High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs

Figure 1.  A cartoon of the bifurcation structure of the traveling waves for a third-order ($M=3$) system: solution branches bifurcate away from the trivial zero-amplitude solution at specific values of the traveling wave speed $c$

Figure 2.  Colliding eigenvalues in the complex plane as a parameter is increased. On the left, two eigenvalues are moving towards each other on the positive imaginary axis, accompanied by a complex conjugate pair on the negative imaginary axis. In the middle, the eigenvalues in each pair have collided. On the right, a Hamiltonian Hopf bifurcation occurs: the collided eigenvalues separate, leaving the imaginary axis (implying that the two Krein signatures were different)

Figure 3.  The graphical interpretation of the collision condition (3.12). The solid curve is the graph of the dispersion relation $\omega(k)$. The slope of the dashed line in the first quadrant is the right-hand side in (3.12). The slope of the parallel dotted line is its left-hand side

Figure 4.  The amplitude vs. $c$ bifurcation plots for the traveling-wave solutions of the generalized KdV equation (3.20). (a) The KdV equation, $n=1$, for the cnoidal wave solutions (3.24). (b) The mKdV equation, $n=2$, for the cnoidal wave solutions (3.26). Lastly, (c) shows the bifurcation plot for the snoidal wave solutions (3.27) of mKdV, $n=2$. Note that all bifurcation branches start at $(-1,0)$, as stated above. Further, for all solutions here the speed $c$ and the amplitude $\to \infty$ as $\kappa\to 1$. This is a consequence of enforcing the $2\pi$-periodicity of the solution, which results in non-smooth limit solution

Figure 5.  (a) The imaginary part of $\lambda_n^{(\mu)}\in (-0.7, 0.7)$ as a function of $\mu\in[-1/4, 1/4)$. Different curves correspond to different half-integer values of $n$. (b) The curves $\Omega(k+n)$, for various (integer) values of $n$, illustrating that collisions occur at the origin only

Figure 6.  (a) The profile of a $2\pi$-periodic small-amplitude traveling wave solution of the Whitham equation (2.1) with $c\approx 0.7697166847$, computed using a cosine collocation method with 128 Fourier modes, see [44]. (b) The stability spectrum of this solution, computed using the Fourier-Floquet-Hill method ^[13] with $128$ modes and 2000 different values of the Floquet parameter $\mu$. The presence of a modulational instability is clear, but no high-frequency instabilities are observed, in agreement with the theory presented. Note that the hallmark bubbles of instability were looked for far outside of the region displayed here

Figure 7.  (a) The dispersion relation for the Whitham equation (curve), together with the line through the origin of slope $\omega(1)/1$, representing the right-hand side of (3.12). (b) The curves $\Omega(k+n)$, for various (integer) values of $n$, illustrating that collisions occur at the origin only

Figure 8.  The graphical interpretation of the collision condition (4.10). The dashed curves are the graphs of the dispersion relations $\omega_1(k)$ and $\omega_2(k)$. The slope of the segment $P_1P_2$ is the right-hand side in (4.10). The collision condition (4.10) seeks points whose abscissas are an integer apart, so that at least one of the segments $P_3P_4$, $P_3P_6$, $P_5P_4$ or $P_5P_6$ is parallel to the segment $P_1P_2$

Figure 9.  (a) The two branches of the dispersion relation for the Sine-Gordon equation. The line segment $P_1 P_2$ has slope $\omega(1)/1$, representing the right-hand side of (4.10). The slope of the parallel line segment $P_3 P_4$ represents the left-hand side of (4.10). (b) The two families of curves $\Omega_1(k+n)$ (red, solid) and $\Omega_2(k+n)$ (black, dashed), for various (integer) values of $n$, illustrating that many collisions occur away from the origin

Figure 10.  (a) A small-amplitude $2\pi$-periodic superluminal solution of the SG equation ($c\approx 1.236084655663$). (b) A blow-up of the numerically computed stability spectrum in a neighborhood of the origin, illustrating the presence of a modulational instability, but the absence of high-frequency instabilities

Figure 11.  The domain for the water wave problem. Here $z=0$ is the equation of the surface for flat water, $z=-h$ is the flat bottom

Figure 12.  (a) The two branches of the dispersion relation for the water wave problem ($g=1$, $h=1$). The line through the origin has slope $\omega_1(1)/1$, representing the right-hand side of (4.10). (b) The two families of curves $\Omega_1(k+n)$ (red, solid) and $\Omega_2(k+n)$ (black, dashed), for various (integer) values of $n$, illustrating that many collisions occur away from the origin. (c) The origin of the high-frequency instability closest to the origin as a function of depth $h$

Figure 13.  (a) A small-amplitude traveling wave solution of the Boussines-Whitham equation (5.1) with $c\approx 1.0498515$. (b) The numerically computed stability spectrum. (c) A blow-up of the stability spectrum in a neighborhood of the origin. (d) A blow-up of the stability spectrum around what appears as a horizontal segment visible in (b) immediately above the longest segment appearing horizontal. More detail is given in the main text