Klein-Gordon equation

The one-dimensional Klein-Gordon equationhas positive-frequency dispersion relation $\omega (k)=c\sqrt{k^2+A^2}$.

For interaction density $g{\varphi }_1{\varphi }_2$, the combinations ${\varphi }_{\pm }={\varphi }_1\pm {\varphi }_2$ diagonalize the equations into independent Klein-Gordon equations with the displayed mass squares. When $0<g<m^2$, all Fourier frequencies are real and nonzero. Initial data in one original field generally excite both normal combinations.

A single spatial Fourier pattern initially displaced in one field excites two different normal frequencies. Their sum and difference generate slowly varying amplitude envelopes in the original fields. Total coupled-field energy is conserved, but the separate uncoupled energies need not be, and zeros of one displacement envelope do not automatically imply exact complete energy transfer.

In a sufficiently strong electrostatic step, the positive-energy incident channel overlaps a negative-kinetic-energy channel on the far side. For a charged scalar field the outgoing antiparticle branch has negative charge current, allowing reflected particle current larger than the incident current while signed charge is conserved. The external source supplies energy for particle-antiparticle creation. This requires a multiparticle interpretation; negative transmitted charge current is not a negative probability.

For the outgoing negative-energy branch $q=-Q$, matching the field and its spatial derivative gives these signed charge-current ratios. Their sum is one although reflection exceeds one. The ideal sharp step has a singular stationary matching denominator at $E=V_0/2$ in the supercritical range. It is not a finite single-particle probability prediction; wavepackets, source preparation and field quantization matter for its physical interpretation.

Set $z=E-V_0+i0$ and continue both roots from the upper half-plane. The right-hand wavenumber is positive above the positive-energy threshold, positive imaginary inside the gap, and negative below the negative-energy threshold. Its group velocity in the last region remains rightward because both momentum and kinetic energy are negative. Choosing a positive real root there instead specifies an incoming wavepacket from the right.

A local scalar nonlinearity preserves Lorentz covariance of the wave operator when the wave speed is one. If $f=F^{\prime }$, the equation follows from $\mathcal{L}=({\partial }_t\varphi {)}^2/2-({\partial }_x\varphi {)}^2/2+F(\varphi )$. The physical potential is $-F$, so its sign must be distinguished from the sign written in the Lagrangian.

For $f(w)=-aw+b{w}^n$, with $a,b>0$ and integer $n>1$, a positive localized static pulse has $C^{n-1}=a(n+1)/(2b)$ and $B=(n-1)\sqrt{a}/2$. Writing the same profile as $[A{\cosh }^2(Bx){]}^{1/(1-n)}$ instead gives $A=2b/[a(n+1)]$. These are different amplitude conventions for the same solution. A Lorentz boost moves the pulse at speed $\tanh \beta$, preserves its peak height and contracts its width by $1/\cosh \beta$. Existence alone does not imply dynamical stability.

A canonical spatially homogeneous field with $V=m^2{\varphi }^2/2$ obeys $\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$. The $3H\dot{\varphi }$ term expresses dilution of field momentum by expansion. Conservation of its stress energy gives this equation wherever $\dot{\varphi }\ne 0$; the canonical field equation is needed to fix an identically stationary interval, while continuity extends a regular solution across isolated turning points.

With inverse-time convention $e^{i\omega t}$, the half-line Klein-Gordon equation ${\sigma }_{tt}-{\sigma }_{rr}+{\mu }^2\sigma =0$ has spatial normal modes $A{e}^{ikr}+B{e}^{-ikr}$, where $k=\sqrt{{\omega }^2-{\mu }^2}>0$ above threshold and $k=i\sqrt{{\mu }^2-{\omega }^2}$ below it. For the branch even in $\omega$, reality above threshold requires $A(\omega )=B(-\omega {)}^{\ast }$. Below threshold boundedness eliminates $B$, and reality requires $A(\omega )=A(-\omega {)}^{\ast }$. At threshold the spatial differential equation instead has the basis $1,r$; boundedness retains only the constant mode. A spectral integral with ordinary integrable functions as amplitudes ignores the isolated threshold points, while distribution amplitudes can represent the surviving monochromatic constant-spatial modes.

For $u_{tt}=u_{xx}-u$ with $u(x,0)=0$ and $u_t(x,0)=e^{-|x|}$, the solution outside $|x|\le t$ is exactly $t{e}^{-|x|}$. On the positive half-line, $t{e}^{-x}$ satisfies both the equation and initial data; the finite propagation speed limits dependence at $x>t$ to positive initial points. Uniqueness on this domain of dependence proves the formula there, and reflection gives the negative half-line. The exterior tail is nonzero because the initial velocity is not compactly supported.

For zero initial velocity and Fourier amplitude $a(k)$, observation along $x=Vt$ with $0\le V<c$ selects the two stationary wavenumbers $\pm k_0$, whereWriting $s=\sqrt{c^2-V^2}$, the leading oscillation has angular frequency $As$ and amplitude proportional to $t^{-1/2}$.

If the leading asymptotic is $C{t}^{-1/2}\cos (\Omega t+\gamma )$ with $C>0$, its upward zero crossings satisfyto leading order.