Overall, note that unlike the weak shear in Figure 10.A, the weak phase in
Figure 13.A shows very little correlation with the geometry of the critical
curve. Finally, we point out that placing the point mass off the horizontal
axis would not lead to a simple rotation of the patterns as in the other
presented plot grids. Instead, the overall halo pattern and the pattern close
to the point mass would remain unchanged. A point mass in a blue region of the
halo would thus locally generate a pair of orange lobes, while in an orange
region it would generate a pair of blue lobes.

####  3.5.8 Weak-phase Deviation Due to the Point Mass

The image-plane color maps presented in Figure 14.A depict the weak-phase
deviation due to the point mass, defined as the difference between the weak
shear of the NFW halo + point-mass lens (shown in Figure 13.A) and the weak
shear of the NFW halo alone (shown in the fourth panel of Figure 2),
$\delta\varphi_{\text{w}}(\bm{x})=\varphi_{\text{w}}-\varphi_{\text{w,\tiny{NFW}}}$.
We correct the difference if necessary by adding or subtracting $\pi$ to keep
$\delta\varphi_{\text{w}}$ in the interval $\left[-\pi/2,\,\pi/2\right]$. The
deviation is also equal to the angle by which the orientation of an image
changes due to the presence of the point mass. Red positive values of
$\delta\varphi_{\text{w}}$ correspond to a counterclockwise change, blue
negative values to a clockwise change in orientation. The color saturation is
scaled linearly for $|\delta\varphi_{\text{w}}|\leq\pi/2000$ and
logarithmically for $|\delta\varphi_{\text{w}}|\geq\pi/2000$. Contours are
plotted for $|\delta\varphi_{\text{w}}|=\pi/20=9^{\circ}$ (dark red and blue),
and for $|\delta\varphi_{\text{w}}|=\pi/200=0.9^{\circ}$ (light red and blue).

When the point mass is located exactly at the center of the halo (bottom row
of Figure 14.A), the lens has axial symmetry and the weak-phase deviation is
zero everywhere, which explains the completely white plots. Note that this
result also reflects the fact that images are oriented radially around a point
mass lying inside the unit-convergence circle, as shown in Section 3.5.7.

All the other plots with an off-center point mass share the same
characteristic color pattern; they differ only in its scale and in color
saturation. The blue and red regions are separated by the dot-dashed contour,
which marks all positions with zero deviation $\delta\varphi_{\text{w}}=0$ or
maximum deviation $|\delta\varphi_{\text{w}}|=\pi/2$. The geometry of the
contour is simple and independent of the mass parameter $\kappa_{\text{P}}$:
it includes the horizontal axis and the $\omega=\pi/2$ circle reaching from
the halo center to the position of the point mass (see Figure 5). Along the
horizontal axis, images are always horizontal inside and vertical outside the
unit-convergence circle, unaffected by the presence of a point mass, as shown
in Section 3.5.7. Hence, this part of the dot-dashed contour corresponds to
$\delta\varphi_{\text{w}}=0$.

Along the $\omega=\pi/2$ circle, there is a right angle between the direction
to the halo center and the direction to the point mass. The shear from the
halo and the shear from the point mass thus act in perpendicular directions.
Going from the halo center along the $\omega=\pi/2$ circle, the halo shear
decreases and the point-mass shear increases, as described in Section 3.1.
Hence, from the halo center to the zero-shear points the halo shear dominates
and the image orientation remains unchanged, so that
$\delta\varphi_{\text{w}}=0$ along this part of the circle. From the zero-
shear points to the point mass, the point-mass shear dominates and the images
are oriented perpendicular to the orientation they would have in absence of
the point mass, so that $|\delta\varphi_{\text{w}}|=\pi/2$ along the remaining
part of the circle. For point-mass positions
$x_{\text{P}}<\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$, there are no zero-
shear points and the entire $\omega=\pi/2$ circle corresponds to
$|\delta\varphi_{\text{w}}|=\pi/2$.

