to increase $s$ by $\Delta s=0.01$. Once this function is positive, the wave
has self-intersected and we have found a bracket to use in Brent’s method to
find $s(\tau)$ such that $\tilde{\xi}(\alpha_{0}(s(\tau),\tau))=\pi$ to
double-precision accuracy. We then compute $\chi(s,\tau)$ at 81 values of $s$,
uniformly spaced between 0 and $s(\tau)$. As a result, the right boundary of
the contour plot corresponds to the maximum amplitude for each $\tau$ where
the air pocket closes to form a bubble.

In the third region of (4.19), an additional step is taken in which two
numerical continuation paths are computed with $s=\hat{\eta}_{1,0}$ held fixed
and $\tau$ decreasing. The specific choices of $s$ are $0.005$ and $0.006$,
with starting points at $\tau\in\\{0.52,0.54\\}$, computed as part of region 2
in (4.19). Once solutions are known with $(\tau,s)$ in the range
$0.05\leq\tau\leq 0.5$ and $s\in\\{0.005,0.006\\}$, we proceed as above to
find the boundary to the right, but with $\Delta s$ decreased to $0.001$ for
the search for the initial bracket for Brent’s method. When the right boundary
is found (where the air pocket pinches off into an air bubble), we compute
$\chi$ at 81 equally spaced points between $s=0.0017$ and the pinch-off
amplitude. We re-iterate that only for $\tau>1/2$ does the wave approach zero-
amplitude when the parameter $s$ decreases to zero. We stop at $s=0.0017$ for
$\tau\leq 1/2$ since a different bifurcation parameter than $\hat{\eta}_{1,0}$
is needed to properly explore the limit as $s\to 0^{+}$ when $\tau\leq 1/2$ on
this sheet of type 1 solutions, and $s=0.0017$ is small enough for the purpose
of plotting $\chi(s,\tau)$.

Figure 10.  Contour plot of $\chi(s,\tau)$ for type 1 gravity-capillary waves
along with representative examples of bifurcation points on the zero contour
of $\chi$ and a secondary branch of quasi-periodic, overhanging traveling
waves bifurcating from solution $F$.

Because it is a two-parameter family, many solutions had to be computed to
generate the contour plot of $\chi(s,\tau)$ in panel (a) of Figure 10. There
are 14661 solutions represented in the plot. The Fourier cutoffs $N_{1}$ and
$N$ used to compute the traveling waves and $\chi$ are as follows:

$\begin{array}[]{c|cccccc}\text{$\tau$
range}&[0.05,0.09]&[0.1,0.19]&[0.2,0.5]&[0.52,0.98]&[1.0,2.45]&[2.5,6.5]\\\
\hline\cr N_{1}&320&256&128&108&72&54\\\ N&384&300&160&128&90&64\end{array}$
(4.20)

We set $M_{1}=3N_{1}$ and $M=3N$ in all cases. We highlight the zero contours
of $\chi(s,\tau)$ in red. There are 4 values of $\tau>0.5$ at which a red
contour line reaches the $\tau$ axis, i.e., where $\chi(0,\tau)=0$. In the
small-amplitude limit, the candidate quasi-periodic bifurcations are plane
waves, enumerated by Fourier lattice points $\vec{l}\in\pm\Lambda$ from
(3.13). Plane waves with wave numbers $k_{1}=1$ and $k_{2}=l_{1}+kl_{2}$ can
co-exist as traveling waves if they are both roots of the dispersion relation

$c^{2}=\frac{g}{k_{j}}+\tau k_{j},\qquad(j=1,2).$ (4.21)

Eliminating $c^{2}$ and setting $g=1$, we have $1+\tau=\frac{1}{k_{2}}+\tau
k_{2}$, or $\tau=1/k_{2}$. In our calculation, we use Bloch theory to restrict
attention to the case $l_{2}\in\\{\pm 1\\}$ when constructing
$\mathcal{J}^{\text{qua}}$ in (3.26). The possible values of $\tau$ that
exceed 1/2, the Wilton ripple cutoff for type 1 waves, are then

$\tau=\frac{1}{1-k}=3.414,\quad\tau=\frac{1}{0+k}=1.414,\quad\tau=\frac{1}{2-k}=0.773,\quad\tau=\frac{1}{1+k}=0.586,$
(4.22)

where $k=1/\sqrt{2}$. These are precisely the locations where the red contour
lines in panel (a) meet the $\tau$ axis. We find that the contours at $1.414$
and $0.773$ form a closed loop, and the contour at 0.586 drops down below
$\tau=0.5$ into region 3 from (4.19), where we used numerical continuation to
extend the range of type 1 waves below the Wilton ripple cutoff.

We now focus on the remaining contour, which begins at $\tau=3.414$. We see in
Figure 10 that this contour extends all the way to the right boundary of the
contour plot, where the type 1 waves self-intersect at a point, trapping an
air bubble below. The solutions labeled $C$, $D$, $E$ and $F$ have values of
$s$ equal to $0.3$, $0.5$, $0.7$ and $0.791$, respectively, and are plotted in
panels (c)–(f). As $s$ increases along the contour, the air pocket at $x=\pi$
(and its $2\pi$-periodic translations) deepens, eventually forming a trapped
bubble. Solution $E$ has formed a deep air pocket, but the free surface is
still a graph. Solution $F$ has overhanging walls that nearly touch. Using
polynomial interpolation from the points on the right boundary with
$6.1\leq\tau\leq 6.5$, we find that the red contour meets the right boundary
at

$(s,\tau)=(\,0.7916457855\;,\;6.297256422\,).$ (4.23)

Solution $F$ is very close to this, with $(s,\tau)=(0.791,6.294747714)$.
Solutions $C$, $D$ and $E$ correspond to $(s,\tau)=(0.3,4.1248233)$,
$(0.5,4.9579724)$ and $(0.7,5.8995675)$, respectively. Each of the waves
$C$–$F$ bifurcates to a quasi-periodic family with basic frequencies $k_{1}=1$
and $k_{2}=k=1/\sqrt{2}$. Indeed, the entire red curve corresponding to
$\chi=0$ gives the intersection of the two-parameter family of periodic type 1
waves and a two-parameter sheet of such QP waves. Fixing $\tau$ or $s$ reduces
the problem to a standard one-parameter bifurcation problem of the type
studied in Section 4.1 above. See [3, 9] for background on the general theory
of multi-parameter bifurcation theory.

