Spatiotemporal wave-packet stability

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.

Modulational instability is growth of long-wavelength perturbations of a uniform or periodic wave due to coupling between its modulation and nonlinear dynamics. In the focusing nonlinear Schrodinger equation, the amplitude and phase perturbations can have imaginary modulation frequencies even though the unperturbed wave is an exact solution.

Linearizing a uniform Kerr-fibre background couples opposite frequency sidebands. With $i{A}_z-{\beta }_2{A}_{TT}/2+\gamma |A{|}^{2}A=0$, the displayed dispersion relation permits exponential growth precisely when ${\beta }_2\gamma <0$ and $0<{\Omega }^2<-4\gamma P/{\beta }_2$. For positive $\gamma$, this is anomalous group velocity dispersion. complex conjugation requires the paired complex-mode ansatz on the unstable branch. amplitude and power growth exponents differ by a factor of two.

Linearization around a pumped background couples the perturbation at positive frequency offset to the conjugate of the negative-offset perturbation. Their matrix eigenvalues determine amplification; assuming two unrelated scalar phase shifts misses the instability.

For the complex Ginzburg–Landau equation $A_t=A+(1+i{c}_1)A_{xx}-(1+i{c}_3)|A{|}^{2}A$, the uniform oscillation has a neutral phase mode whose growth rate is $\lambda =-(1+c_1{c}_3)k^2+O(k^4)$. Negative phase diffusion therefore causes a long-wavelength modulational instability. The Benjamin-Feir stability condition is the complementary positive-diffusion condition. On a finite domain an unstable wavenumber must actually be allowed by the boundary conditions.

For $-i{A}_t+\chi A_{xx}-\mu A+\delta A|A{|}^2=0$ with positive coefficients, linearize around $A=\sqrt{\mu /\delta }$ using real amplitude and phase perturbations. Their two-by-two dispersion relation gives the displayed equation. The modulational instability band is $0<K^2<2\mu /\chi$, with maximum growth rate $\mu$ at $K^2=\mu /\chi$.

A criterion for absolute wave-packet instability based on analytic continuation of the spatial roots of a dispersion relation. Start the temporal inversion contour above the spectrum, then lower it while deforming the spatial contour. A spatial pinch point occurs when branches originating in opposite spatial half-planes obstruct that deformation. A double root $D=D_k=0$ is only an algebraic candidate. The opposite-half-plane condition is stated in the primary study doi.org/10.1017/jfm.2016.195, section 4.2.

For $\omega (k)=i\alpha k^2-\beta k^3-i\gamma$, real-wavenumber growth is $\mathrm{Re}\alpha k^2-\mathrm{Im}\beta k^3-\mathrm{Re}\gamma$. A nonzero cubic coefficient is unbounded above on one side; after it vanishes, a positive quadratic coefficient is unbounded above on both sides. The displayed conditions are therefore necessary and sufficient for a finite temporal upper bound, which is then $-\mathrm{Re}\gamma$. The bound permits the initial causal temporal contour required by the Briggs-Bers criterion.

For the whole-line equation $A_t+a_1{A}_{xx}+a_2{A}_{xxxx}+a_{3}A=0$ with $\mathrm{Re}{a}_2>0$, the dispersion relation is $s=a_1{k}^2-a_2{k}^4-a_3$. The temporal Laplace transform of the impulse response at zero is the real-k integral of $[s+a_3-a_1{k}^2+a_2{k}^4{]}^{-1}/(2\pi )$. Factor the denominator as $a_2(k^2+{\kappa }_1^2)(k^2+{\kappa }_2^2)$ with $\mathrm{Re}{\kappa }_j>0$ for large $\mathrm{Re}s$. Partial fractions give the displayed resolvent. Square-root sheets are chosen by continuation from this causal large-s region; choosing roots independently can introduce false spatial pinch points.

Put $z=a_1/\sqrt{a_2}$ using the root with positive real part. The additional resolvent singularity at $s+a_3=z^2/4$ lies on the physical square-root sheet only when $\mathrm{Re}z>0$; otherwise the apparent saddle is on the other sheet. The singularity at $s=-a_3$ remains, including a quartic degeneracy when $a_1=0$. The rightmost physical singularity determines the exponential fixed-position growth rate. Compare it with the temporal maximum $-\mathrm{Re}{a}_3+\max (\mathrm{Re}{a}_1,0{)}^2/(4\mathrm{Re}{a}_2)$ to distinguish absolute wave-packet instability from convective wave-packet instability. Equality cases require their algebraic prefactors.

A collision of spatial branches which pinches the spatial inversion contour as the temporal contour is lowered from above all singularities. The branches must originate on opposite sides of the spatial contour. A collision of two upper-half-plane branches or two lower-half-plane branches can solve the double-root equations without causing absolute wave-packet instability.

For these constants, every real-wavenumber growth rate is $-k^2-1/10<0$. The real-contour impulse integral decays with envelope $e^{-t/10}{t}^{-1/2}$ at fixed position. Yet the stationary candidate $k=-2i/3$ has $\mathrm{Im}\omega =4/27-1/10>0$. It cannot be a contributing spatial pinch point; otherwise it would contradict the real-contour growth bound. This explicit example shows why solving the algebraic saddle equations is insufficient for absolute wave-packet instability.

The real dispersion above is temporally stable for every real $k$, but its analytic continuation has stationary points at $k=\pm i$ with $\omega =i/2$. Writing $\omega =i\Omega$ gives $k^2=-1\pm \sqrt{1/2-\Omega }$; as $\Omega$ decreases to $1/2$, both branches at $+i$ approach from the upper half-plane and both at $-i$ from the lower. These are not spatial pinch points. This provides a concrete counterexample to treating growing algebraic double roots as sufficient for absolute wave-packet instability.

A localized disturbance grows along some moving rays but decays at its original fixed location. This wave-packet meaning differs from buoyancy-driven stellar convection and is therefore given a distinct canonical title. Changing the observation frame can change the absolute/convective classification. The Briggs-Bers criterion uses the laboratory-frame impulse response to distinguish this case from absolute wave-packet instability.

A localized perturbation has positive asymptotic exponential growth at a fixed observation point in the chosen reference frame. The relevant spatial pinch point of the Briggs-Bers criterion has positive imaginary frequency. Merely finding a growing temporal mode or a formal complex saddle is not sufficient; the saddle must contribute to the localized impulse response.

With normal modes $e^{ikx-i\omega t}$, a finite upper bound on $\mathrm{Im}\omega$ allows the inverse temporal Laplace transform contour to start above all temporal singularities. An unbounded growth rate at high real wavenumber prevents the standard causal Green function construction used in the Briggs-Bers criterion. For $\mathrm{Im}\omega =\alpha k^2-\beta k^4-\gamma$ with nonzero real $\beta$, the bound holds exactly when $\beta >0$.