Above the horizontal axis and outside the $\omega=\pi/2$ circle, the weak-
shear deviation is positive meaning that in this region the image orientation
changes counterclockwise. Conversely, under the axis and outside the circle,
the deviation is negative and the image orientation changes clockwise. Inside
the $\omega=\pi/2$ circle, the sign of the deviation in either half-plane is
switched and image orientations change in the opposite sense. At the center of
the halo and at the location of the point mass, four regions of alternating
positive and negative deviation meet.

Deviations $|\delta\varphi_{\text{w}}|$ peak close to the point mass
(particularly along the $\omega=\pi/2$ circle), and fall rapidly with
increasing distance from it. The color pattern near the point mass indicates
that images are oriented tangentially around it (when it lies outside the
unit-convergence circle) or radially around it (when it lies inside the unit-
convergence circle). The color pattern around the halo center indicates that
the point mass orients the images more horizontally there. When zero-shear
points are present, zero deviations occur in the horizontal and vertical
directions from the center and strongest deviations occur along the diagonals.
In the absence of zero-shear points, for
$x_{\text{P}}<\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$, zero deviations
occur in the horizontal directions and strongest
$|\delta\varphi_{\text{w}}|=\pi/2$ deviations occur in the vertical
directions.

The areas with high deviation $|\delta\varphi_{\text{w}}|$ extend further from
the point mass for greater values of $\kappa_{\text{P}}$, as seen in the
central and right columns of Figure 14.A. For point masses far from the halo
center (in the top rows), all contours form four-lobed butterfly-like shapes.
For point masses closer to the halo center, the left lobes of the contours
become pointy (e.g., $x_{\text{P}}=0.2$ in the central column) and eventually
they extend to the halo center (e.g., $x_{\text{P}}=0.15$ in the central
column). To illustrate the area affected by the point mass, we estimate the
extent of the contours at the moment of separation of the critical curve
surrounding the point mass from the perturbed NFW tangential critical curve,
i.e., between $x_{\text{P}}=0.20$ and $x_{\text{P}}=0.25$ for the sub-critical
and critical cases, and between $x_{\text{P}}=0.25$ and $x_{\text{P}}=0.30$ in
the super-critical case. In all three cases, the inner
$|\delta\varphi_{\text{w}}|=\pi/20$ contours extend about four Einstein radii,
while the outer $|\delta\varphi_{\text{w}}|=\pi/200$ contours extend as far as
twelve Einstein radii from the point mass.

The changes in the contours for different point-mass positions can be studied
in more detail in Figure 14.B. The abrupt change from zero deviation to
$|\delta\varphi_{\text{w}}|=\pi/2$ along the $\omega=\pi/2$ circle at the
zero-shear points can be best seen in the presence of the small critical-curve
loops surrounding them, e.g., for $x_{\text{P}}=0.12$. Note also the very
small extent of the colored regions for the lowest separations $x_{\text{P}}$.

Similarly to Figure 13.A, the weak-phase deviation in Figure 14.A shows very
little correlation with the geometry of the critical curve. However, unlike
Figure 13.A, it also shows no influence of the unit-convergence circle.

##  4 Discussion

The examples of images formed by the NFW halo + point-mass lens in the bottom
right panel of Figure 1 can be compared with the results presented in Section
3.5, specifically with the corresponding $x_{\text{P}}=0.2$,
$\kappa_{\text{P}}\approx 2.714\cdot 10^{-4}$ panels of the plot grids. In
particular, the weak-shear map in Figure 10.A shows the image flattening, the
weak-phase map in Figure 13.A shows the image orientation, and the weak-shear-
deviation-from-shear map in Figure 12.A shows the relative shear error that
would be incurred by assuming the weak-lensing relation between image
distortion and shear.

A few points should be noted regarding the interpretation of such comparisons.
First, Figures 7.A–14.A present maps of their respective quantities for point-
like sources. For images of extended sources such as those shown in Figure 1,
one has to consider the full variation of the quantities within the area of
the image. For example, an image lying on the unit-convergence circle will
have its inner part extended radially and its outer part tangentially, as seen
from Figure 13.A.

