Derrick's theorem

Derrick's theorem is an argument due to a 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,[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 time-periodic 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

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.CS1 maint: multiple names: authors list (link)
3. Pokhozhaev, S.I. (1965). "On the eigenfunctions of the equation ${\displaystyle \Delta u+\lambda f(u)=0}$". 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. 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: 184–205. arXiv:math/0611559. doi:10.1016/j.jde.2007.06.006.
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.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.