Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 79 / 1 / b

The Briggs-Bers technique starts with a temporal Laplace transform and spatial Fourier transform of a localized disturbance. The temporal inversion contour must initially lie above every temporal singularity. Lower it while deforming the spatial contour away from the roots of the dispersion relation. A collision of spatial branches originating on opposite sides of that contour can create a spatial pinch point. A contributing pinch with $\mathrm{Im}{\omega }_{\ast }>0$ gives absolute wave-packet instability: exponential growth at a fixed position. Growth that is transported away while a fixed position decays is convective wave-packet instability. Solving $D=D_k=0$ finds candidate double roots, but does not by itself establish a pinch.

Substitution of the normal mode givesFor real wavenumber $k$, the temporal growth rate isA nonzero real cubic coefficient in this expression makes it unbounded above on one end of the real axis. Once that coefficient vanishes, a positive quadratic coefficient also makes it unbounded above. Hence the bounded temporal growth condition for cubic dispersion isThe maximum is then $-\mathrm{Re}\gamma$, attained at $k=0$ and, when $\mathrm{Re}\alpha =0$, at all real $k$. A finite upper bound is necessary for choosing the initial temporal inversion contour; arbitrarily fast high-wavenumber growth prevents the usual Briggs-Bers criterion from defining that causal inversion for generic localized data.

For $\beta \ne 0$, the stationary candidates satisfysoEach candidate must pass the spatial-branch pinch test before its imaginary frequency is used. In particular, under the finite-growth conditions the real-wavenumber evolution has multiplier modulus at most $e^{-t\mathrm{Re}\gamma }$. For integrable transformed initial data this bounds its fixed-position amplitude by a constant times that exponential. Thus a further necessary condition isThe false complex saddle in dissipative cubic dispersion illustrates why the second formal candidate cannot override this bound. For $\mathrm{Re}\alpha <0$, the neighborhood $k=0$ gives a fixed-position impulse response proportional to $e^{-\gamma t}/\sqrt{t}$: the cubic term is smaller on its $k=O(t^{-1/2})$ scale. It therefore supplies the usual growing pinch when $\mathrm{Re}\gamma <0$. The purely dispersive cases have stationary-phase algebraic prefactors instead. If $\beta =0$, only the quadratic stationary point remains; if both derivative coefficients vanish, the equation is local multiplication and its growth is directly $e^{-\gamma t}$.

Write $r=\sqrt{\mu /\delta }$. At first order the amplitude-phase perturbation is $A=r+\epsilon (A_1+ir\theta )+O({\epsilon }^2)$. Expanding the cubic term in the focusing nonlinear Schrodinger equation gives $A|A{|}^2=r^3+\epsilon (3r^2{A}_1+i{r}^3\theta )+O({\epsilon }^2)$. The constant terms cancel because $\delta r^2=\mu$. Separating real and imaginary parts yieldsFor the Fourier normal modes, these equations give $-i\Omega {\hat{A}}_1=-\chi r{K}^2\hat{\theta }$ and $-ir\Omega \hat{\theta }+(2\mu -\chi K^2){\hat{A}}_1=0$. Their determinant vanishes exactly whenThe focusing nonlinear Schrodinger modulation dispersion is negative for $0<K^2<2\mu /\chi$. One of the two frequencies then has positive imaginary part and grows exponentially: the constant state has modulational instability. The growth rate is $\sqrt{\chi K^2(2\mu -\chi K^2)}$, maximized at $K^2=\mu /\chi$ with value $\mu$. Modes beyond the unstable band have real frequencies. The zero-wavenumber phase mode is neutral, with possible linear phase drift from a uniform amplitude perturbation.