Second, when studying the changes in image shape and orientation due to the
point mass using Figure 1 with Figure 11.A or Figure 14.A, a direct comparison
can be made when there is at least a partial overlap of the images formed by
halos with and without the point mass. In the bottom right panel of Figure 1,
the left and central images have a large overlap with the images in the top
right panel, the lower right image has a smaller overlap, and the two images
close to the point mass have no overlap. For these two images, Figure 11.A and
Figure 14.A show the deviations from images formed at the same positions by
the halo-only lens, but for different source positions. In this particular
case, the sources would lie above the horizontal axis at different radial
distances in the top left panel of Figure 1. In fact, even in the case of
overlapping images, the overlapping parts may be images of different parts of
the source. Thus, Figure 11.A and Figure 14.A compare the properties of images
formed by the two lens models at a same position in the image plane of sources
at different positions in the source plane. They do not compare the properties
of images of a fixed source formed by a halo with and without a point mass.

Third, in this work we do not explore the changes in image positions due to
the point mass. This would be difficult to present in general, if only due to
the change in the number of images. However, it could be done in a
perturbative regime, by studying the displacement and distortion of particular
images of a fixed source due to the presence of a point mass. This approach
was taken by Wagner (2018), who studied general perturbations of large arc-
like images along the tangential critical curve of axisymmetric lenses. One
provided example shows the influence of a point-mass perturber on the radii
and lengths of arcs formed by a singular isothermal sphere lens. Even though
NFW halos have different lensing properties, the generic pattern of angular
distortions seen for example in the $x_{\text{P}}=0.25$,
$\kappa_{\text{P}}=10^{-4}$ panel of Figure 14.A indicates that an arc-like
image between the tangential critical curve and the point mass inside the
$\omega=\pi/2$ circle would be straightened by its influence. Hence, its
radius of curvature would be increased, in agreement with the example in
Wagner (2018). Overall, image-plane maps such as those in Figures 11.A and
14.A are suitable for assessing the influence of compact masses in observed
lensing clusters or galaxies where the image positions are fixed.

The results presented in this work correspond to one fiducial value of the
halo convergence parameter $\kappa_{\text{s}}$. The variation of the critical
curves and the unit-convergence circle with $\kappa_{\text{s}}$, as well as
the variation of $\kappa_{\text{s}}$ and $\kappa_{\text{P}}$ with source
redshift are discussed in Paper I. The plots presented above in Sections
3.5.1–3.5.8 can be expected to follow the combined geometry of the critical
curve, the unit-convergence circle, and the $\omega=\pi/2$ circle. Note that
the last mentioned circle is given purely by the value of $x_{\text{P}}$,
which is independent of the source redshift and the halo convergence
parameter.

Some of the obtained results have more general relevance than just for a point
mass embedded in a spherically symmetric NFW halo. These include some of the
analytic results, such as the shear of a combination of two mass distributions
described in Section 3.1. Among the numerical results shown in Figures
7.A–14.A, the patterns seen around the point mass at sufficient separation
from the halo critical curves will be very similar for other halo mass
distributions with low spatial variation on the scale of the point-mass
Einstein radius (e.g., Chang & Refsdal, 1984), as discussed further below.
Best seen in the top left plots for $x_{\text{P}}=0.30$,
$\kappa_{\text{P}}=10^{-4}$, the patterns are relevant not only for point
masses separated from other mass concentrations, but also for extended bodies
with compact mass distributions that do not extend significantly beyond their
Einstein radius.

The applicability of the studied lens model to the astrophysical scenarios of
a galaxy within a galaxy cluster, of a satellite galaxy within a galactic
halo, and of a massive black hole in a galactic halo is discussed in Paper I.
In addition, possible extensions toward more advanced models are pointed out
there, such as replacing the point mass by an extended mass distribution. For
distributions that do not extend significantly beyond their total-mass
Einstein radius, the lensing quantities in the surroundings will not differ
significantly from the patterns seen in Figures 7.A–14.A. However, for more
extended distributions the patterns will be affected more substantially.
Nevertheless, at large separations the lensing impact of any compact object
may be approximated by that of a point mass.

