# Klein-Gordon/Maxwell Equation: dissipative or dispersive?

In Aspects of Symmetry, Coleman says (p. 185)

''Most of the simple field theories with which we are familiar have the property that all of their non-singular solutions of finite total energy are dissipative. This is the case for Maxwell's equations, the Klein-Gordon equation, etc.''

However, isn't it the case that the mentioned equations for regular initial data are dispersive? For example, we take the vacuum Maxwell equations - they will have the form (for the electric/magnetic field)

$$$$y_{xx}-c^{-2}y_{tt}=0.$$$$

If we take $$y(x,0)=\sin x$$ and $$\partial_t y(x,0)=(1/\epsilon)\sin x$$ for the electric field and mutatis mutandis for the magnetic field, we obtain the electromagnetic wave - a dispersive solution.

• This is unclear -- what do you think the definitions of the words "dispersive" and "dissipative" are? Jun 7, 2019 at 18:56
• I'm not OP but "dissipative" usually means a solution that does not conserve energy, like in a system with a damping term. "Dispersive" sounds more like what the 3D KG equation and the Maxwell equations act like - spreading the energy density across an ever bigger volume as time goes on, while the total remains constant. Contrast the massless Klein-Gordon equation in 1D where if your solution begins as a lump it stays as a lump and moves onwards without spreading. Jun 7, 2019 at 18:59
• Maybe he meant something as, "Energy in any given closed subspace diminishes to zero when $t\to\infty$"? Since there's no ideal EMW, or free space, your counter-example wouldn't matter. Jun 7, 2019 at 19:38

This is the general solution of wave equation with speed $$c$$ or Klein-Gordon Equation with Dissipation (KGD): $$\Box y + a \dot y + b y = 0.$$ For $$a=0$$ and $$b=0$$ you have the plain d'Alembert equation where the waves travel at $$c$$.
For just $$a=0$$ you have the K-G equation where the waves' group speed is $$c$$ but the single waves propagate at different speeds depending on $$b$$ and so the waves' group is dispersed
For just $$b=0$$ you have Viscoelastic Maxwell Equation, a dumped oscillator where the waves propagate at $$c$$ but loosing amplitude, i.e. is energy is dissipated.