To explore the existence of spatially quasi-periodic, overhanging traveling
waves, we follow the bifurcation branch from solution $F$ using our numerical
continuation algorithm. We choose $N_{1}=N_{2}=64$, $M_{1}=M_{2}=150$ and use
$q^{\text{bif}}\pm 10^{-5}\big{(}0,0,\dot{\eta}^{\text{qua}}\big{)}$ to jump
from the periodic branch to the quasi-periodic branch. The largest Fourier
coefficient of $\dot{\eta}^{\text{qua}}$ is
$\dot{\hat{\eta}}^{\text{qua}}_{0,1}\approx 0.6901$. Thus,
$\theta=\hat{\eta}_{0,1}$ is a natural choice for the continuation parameter
on the QP branch. We hold $\tau$ fixed at $\tau^{\text{bif}}$ in this search.
We are able to compute the quasi-periodic continuation path until $\theta$
reaches $\pm 0.1$. The corresponding solution with positive $\theta$, labeled
$G$ in panel (b), is plotted in panel (g) of Figure 10. For this solution,
$\hat{\eta}_{1,0}\approx 0.7529$ and the objective function $f$ is minimized
to $3.7\times 10^{-25}$. In panel (b) of Figure 10, we plot the Fourier
coefficients $\hat{\eta}_{1,0}$ and $\hat{\eta}_{0,1}$ of the bifurcated
quasi-periodic solutions. Along the quasi-periodic branch, as $|\theta|$
increases, $|\hat{\eta}_{1,0}|$ decreases. We also observe that the plot is
symmetric with respect to the vertical line $\hat{\eta}_{0,1}=0$; this is
because the quasi-periodic solutions with negative $\theta$ can be obtained
from ones with positive $\theta$ through a spatial shift in $\alpha_{2}$:
$(\alpha_{1},\alpha_{2})\mapsto(\alpha_{1},\alpha_{2}+\pi)$. Solutions $G$ and
$G^{\prime}$ are related in this way.

In panel (g) of Figure 10, we compare the wave profile $\zeta=\xi+i\eta$ of
solution $F$, which is periodic with amplitude $s=0.791$, and solution $G$,
which is quasi-periodic with $\theta=0.1$. We observe that the peaks and
troughs of the QP solution appear in a non-periodic pattern. The peaks of
solution $G$ are above those of solution $F$ near $\xi=0,6\pi$ and below near
$\xi=2\pi,4\pi,10\pi$; the troughs of $G$ are on the left of those of $F$ near
$\xi=5\pi,11\pi$ and on the right near $\xi=3\pi,7\pi,9\pi$. Beyond the plot
window shown, the deviation of solution $G$ from $F$ will continue to be
different from one peak and trough to the next, never exactly repeating over
the real line. We zoom in on the troughs of the two solutions near $\xi=5\pi$
and observe that the trough of solution $G$ is asymmetrical and wider than
that of the periodic solution $F$. Both solutions are not self-intersecting.
Moreover, solution $G$ is further from self-intersecting than solution $F$.
This may be related to the result in panel (b) that increasing $|\theta|$
causes $\hat{\eta}_{1,0}$ to decrease, and decreasing $\hat{\eta}_{1,0}$ on
the periodic branch increases the gap between the overhanging walls.
Specifically, solution $G$ has $\hat{\eta}_{1,0}\approx 0.7529$ while solution
$F$ has $\hat{\eta}_{1,0}\approx 0.7916$. Nevertheless, this quasi-periodic
solution does exhibit overhanging regions, as shown in the inset plot.

##  5\. Conclusion

We have shown that a signed version, $\chi$, of the smallest singular value of
the Jacobian, $\sigma_{\text{min}}$, serves as an excellent test function to
locate branch points in equilibrium problems. While $\sigma_{\text{min}}$ has
a slope discontinuity at each of its zeros, the existence of an analytic or
smooth SVD ensures that the function becomes smooth when a sign is included on
one side of the bifurcation point. We show that this sign factor can be
defined as the product of the determinants of the orthogonal matrices
containing the left and right singular vectors in the standard SVD, which
alleviates the need to actually compute an analytic SVD. We also show how to
compute this sign efficiently from the bidiagonal matrix obtained in the first
phase of the standard SVD algorithm. It is not necessary to form the matrices
of singular vectors, compute their determinants, or compute the determinant of
the Jacobian itself.

One benefit of using $\chi$ as a test function is that root bracketing
algorithms such as Brent’s method can then be used to locate bifurcation
points. This is simpler than the Newton-type method proposed by Shen [69] to
locate zeros of $\sigma_{\text{min}}$ or by various authors [49, 5, 14, 33,
16] to solve minimally extended systems. Within the constraints and philosophy
of Remark 3.9, our method is as efficient as these alternative approaches. We
also proposed a polynomial interpolation approach using Chebyshev polynomials,
which relies on the smoothness of $\chi$ to achieve spectral accuracy. In
multi-parameter bifurcation problems such as the gravity-capillary problem of
Section 4.2, the zero level set of $\chi$ can be used to visualize and compute
the intersection of the primary family of solutions with the secondary family
of solutions. This would not work well using $\sigma_{\text{min}}$ or $\psi$
from (1.2) instead of $\chi$. We also showed that $\chi$ stops changing on
further mesh refinement once the singular vectors corresponding to the
smallest singular values are resolved. As a result, the plots in Figure 3 and
the contour plot of Figure 10 are smooth even though many different mesh sizes
are used in the underlying calculations plotted. Other test functions such as
the determinant of the Jacobian or $\psi(s)$ in (1.2) do not have this mesh
independence property, and some behave poorly on large-scale discretizations
of infinite dimensional problems.