The explored model can be extended also by altering the properties of the NFW
halo. On the one hand, one may change its central properties by adding a core
radius, or by changing its density divergence (Evans & Wilkinson, 1998). Such
changes would alter the radii of the halo critical curves and caustics, the
reference plots in Figure 2, and the critical value $\kappa_{\text{PC}}$ of
the mass parameter of the point mass. This would impact the presented results
primarily for point-mass positions close to the halo center. Regardless of the
nature of the alterations, for sufficiently large distances the point-mass
critical-curve loop disconnects from the halo critical curves. In this distant
regime the patterns in the vicinity of the point mass will have the same
character as seen in the figures.

On the other hand, one may abandon spherical symmetry and study the effect of
a massive object embedded in a more realistic elliptical NFW halo. The lensing
properties of an NFW halo with an elliptically symmetric mass distribution are
poorly studied due to the lack of simple analytic expressions to describe them
(Oguri, 2021). Nevertheless, for low values of the halo ellipticity the
properties can be approximated by those of the pseudo-elliptical model, which
has an elliptically symmetric lens potential (Golse & Kneib, 2002; Meneghetti
et al., 2003; Dúmet-Montoya et al., 2012). In this model the critical curve
typically consists of two nested oval loops instead of the two circles of the
spherical model. The point-like tangential caustic of the spherical model is
replaced by a four-cusped loop; the circular radial caustic by an oval loop.
In this model one may expect an even more complex dependence of the critical
curves and caustics of the combined lens on the mass and two-dimensional
position of the point mass than in the spherical case described in Paper I.

In the elliptical model, the unit-convergence circle which plays a key role in
the spherical model when studying the geometry of images is replaced by a
unit-convergence ellipse. The grids of the plots corresponding to those in
Figures 7.A–14.A would be complicated by an additional parameter, the angular
position of the point mass with respect to the axes of the elliptical halo.
Nevertheless, the main factors driving the patterns described in the text,
such as the existence and location of zero-shear points or the geometry of the
perturbations of the critical curve, would remain the same. The asymptotic
patterns would have the same nature as seen in the spherical model, as
mentioned above. However, they would be less symmetric and the extent of the
contours would additionally depend on the angular position within the halo.

The presented single-point-mass results will be useful for interpreting the
properties of cluster lens models with multiple (point) masses embedded in an
NFW halo. Masses that are sufficiently separated from the perturbed NFW halo
critical curves as well as from other masses should display similar patterns
in their vicinity, as discussed above. Each of these masses will also produce
deviation patterns near the halo center similar to those seen in the top left
plots in Figures 8.A, 11.A, or 14.A. Due to the directional dependence of
these patterns, their superposition for a sufficient number of isotropically
distributed masses would drive the amplitude of the central deviations to
zero. Strong differences can be expected when one or more of the masses are
positioned close to the halo center, or when two or more neighboring masses
are mutually separated by less than a few Einstein radii. These situations
cannot be simply extrapolated from the results presented in this work and
their study requires direct simulations.

##  5 Summary

In this paper we proceeded in our study of gravitational lensing by a compact
massive object in a dark matter halo. In Paper I we analyzed the critical
curves and caustics of a lens consisting of a point mass embedded in a
spherical NFW halo. Here we concentrated on the shear and phase of the same
lens model, focusing on their relation to the geometry of images formed by the
lens.

In Section 2.1, we described the properties of the shear and phase of a lens
consisting only of a NFW halo. In order to study the images, we used the
eigenvalue decomposition of the inverse matrix of a general lens-equation
Jacobian matrix presented in Equation (21). Based on it, we introduced the
convergence–shear diagram (CS diagram) in Appendix A, which illustrates
concisely the connection between arbitrary convergence and shear values and
the geometry of an image corresponding to them. Specific lens models occupy
characteristic regions in the diagram, which then define the properties of all
images that could be formed by the lens. We described the properties of images
formed by the NFW halo by reading them off the CS diagram in Section 2.3. In
Section 2.4 we defined the weak shear and weak phase as the shear and phase
values obtained by assuming weak-lensing relations involving the semiaxes of
images and their orientation.

