. Then   satisfies the relation

 

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

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We may write the equation   in the Hamiltonian form  ,  , where   are functions of  , the Hamilton function is given by

 

and  ,   are the variational derivatives of  .

Then the stationary solution   has the energy   and satisfies the equation

 

with   denoting a variational derivative of the functional  . Although the solution   is a critical point of   (since  ), Derrick's argument shows that   at  , hence   is not a point of the local minimum of the energy functional  . Therefore, physically, the solution   is expected to be unstable. A related result, showing non-minimization of the energy of localized stationary states (with the argument also written for  , although the derivation being valid in dimensions  ) was obtained by R. H. Hobart in 1963.[4]

Relation to linear instability

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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

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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 time-periodic solitary wave   with frequency   may be orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied.

See also

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References

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  1. ^ G. H. Derrick (1964). "Comments on nonlinear wave equations as models for elementary particles". J. Math. Phys. 5 (9): 1252–1254. Bibcode:1964JMP.....5.1252D. doi:10.1063/1.1704233.
  2. ^ Berestycki, H. and Lions, P.-L. (1983). "Nonlinear scalar field equations, I. Existence of a ground state". Arch. Rational Mech. Anal. 82 (4): 313–345. Bibcode:1983ArRMA..82..313B. doi:10.1007/BF00250555. S2CID 123081616.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. ^ Pokhozhaev, S. I. (1965). "On the eigenfunctions of the equation  ". Dokl. Akad. Nauk SSSR. 165: 36–39.
  4. ^ R. H. Hobart (1963). "On the instability of a class of unitary field models". Proc. Phys. Soc. 82 (2): 201–203. Bibcode:1963PPS....82..201H. doi:10.1088/0370-1328/82/2/306.
  5. ^ P. Karageorgis and W. A. Strauss (2007). "Instability of steady states for nonlinear wave and heat equations". J. Differential Equations. 241 (1): 184–205. arXiv:math/0611559. Bibcode:2007JDE...241..184K. doi:10.1016/j.jde.2007.06.006. S2CID 18889076.
  6. ^ Вахитов, Н. Г. and Колоколов, А. А. (1973). "Стационарные решения волнового уравнения в среде с насыщением нелинейности". Известия высших учебных заведений. Радиофизика. 16: 1020–1028.{{cite journal}}: CS1 maint: multiple names: authors list (link) N. G. Vakhitov and A. A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16 (7): 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343. S2CID 123386885.