We use this method to compute, for the first time, quasi-periodic traveling
gravity waves with zero surface tension and overhanging traveling gravity-
capillary waves. The former example yields traveling waves that still make
sense at the scale of the ocean, where the length scale of capillary waves is
so much smaller than that of gravity waves (by a factor of $10^{-7}$) that one
can set $\tau=0$. Quasi-periodic pure gravity waves do not persist to zero
amplitude, which motivated us to search for quasi-periodic bifurcations from
large-amplitude periodic waves. The latter example showcases the use of
$\chi(s,\tau)$ to study two-parameter bifurcation problems in which the
primary sheet is parameterized over a region with one side bounded by singular
solutions. In our case, the right boundary of the contour plot in Figure 10
corresponds to type 1 waves that self-intersect to form an air pocket. This
contour plot makes it easy to visualize how the secondary two-parameter family
of quasi-periodic solutions fits together with the primary two-parameter
family of periodic traveling waves.

Once bifurcations are found, we use numerical continuation to explore the
secondary branches of quasi-periodic solutions. This becomes computationally
expensive, especially in the case of the pure gravity wave problem. We
formulated the problem of finding solutions on the secondary branch as an
overdetermined nonlinear least squares problem and implemented a parallel
algorithm employing MPI and ScaLapack to carry out the trust-region
minimization steps of the Levenberg-Marquardt method. The largest-amplitude
solution we computed, namely solution $B$ of Figures 5 and 6, required solving
for $N_{\text{tot}}=53398$ independent Fourier modes in its 2D torus
representation. We used the Savio cluster at UC Berkeley and the Lawrencium
cluster at Lawrence Berkeley National Laboratory for these calculations.

An interesting feature of these solutions is that the 2D Fourier modes
$\hat{\eta}_{j_{1},j_{2}}$ continue to exhibit visible effects of resonance
near the line $j_{1}+kj_{2}=0$ even though these are large-amplitude solutions
far from linear water wave theory. We explain this by noting that these modes
correspond to long wavelengths when the torus function is restricted to the
characteristic line $\alpha_{1}=\alpha$, $\alpha_{2}=k\alpha$, and the Euler
equations are not strongly affected by long wavelength perturbations. We also
identified a symmetry connecting solutions on one side of each secondary
bifurcation branch to the other. In particular, solution $B^{\prime}$ in
Figure 5 is related to solution $B$ via
$\eta_{B^{\prime}}(\alpha_{1},\alpha_{2})=\eta_{B}(\alpha_{1},\alpha_{2}+\pi)$.
The same symmetry was found in the gravity-capillary problem, e.g., solutions
$G$ and $G^{\prime}$ in Figure 10 are also related by this transformation.
Figures 9 and 10 also show that when a large-amplitude periodic traveling
water wave is perturbed to create a quasi-periodic traveling wave, each crest
and trough of the infinite wave train will undergo a different perturbation,
so that no two are exactly the same. Nevertheless, they fit together to form a
single traveling wave profile extending over the real line.

We took advantage of Bloch theory to express the Jacobian as a direct sum of
operators mapping $(\mathcal{X}_{\sigma}^{\text{per}},\mathbb{R},\mathbb{R})$
to $X_{\sigma}^{\text{per}}$ and $(\mathcal{X}_{\sigma}^{(l_{2})},0,0)$ to
$\mathcal{X}_{\sigma}^{(l_{2})}$. This greatly reduces the number of rows and
columns of $\mathcal{J}^{\text{qua}}$ in (3.26) since $l_{2}$ could be set to
1 rather than varying over some range $-N_{2}\leq l_{2}\leq N_{2}$. Bloch
theory is also useful for studying the stability of traveling waves to
subharmonic perturbations [57, 60, 59], and indeed the present work of
searching for null vectors of the Jacobian of the traveling wave equations can
be thought of as a special case of looking for zero eigenvalues of the dynamic
stability problem. In the present paper, we have shown that perturbations in
null directions of the linearization lead to branches of spatially quasi-
periodic traveling waves for the full water wave equations. In future work, it
would be interesting to investigate the Benjamin-Feir instability [11, 84] and
other unstable subharmonic perturbations [29, 71, 73, 62] by evolving them
beyond the realm of validity of Bloch stability theory using the dynamic
version [82] of our spatially quasi-periodic water wave framework. Linearly
stable subharmonic perturbations would also be interesting to investigate as
they may lead to solutions of the full water wave equations that are quasi-
periodic in time [13, 10, 12, 44, 79] as well as space.

Acknowledgements. This work was supported in part by the National Science
Foundation under award number DMS-1716560 and by the Department of Energy,
Office of Science, Applied Scientific Computing Research, under award number
DE-AC02-05CH11231. This research used the Lawrencium computational cluster
resource provided by the IT Division at the Lawrence Berkeley National
Laboratory and the Savio computational cluster provided by the Berkeley
Research Computing program at the University of California, Berkeley
(supported by the UC Berkeley Chancellor, Vice Chancellor for Research, and
Chief Information Officer).

Declaration of Interest. The authors report no conflict of interest.