We followed the same outline for the NFW halo + point-mass lens in Section 3.
In particular, we derived the formula for the shear in Equation (31) which
provides a geometric interpretation in terms of the halo and point-mass shears
and the viewing angle $\omega$ of the line segment separating them. The
formula which is valid for combinations of other axisymmetric lenses is a
special case of the more general formula in Equation (36) for the shear of a
combination of two arbitrary lenses. For the NFW halo + point-mass lens, we
discuss the appearance and location of zero-shear points in Section 3.1 and
describe the conditions under which they form umbilic points in Section 3.2.

Figures 7.A–14.A illustrate the main results in terms of image-plane maps of
different lens characteristics and CS diagrams, all presented for the same
grid of point-mass parameter combinations as the grid used in Paper I.
Important features and trends seen in the figures are described in the
corresponding Sections 3.5.1–3.5.8. As discussed in Section 4, the obtained
results have broader implications beyond the specific properties of the
studied lens model.

We thank the anonymous referee for comments and suggestions that helped
improve the manuscript. Work on this project was supported by Charles
University Grant Agency project GA UK 1000218.

##  Appendix A IMAGE GEOMETRY AS A FUNCTION OF CONVERGENCE AND SHEAR

The mapping from a source to its image is described locally by the inverse of
the lens-equation Jacobian matrix. The eigenvalue decomposition of this
inverse matrix $\mathbb{A}$ presented in Equation (21) shows that the full
geometry of an image of a small source, including its shape, size, and
orientation with respect to the phase, is given by the lens convergence and
shear at the position of the image. We demonstrate here the connection between
the geometry of an image and its position in a general convergence–shear
diagram.

The properties of an image appearing at position $\bm{x}$ in the image plane
can be determined from the convergence–shear diagram (CS diagram) in Figure 15
using the combination of the local convergence $\kappa(\bm{x})$ and shear
$\gamma(\bm{x})$, and the eigenvalues introduced in Equation (20) computed
from them. The eigenvalue $\lambda_{\parallel}$, which defines the scaling
factor in the direction of the phase, is constant in the diagram along
straight lines with slope $-1$. It starts at $1$ at the origin of the plot,
and increases to $\infty$ at the solid red line, which corresponds to critical
curves (tangential in the case of axially symmetric lenses). Above it
$\lambda_{\parallel}$ changes discontinuously to $-\infty$, with the negative
sign indicating that the image is flipped in the direction of the phase. The
value of $\lambda_{\parallel}$ increases to $-1$ along the dashed red line,
above which the flipped image is contracted rather than expanded in the
direction of the phase. Further beyond the top right corner of the diagram,
$\lambda_{\parallel}$ increases asymptotically to $0$. Overall, below the
dashed red line the image is expanded in the direction of the phase, while
above it the image is contracted in the direction of the phase.

Figure 15: Convergence–shear diagram (CS diagram) illustrating the geometry of
lensed images as determined by the $(\kappa,\gamma)$ combination at their
position. The color map and the purple-labeled hyperbolic contours show the
values of $\mathrm{det}\,\mathbb{A}$; its absolute value yields the
magnification and its sign yields the parity of the image (positive in green,
negative in pink areas). Unit magnification occurs at the origin of the
diagram and along the bold black hyperbolae; outside the region delimited by
them images are demagnified. The black-labeled straight-line contours
correspond to constant flattening of the image, given by Equation (22).
Undistorted images occur along the $\gamma=0$ axis and the bold black
$\kappa=1$ line. The solid red and solid blue lines correspond to critical-
curve points; the combination $(\kappa,\gamma)=(1,0)$ specifically to umbilic
points. Unit absolute values of the eigenvalues of $\mathbb{A}$ occur along
the dashed diagonals: $\lambda_{\parallel}=-1$ (red); $\lambda_{\perp}=1$
(blue from origin); $\lambda_{\perp}=-1$ (blue from $\kappa=2$). For
$\kappa<1$ images are elongated in the direction of the phase; for $\kappa>1$
perpendicular to the phase. For more details, see Appendix A.