##  Appendix A The effects of floating-point errors on the smallest singular
value

The SVD algorithm is backward stable [32], which leads to a well-known
estimate [31]

$|\hat{\sigma}_{i}-\sigma_{i}|\leq p(n)\|\mathcal{J}\|\varepsilon,\qquad 1\leq
i\leq n,$ (A.1)

where $\\{\hat{\sigma}_{i}\\}$ are the numerically computed singular values of
the $n\times n$ matrix $\mathcal{J}$, $p(n)$ is a slowly growing function of
the matrix dimension, $\|\mathcal{J}\|$ is the 2-norm, and $\varepsilon$ is
machine precision. Following the approach of [78], one can show that $p(n)\leq
O(n^{2.5})$ using standard backward stability estimates for Householder
transformations in the bidiagonalization phase [51, 32]; however, this is
pessimistic. In particular, it assumes worst-case $O(n\varepsilon)$ errors
when summing $n$ numbers. In practice [8], $p(n)$ is often taken to be 1. In
section 4.1, the condition numbers of $\mathcal{J}^{\text{qua}}$ for solutions
$A$ and $E$ are 1636 and 69074, respectively. Taking $p(n)=1$, this leads to
expected errors in $\chi(s)$ around $(1636)(2.2\times 10^{-16})=3.6\times
10^{-13}$ and $(69074)(2.2\times 10^{-16})=1.5\times 10^{-11}$, respectively.
Using Brent’s method, we reduced $|\chi(s)|$ to $2.9\times 10^{-15}$ for
solution $A$. However, Brent’s method will report the result in which
floating-point errors combine most favorably to minimize $|\chi(s)|$, so this
value likely over-predicts the accuracy actually obtained. Indeed, if we
increase the size of $\mathcal{J}^{\text{qua}}$ from 1537 to 2049 and
recompute $\chi$ without re-optimizing via Brent’s method, we obtain
$|\chi|=1.5\times 10^{-14}$, which is five times bigger. The flattening of the
high-frequency Chebyshev modes $\check{\chi}_{m}$ in panel (d) of Figure 3
suggests floating-point errors around $10^{-11}$ or $10^{-12}$, which is
consistent with the above condition number estimate. Chebyshev interpolation
seems less prone than Brent’s method to optimizing beyond the actual error, so
the minimum value of $|\chi(s)|$ obtained in problem $E$, namely $2.1\times
10^{-12}$, may be accurate. Additional calculations would have to be done in
quadruple-precision to fully quantify the effects of floating-point arithmetic
in double-precision, but this is beyond the scope of the present work.

##  Appendix B Proof of Theorem 3.2

Let us restate the theorem in a slightly more general form that simplifies the
proof. Theorem 3.2 is recovered by setting $\rho=\rho_{1}/2$ and
$\sigma_{1}=\sigma$.

######  Theorem B.1.

Suppose $q^{\text{per}}=(\eta,\tau,b)$ with
$\eta\in\mathcal{V}_{\sigma}^{\text{per}}$ for some $\sigma>0$. Suppose also
that $\eta$ is real-valued and the resulting $J(\alpha_{1},\alpha_{2})$ in
(2.9), which is independent of $\alpha_{2}$, is non-zero for every
$\alpha_{1}\in\mathbb{T}$. Then there exists $\rho_{1}\in(0,\sigma]$ such that
$D_{q}\mathcal{R}[q^{\text{per}}]$ is a bounded operator from
$\big{(}\mathcal{V}_{\sigma_{1}},\mathbb{C},\mathbb{C}\big{)}$ to
$\mathcal{V}_{\rho}$ provided that $0<\rho<\rho_{1}$ and $\sigma_{1}>\rho$.

###### Proof.

Since $\eta\in\mathcal{V}_{\sigma}^{\text{per}}$, we know
$\eta(\alpha_{1},\alpha_{2})=\tilde{\eta}(\alpha_{1})$ is independent of
$\alpha_{2}$, and $\tilde{\eta}(\alpha)$ is real analytic and $2\pi$ periodic.
All the torus functions without a dot in (2.34) are then independent of
$\alpha_{2}$, and can be replaced by the corresponding 1d extracted function
(adding a tilde), evaluated at $\alpha_{1}$. The torus operators
$\partial_{\alpha}$ and $H$ are replaced by their 1d variants when this is
done. For example, $J(\alpha_{1},\alpha_{2})=\tilde{J}(\alpha_{1})$, where
$\tilde{J}(\alpha)=(1+H\partial_{\alpha}\tilde{\eta})^{2}+(\partial_{\alpha}\tilde{\eta})^{2}$.
Since $\tilde{J}(\alpha)$ is continuous and assumed non-zero at each
$\alpha\in[0,2\pi]$, it is bounded away from zero. Thus, each of the functions

$\begin{gathered}1/\big{(}2\tilde{J}\big{)},\quad\tilde{\kappa},\quad
b/\big{(}2\tilde{J}^{2}\big{)},\quad(1+H\partial_{\alpha}\tilde{\eta}),\quad\partial_{\alpha}\tilde{\eta},\quad
3\tilde{\kappa}/\big{(}2\tilde{J}\big{)},\quad\tilde{J}^{-3/2},\quad\partial_{\alpha}^{2}\tilde{\eta},\quad
H\partial_{\alpha}^{2}\tilde{\eta}\end{gathered}$ (B.1)

that appears in (2.34) (after setting $\xi=H\eta$) is real analytic and
$2\pi$-periodic. As a result (see, e.g., Lemma 5.6 of [19]) the 1d Fourier
coefficients of each of these functions decay exponentially. Thus, there exist
$C,\rho_{1}>0$ such that $Ce^{-\rho_{1}|j|}$ is a common bound on each of
these sets of Fourier coefficients, i.e., if
$\tilde{\gamma}(\alpha)=\sum_{j}\hat{\gamma}_{j}e^{ij\alpha}$ represents any
of the functions in (B.1), then $|\hat{\gamma}_{j}|\leq Ce^{-\rho_{1}|j|}$ for
$j\in\mathbb{Z}$. Now fix $\rho$ in the range $0<\rho<\rho_{1}$ and let
$\sigma_{1}>\rho$.