The scaling factor in the direction perpendicular to the phase is defined by
the eigenvalue $\lambda_{\perp}$, which is constant in the diagram along
straight lines with slope $1$. Along the dashed blue line passing through the
origin we find $\lambda_{\perp}=1$. Above it the image is contracted in the
direction perpendicular to the phase, and $\lambda_{\perp}$ decreases
asymptotically to $0$ above the top left of the diagram. Below the dashed blue
line passing through the origin $\lambda_{\perp}$ increases to $\infty$ at the
solid blue line, which corresponds to critical curves (radial in the case of
axially symmetric lenses). Below it $\lambda_{\perp}$ changes discontinuously
to $-\infty$, with the negative sign indicating that the image is flipped in
the direction perpendicular to the phase. The value of $\lambda_{\perp}$
increases to $-1$ along the right dashed blue line, below which the flipped
image is contracted in the direction perpendicular to the phase. Further
beyond the bottom right corner of the diagram, $\lambda_{\perp}$ increases
asymptotically to $0$. Overall, in the band between the dashed blue lines the
image is expanded in the direction perpendicular to the phase, while outside
it the image is contracted in the direction perpendicular to the phase.

The described scalings in the two perpendicular directions can be combined to
yield information about the orientation, shape, and size of the image.
Comparing their absolute values for a non-zero shear, $|\lambda_{\parallel}|$
is larger for $\kappa<1$ and $|\lambda_{\perp}|$ is larger for $\kappa>1$.
Hence, the vertical solid black line at $\kappa=1$ divides images by the
orientation of their distortion, i.e., the orientation of the major axis of an
elliptical image of a small circular source. To the left of the line, images
are oriented in the direction $\varphi$ (parallel to the phase), while to the
right of the line, images are oriented in the direction $\varphi+\pi/2$
(perpendicular to the phase).

The distortion of the shape can be quantified by the flattening, computed from
Equation (22). In the diagram in Figure 15, the flattening $f$ is constant
along the straight lines radiating from the point $(\kappa,\gamma)=(1,0)$,
labeled by their $f$ value at the outer edge of the plot. The horizontal axis
corresponds to $f=0$, i.e., there is no distortion for zero shear. Clockwise
from the direction to the origin of the plot, $f$ values along the lines
increase in steps of $0.25$ to $1$ at the solid red line, corresponding to
maximum flattening in the direction of the phase at the tangential critical
curve. For the following lines, $f$ decreases in steps of $0.25$ to $0$ at the
vertical $\kappa=1$ line, along which there is no distortion either.
Continuing clockwise, $f$ increases to $1$ at the solid blue line,
corresponding to maximum flattening in the direction perpendicular to the
phase at the radial critical curve. The flattening along the following lines
decreases back to $0$ along the horizontal axis.

The change in size of the image is given by the absolute value of the product
of the two scale factors. Since they are eigenvalues of $\mathbb{A}$, their
product is equal to its determinant,

$\mathrm{det}\,\mathbb{A}(\bm{x})=\lambda_{\parallel}(\bm{x})\,\lambda_{\perp}(\bm{x})=\left\\{\,\left[1-\kappa(\bm{x})\right]^{2}-\gamma^{2}(\bm{x})\,\right\\}^{-1}\,,$
(A1)

which is the inverse of the Jacobian $\mathrm{det}\,J(\bm{x})$ from Equation
(19). For an image at $\bm{x}$, the sign of $\mathrm{det}\,\mathbb{A}(\bm{x})$
yields the parity and its absolute value $|\mathrm{det}\,\mathbb{A}(\bm{x})|$
yields the (point-source) magnification, the ratio of solid angles subtended
by the image and by the source. The values of $\mathrm{det}\,\mathbb{A}$ are
indicated by the purple-labeled hyperbolic contours and the color map in
Figure 15. Shades of pink above the critical-curve lines indicate negative
$\mathrm{det}\,\mathbb{A}$, i.e., all images here are mirror images with
negative parity. Positive-parity images lie in the green regions below the
critical-curve lines.