We need to show that $D_{q}\mathcal{R}\big{[}q^{\text{per}}\big{]}$ is a
bounded operator from $(\mathcal{V}_{\sigma_{1}},\mathbb{C},\mathbb{C})$ into
$\mathcal{V}_{\rho}$. It suffices to show that the restrictions to the
subspaces $(0,\mathbb{C},\mathbb{C})$ and $(\mathcal{V}_{\sigma_{1}},0,0)$ are
bounded. In the first case, we have

$D_{q}\mathcal{R}\big{(}q^{\text{per}}\big{)}\big{(}0,\dot{\tau},\dot{b}\big{)}=P\Big{[}\dot{b}/\big{(}2J\big{)}-\kappa\dot{\tau}\Big{]}.$
(B.2)

Note that $1/(2J)$ and $\kappa$ are independent of $\alpha_{2}$. Letting
$\gamma(\alpha_{1},\alpha_{2})=\tilde{\gamma}(\alpha_{1})$ represent either of
these functions, we have

$\|\gamma\|_{\mathcal{V}_{\rho}}^{2}\leq
C^{2}+2C^{2}\sum_{j=1}^{\infty}e^{-2(\rho_{1}-\rho)j}=C^{2}+\frac{2C^{2}}{e^{2(\rho_{1}-\rho)}-1}\leq
C^{2}+\frac{C^{2}}{\rho_{1}-\rho}=A^{2}<\infty,$ (B.3)

where the last equality defines $A$. So the $\mathcal{V}_{\rho}$ norm of the
left-hand side of (B.2) is bounded by $A(|\dot{b}|+|\dot{\tau}|)\leq
A\sqrt{2}(\dot{b}^{2}+\dot{\tau}^{2})^{1/2}$.

Now let $\dot{\eta}\in\mathcal{V}_{\sigma_{1}}$. We need to compute
$\dot{R}=D_{q}\mathcal{R}\big{[}q^{\text{per}}\big{]}\big{(}\dot{\eta},0,0\big{)}$
and bound its norm in $\mathcal{V}_{\rho}$ by a constant times
$\|\dot{\eta}\|_{\mathcal{V}_{\sigma_{1}}}$. To compute $\dot{R}$, we evaluate
(2.34) in two steps. First the terms

$\big{(}\partial_{\alpha}\dot{\xi}\big{)}=\big{(}H\partial_{\alpha}\dot{\eta}\big{)},\qquad\big{(}\partial_{\alpha}\dot{\eta}\big{)},\qquad\big{(}\partial_{\alpha}^{2}\dot{\eta}\big{)},\qquad\big{(}\partial_{\alpha}^{2}\dot{\xi}\big{)}=\big{(}H\partial_{\alpha}^{2}\dot{\eta}\big{)}$
(B.4)

are computed from $\dot{\eta}$. The symbols of $H\partial_{\alpha}$,
$\partial_{\alpha}$, $\partial_{\alpha}^{2}$ and $H\partial_{\alpha}^{2}$ are,
respectively, $|l_{1}+kl_{2}|$, $i(l_{1}+kl_{2})$, $-(l_{1}+kl_{2})^{2}$ and
$i(l_{1}+kl_{2})|l_{1}+kl_{2}|$, so these operators are bounded from
$\mathcal{V}_{\sigma_{1}}$ to $\mathcal{V}_{\rho}$ since $\sigma_{1}>\rho$.
For example,

$\big{\|}H\partial_{\alpha}\dot{\eta}\big{\|}^{2}_{\mathcal{V}_{\rho}}=\sum_{l_{1},l_{2}}\Big{|}|l_{1}+kl_{2}|\hat{\eta}_{l_{1},l_{2}}e^{\rho(|l_{1}|+|l_{2}|)}\Big{|}^{2}\leq
B^{2}\sum_{l_{1},l_{2}}\Big{|}\hat{\eta}_{l_{1},l_{2}}e^{\sigma_{1}(|l_{1}|+|l_{2}|)}\Big{|}^{2}=B^{2}\big{\|}\dot{\eta}\big{\|}^{2}_{\mathcal{V}_{\sigma_{1}}},$
(B.5)

where $B^{2}=\max(1,k)\max_{x\geq
0}xe^{-(\sigma_{1}-\rho)x}=\max(1,k)/[e(\sigma_{1}-\rho)]<\infty$. The second
step is to consider

$\dot{\mathcal{R}}=P\bigg{[}-\frac{b}{2J^{2}}\dot{J}+g\dot{\eta}-\tau\dot{\kappa}\bigg{]}.$
(B.6)

The projection $P$ is bounded on $\mathcal{V}_{\rho}$, and
$\|g\dot{\eta}\|_{\mathcal{V}_{\rho}}\leq
g\|\dot{\eta}\|_{\mathcal{V}_{\sigma_{1}}}$. Substitution of $\dot{J}$ and
$\dot{\kappa}$ in (2.34) into (B.6) yields sums of products containing two
factors from the list (B.1) and one factor from the list (B.4). From the first
step, we know each term in the list (B.4) is bounded in $\mathcal{V}_{\rho}$
by a constant times $\|\dot{\eta}\|_{\mathcal{V}_{\sigma_{1}}}$. The following
lemma shows that multiplication by any of the terms in (B.1) is a bounded
operator on $\mathcal{V}_{\rho}$. Multiplying by two of them is then also
bounded, which concludes the proof. ∎

######  Lemma B.2.

Suppose $\tilde{\gamma}(\alpha)=\sum_{j}\hat{\gamma}_{j}e^{ij\alpha}$ with
$|\hat{\gamma}_{j}|\leq Ce^{-\rho_{1}|j|}$ for positive constants $C$ and
$\rho_{1}$, and fix $\rho\in(0,\rho_{1})$. Then multiplication by
$\gamma(\alpha_{1},\alpha_{2})=\tilde{\gamma}(\alpha_{1})$ is a bounded
operator on $\mathcal{V}_{\rho}$.

###### Proof.

Let $u\in\mathcal{V}_{\rho}$ and define
$v(\alpha_{1},\alpha_{2})=\tilde{\gamma}(\alpha_{1})u(\alpha_{1},\alpha_{2})$.
The Fourier modes of $v$ are related to those of $u$ by convolution along
horizontal slices,
$\hat{v}_{l_{1},l_{2}}=\sum_{j}\hat{\gamma}_{l_{1}-j}\hat{u}_{j,l_{2}}$. Using
the bound on $|\hat{\gamma}_{j}|$, we have

$\displaystyle e^{\rho(|l_{1}|+|l_{2}|)}\big{|}\hat{v}_{l_{1},l_{2}}\big{|}$
$\displaystyle\leq\sum_{j}Ce^{-\rho_{1}|l_{1}-j|}e^{\rho(|l_{1}|-|j|)}\left(e^{\rho(|j|+|l_{2}|)}\big{|}\hat{u}_{j,l_{2}}\big{|}\right)$
(B.7)
$\displaystyle\leq\sum_{j}Ce^{-(\rho_{1}-\rho)|l_{1}-j|}e^{-\rho\big{(}|l_{1}-j|-|l_{1}|+|j|\big{)}}\left(e^{\rho(|j|+|l_{2}|)}\big{|}\hat{u}_{j,l_{2}}\big{|}\right).$

Since $|l_{1}|\leq|l_{1}-j|+|j|$, we can use Young’s inequality to conclude

$\sum_{l_{1}}e^{2\rho(|l_{1}|+|l_{2}|)}\big{|}\hat{v}_{l_{1},l_{2}}\big{|}^{2}\leq
A^{2}\sum_{j}e^{2\rho(|j|+|l_{2}|)}\big{|}\hat{u}_{j,l_{2}}\big{|}^{2},\qquad(l_{2}\in\mathbb{Z}),$
(B.8)

where $A=\sum_{m}Ce^{-(\rho_{1}-\rho)|m|}\leq C+2C/(\rho_{1}-\rho)<\infty$.
Summing (B.8) over $l_{2}$ and square rooting gives
$\|v\|_{\mathcal{V}_{\rho}}\leq A\|u\|_{\mathcal{V}_{\rho}}$, as claimed. ∎

## References

  * [1] B. F. Akers, D. M. Ambrose, and J. D. Wright.  Gravity perturbed Crapper waves.  Proc. R. Soc. A, 470:20130526, 2014.
  * [2] B. F. Akers and W. Gao.  Wilton ripples in weakly nonlinear model equations.  Commun. Math. Sci., 10(3):1015–1024, 2012.
  * [3] J. C. Alexander and S. S. Antman.  Global and local behavior of bifurcating multi-dimensional continua of solutions for multiparameter nonlinear eigenvalue problems.  Arch. Ration. Mech. Anal., 76:339–354, 1981.
  * [4] E. L. Allgower and K. Georg.  Numerical continuation methods: an introduction, volume 13.  Springer Science & Business Media, 2012.
  * [5] E. L. Allgower and H. Schwetlick.  A general view of minimally extended systems for simple bifurcation points.  Z. Angew. Math. Mech., 77:83–97, 1997.
  * [6] D. M. Ambrose and J. Wilkening.  Computation of time-periodic solutions of the Benjamin–Ono equation.  J. Nonlinear Sci., 20(3):277–308, 2010.
  * [7] C. J. Amick, L. E. Fraenkel, and J. F. Toland.  On the Stokes conjecture for the wave of extreme form.  Acta Math., 148(1):193–214, 1982.
  * [8] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen.  LAPACK Users’ Guide.  SIAM, Philadelphia, third edition, 1999.
  * [9] S. S. Antman.  Nonlinear Problems of Elasticity.  Springer, New York, 2nd edition, 2004.
  * [10] P. Baldi, M. Berti, E. Haus, and R. Montalto.  Time quasi-periodic gravity water waves in finite depth.  Invent. Math., 214(2):739–911, 2018.
  * [11] T. B. Benjamin and J. Feir.  The disintegration of wave trains on deep water.  J. Fluid Mech., 27(3):417–430, 1967.
  * [12] M. Berti, L. Franzoi, and A. Maspero.  Traveling quasi-periodic water waves with constant vorticity.  Arch. Rational Mech. Anal., 240:99–202, 2021.
  * [13] M. Berti and R. Montalto.  Quasi-periodic standing wave solutions of gravity-capillary water waves, volume 263 of Memoirs of the American Mathematical Society.  American Mathematical Society, 2016.
  * [14] W.-J. Beyn, W. Kleß, and V. Thümmler.  Continuation of low-dimensional invariant subspaces in dynamical systems of large dimension.  In B. Fiedler, editor, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pages 47–72. Springer, Berlin, 2001.
  * [15] D. Bindel, J. Demmel, and M. Friedman.  Continuation of invariant subspaces in large bifurcation problems.  SIAM J. Sci. Comput., 30(2):637–656, 2008.
  * [16] D. Bindel, M. Friedman, W. Govaerts, J. Hughes, and Y. Kuznetsov.  Numerical computation of bifurcations in large equilibrium systems in matlab.  J. Comput. Appl. Math., 261:232–248, 2014.
  * [17] R. P. Brent.  Algorithms for minimization without derivatives.  Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1973.
  * [18] T. Bridges and F. Dias.  Spatially quasi-periodic capillary-gravity waves.  Contemp. Math., 200:31–46, 1996.
  * [19] H. Broer and F. Takens.  Dynamical Systems and Chaos, volume 172 of Applied Mathematical Sciences.  Springer, New York, 2011.
  * [20] B. Buffoni, E. Dancer, and J. Toland.  The regularity and local bifurcation of steady periodic water waves.  Arch. Ration. Mech. Anal., 152(3):207–240, 2000.
  * [21] B. Buffoni, E. Dancer, and J. Toland.  The sub-harmonic bifurcation of Stokes waves.  Arch. Ration. Mech. Anal., 152(3):241–271, 2000.
  * [22] A. Bunse-Gerstner, R. Byers, V. Mehrmann, and N. K. Nichols.  Numerical computation of an analytic singular value decomposition of a matrix valued function.  Numer. Math., 60:1–39, 1991.
  * [23] B. Chen and P. Saffman.  Steady gravity-capillary waves on deep water – 1. weakly nonlinear waves.  Stud. Appl. Math., 60(3):183–210, 1979.
  * [24] B. Chen and P. Saffman.  Numerical evidence for the existence of new types of gravity waves of permanent form on deep water.  Stud. Appl. Math., 62(1):1–21, 1980.
  * [25] W. Choi and R. Camassa.  Exact evolution equations for surface waves.  J. Eng. Mech., 125(7):756–760, 1999.
  * [26] S.-N. Chow and Y.-Q. Shen.  Bifurcations via singular value decompositions.  Appl. Math. Comput., 28:231–245, 1988.
  * [27] M. G. Crandall and P. H. Rabinowitz.  Bifurcation from simple eigenvalues.  J. Funct. Anal., 8(2):321–340, 1971.
  * [28] G. D. Crapper.  An exact solution for progressive capillary waves of arbitrary amplitude.  J. Fluid Mech., 2:532–540, 1957.
  * [29] B. Deconinck and K. Oliveras.  The instability of periodic surface gravity waves.  J. Fluid Mech., 675:141–167, 2011.
  * [30] J. Demmel, L. Dieci, and M. Friedman.  Computing connecting orbits via an improved algorithm for continuing invariant subspaces.  SIAM J. Sci. Comput., 22:81––94, 2001.
  * [31] J. Demmel and W. Kahan.  Accurate singular values of bidiagonal matrices.  SIAM J. Sci. Stat. Comput., 11(5):873–912, 1990.
  * [32] J. W. Demmel.  Applied Numerical Linear Algebra.  SIAM, Philadelphia, 1997.
  * [33] A. Dhooge, W. Govaerts, and Y. Kuznetsov.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs.  ACM Trans. Math. Software, 29:141–164, 2003.
  * [34] L. Dieci and T. Eirola.  On smooth decompositions of matrices.  SIAM J. Matrix Anal. Appl., 20:800–819, 1999.
  * [35] L. Dieci and M. Friedman.  Continuation of invariant subspaces.  Numer. Linear Algebra Appl., 8:317–327, 2001.
  * [36] V. D. Djordjevic and L. G. Redekopp.  On two-dimensional packets of capillary-gravity waves.  J. Fluid Mech., 79:703–714, 1977.
  * [37] A. Dyachenko.  On the dynamics of an ideal fluid with a free surface.  Dokl. Math., 63(1):115–117, 2001.
  * [38] A. I. Dyachenko, E. A. Kuznetsov, M. Spector, and V. E. Zakharov.  Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping).  Physics Letters A, 221(1-2):73–79, 1996.
  * [39] A. I. Dyachenko, V. E. Zakharov, and E. A. Kuznetsov.  Nonlinear dynamics of the free surface of an ideal fluid.  Plasma Physics Reports, 22(10):829–840, 1996.
  * [40] S. Dyachenko, P. Lushnikov, and A. Korotkevich.  Branch cuts of Stokes wave on deep water. part I: Numerical solution and Padé approximation.  Stud. in Appl. Math, 2016.
  * [41] S. Dyachenko and A. C. Newell.  Whitecapping.  Studies Appl. Math., 137:199–213, 2016.
  * [42] S. A. Dyachenko, P. M. Lushnikov, and A. O. Korotkevich.  Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Padé approximation.  Stud. Appl. Math., 137(4):419–472, 2016.
  * [43] I. A. Dynnikov and S. P. Novikov.  Topology of quasi-periodic functions on the plane.  Russ. Math. Surv., 60(1):1, 2005.
  * [44] R. Feola and F. Giuliani.  Quasi-periodic traveling waves on an infinitely deep perfect fluid under gravity, 2020.  arXiv:2005.08280.
  * [45] K. V. Fernando and B. N. Parlett.  Accurate singular values and differential qd algorithms.  Numer. Math., 67:191–229, 1994.
  * [46] M. Friedman.  Improved detection of bifurcations in large nonlinear systems via the continuation of invariant subspaces algorithm.  Int. J. Bifurcation Chaos Appl. Sci. Eng., 11:2277–2285, 2001.
  * [47] I. S. Gandzha and V. P. Lukomsky.  On water waves with a corner at the crest.  Proc. R. Soc. A, 463:1597–1614, 2007.
  * [48] W. J. F. Govaerts.  Numerical Methods for Bifurcations of Dynamical Equilibria.  SIAM, Philadelphia, 2000.
  * [49] A. Griewank and G. Reddien.  Characterization and computation of generalized turning points.  SIAM J. Numer. Anal., 21:176–185, 1984.
  * [50] P. Guyenne and E. I. P. arău.  Computations of fully nonlinear hydroelastic solitary waves on deep water.  J. Fluid Mech., 713:307–329, 2012.
  * [51] N. J. Higham.  Accuracy and stability of numerical algorithms.  SIAM, Philadelphia, 1996.
  * [52] G. Iooss, P. I. Plotnikov, and J. F. Toland.  Standing waves on an infinitely deep perfect fluid under gravity.  Arch. Rat. Mech. Anal., 177:367–478, 2005.
  * [53] T. Kato.  Perturbation theory for linear operators.  Springer, Berlin, 1980.
  * [54] C. Kittel.  Introduction to Solid State Physics.  John Wiley and Sons, New York, 8th edition, 2005.
  * [55] S. G. Krantz.  Several Complex Variables.  AMS Chelsea Publishing, Providence, RI, 2nd edition, 2001.
  * [56] Y. A. Kuznetsov.  Elements of Applied Bifurcation Theory.  Springer-Verlag, New York, 2nd edition, 1998.
  * [57] M. S. Longuet-Higgins.  The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics.  Proc. R. Soc. Lond. A, 360:489–505, 1978.
  * [58] M. S. Longuet-Higgins and M. J. H. Fox.  Theory of the almost-highest wave. Part 2. Matching and analytic extension.  J. Fluid Mech., 85(4):769–786, 1978.
  * [59] R. S. MacKay and P. G. Saffman.  Stability of water waves.  Proc. R. Soc. Lond. A, 406:115–125, 1986.
  * [60] J. W. McLean.  Instabilities of finite-amplitude water waves.  J. Fluid Mech., 114:315–330, 1982.
  * [61] J. Moser.  On the theory of quasiperiodic motions.  Siam Review, 8(2):145–172, 1966.
  * [62] S. Murashige and W. Choi.  Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping.  J. Fluid Mech., 885:A41:1–27, 2020.
  * [63] J. Nocedal and S. J. Wright.  Numerical Optimization.  Springer, New York, 1999.
  * [64] P. Plotnikov and J. Toland.  Nash-moser theory for standing water waves.  Arch. Rat. Mech. Anal., 159:1–83, 2001.
  * [65] P. I. Plotnikov.  Nonuniqueness of solutions of the problem of solitary waves and bifurcation of critical points of smooth functionals.  Math. USSR-Izvestiya, 38(2):333, 1992.
  * [66] V. P. Ruban.  Water waves over a time-dependent bottom: exact description for 2D potential flows.  Phys. Letters A, 340:194–200, 2005.
  * [67] L. W. Schwartz and J.-M. Vanden-Broeck.  Numerical solution of the exact equations for capillary–gravity waves.  J. Fluid Mech., 96:119–139, 1979.
  * [68] H. Schwetlick and U. Schnabel.  Iterative computation of the smallest singular value and the corresponding singular vectors of a matrix.  Linear Algegra Appl., 371:1–30, 2003.
  * [69] Y.-Q. Shen.  Computation of a simple bifurcation point using one singular value decomposition nearby.  Computing, 58:335–350, 1997.
  * [70] G. G. Stokes.  On the theory of oscillatory waves.  Trans. Cambridge Philos. Soc., 1880.
  * [71] R. Tiron and W. Choi.  Linear stability of finite-amplitude capillary waves on water of infinite depth.  J. Fluid Mech., 696:402, 2012.
  * [72] J. F. Toland et al.  Stokes waves.  Topological Methods in Nonlinear Analysis, 7(1):1–48, 1996.
  * [73] O. Trichtchenko, B. Deconinck, and J. Wilkening.  The instability of Wilton’s ripples.  Wave Motion, 66:147–155, 2016.
  * [74] H. Uecker, D. Wetzel, and J. Rademacher.  Pde2path - a Matlab package for continuation and bifurcation in 2D elliptic systems.  Numerical Mathematics: Theory, Methods and Applications, 7(1):58–106, 2014.
  * [75] J.-M. Vanden-Broeck.  Gravity–Capillary Free–Surface Flows.  Cambridge University Press, Cambridge, 2010.
  * [76] J.-M. Vanden-Broeck.  On periodic and solitary pure gravity waves in water of infinite depth.  J. Eng. Math., 84(1):173–180, 2014.
  * [77] Z. Wang, J. Vanden-Broeck, and P. Milewski.  Two-dimensional flexural–gravity waves of finite amplitude in deep water.  IMA Journal of Applied Mathematics, 78(4):750–761, 2013.
  * [78] J. Wilkening.  An algorithm for computing Jordan chains and inverting analytic matrix functions.  Linear Algebra Appl., 427:6–25, 2007.
  * [79] J. Wilkening.  Traveling-standing water waves.  Fluids, 6:187:1–35, 2021.
  * [80] J. Wilkening and J. Yu.  Overdetermined shooting methods for computing standing water waves with spectral accuracy.  Computational Science & Discovery, 5(1):014017, 2012.
  * [81] J. Wilkening and X. Zhao.  Quasi-periodic travelling gravity-capillary waves.  J. Fluid Mech., 915:A7:1–35, 2021.
  * [82] J. Wilkening and X. Zhao.  Spatially quasi-periodic water waves of infinite depth.  J. Nonlinear Sci., 31:52:1–43, 2021.
  * [83] J. Wilton.  On ripples.  Philosophical Magazine Series 6, 29(173):688–700, 1915.
  * [84] V. E. Zakharov.  Stability of periodic waves of finite amplitude on the surface of a deep fluid.  J. Appl. Mech. Tech. Phys., 9(2):190–194, 1968.
  * [85] J. Zufiria.  Non-symmetric gravity waves on water of infinite depth.  J. Fluid Mech., 181:17–39, 1987.