The origin of the diagram with zero convergence and shear has unit
magnification and positive parity, $\mathrm{det}\,\mathbb{A}=1$. Proceeding
from the origin, contours are plotted for magnifications increasing in powers
of two, corresponding to
$\mathrm{det}\,\mathbb{A}\in\left\\{2,4,8,16\right\\}$. Along the solid red
and solid blue lines the magnification is infinite; above them the sign of
$\mathrm{det}\,\mathbb{A}$ flips to negative. Going upward from the critical-
curve lines, the magnifications along the plotted hyperbolas decrease in
powers of two, corresponding to
$\mathrm{det}\,\mathbb{A}\in\left\\{-16,-8,-4,-2,-1,-0.5\right\\}$. The bold
black hyperbola passing through the point $(\kappa,\gamma)=(1,1)$ thus
corresponds to unit magnification. All images above it are demagnified. Below
the solid blue radial-critical-curve line the sign of
$\mathrm{det}\,\mathbb{A}$ flips back to positive. Proceeding from it to the
right, contours are plotted for
$\mathrm{det}\,\mathbb{A}\in\left\\{16,8,4,2,1,0.5\right\\}$. The bold black
hyperbola passing through the point $(\kappa,\gamma)=(2,0)$ thus also
corresponds to unit magnification. All images to its right are demagnified.

Note that the values of $\lambda_{\parallel}$ along diagonal lines with slope
$-1$ are equal to the value of $\mathrm{det}\,\mathbb{A}$ at their
intersection with the dashed blue line starting from the origin. The values of
$\lambda_{\perp}$ along diagonal lines with slope $1$ are equal to
$-\mathrm{det}\,\mathbb{A}$ at their intersection with the dashed red line.

The two perpendicular lines in the diagram corresponding to undistorted images
differ by the sign of $\mathrm{det}\,\mathbb{A}$. Images along the horizontal
axis (with $\gamma=0$) have positive parity; they are magnified for $\kappa<2$
and demagnified for $\kappa>2$. Images along the vertical bold line (with
$\kappa=1$) are mirror images with negative parity; they are magnified for
$\gamma<1$ and demagnified for $\gamma>1$. Clearly, the point
$(\kappa,\gamma)=(1,0)$ lying at the intersection of these lines has special
significance. At this point, the entire Jacobian matrix given by Equation (18)
is equal to zero, and its inverse $\mathbb{A}$ is thus undefined. If such
points exist in the image plane of a gravitational lens, they define the
position of critical-curve umbilic points (Schneider et al. 1992; Paper I).
The properties of images in their vicinity depend on higher-order derivatives
of the lens equation.

The structure of the CS diagram shows that in the general non-critical case
there are four different $(\kappa,\gamma)$ combinations that lead to the same
combination of $(f,|\mathrm{det}\,\mathbb{A}|)$, i.e., an image of the same
shape and size. Two of these have positive and two have negative parity, as
illustrated in Figure 3. In the case of zero flattening, there are three
different $(\kappa,\gamma)$ combinations with only one negative-parity image.

For any specific gravitational lens, the range of
$(\kappa(\bm{x}),\gamma(\bm{x}))$ combinations occurring in its image plane
defines a region in the diagram which demonstrates the properties of all
possible images formed by the lens. For a lens with an axially symmetric mass
distribution the region is one-dimensional, described by the curve
$\left(\kappa(x),\gamma(x)\right)$ with the radial position $x$ varying from
$0$ to $\infty$. For a spherical NFW halo this case is described in Section
2.3 and illustrated in Figure 4; for a NFW halo with a centrally positioned
point mass see the bottom row of Figure 9.A. For more asymmetric lenses with a
(non-constant) continuous mass distribution, the region in the diagram is two-
dimensional, as shown for the spherical NFW halo with an off-center point mass
in Figure 6 and 9.A (except the bottom row).

For lenses consisting of point masses without continuous matter (e.g., stars
and stellar systems in Galactic microlensing), the corresponding one-
dimensional region is the $\kappa=0$ vertical axis, with $\gamma\to\infty$ at
the positions of the masses and $\gamma\to 0$ far from them. For quasar
microlensing, in which point masses are combined with a constant background
convergence $\kappa_{0}$ and shear $\gamma_{0}$, the one-dimensional region
lies along the $\kappa=\kappa_{0}$ vertical line. Finally, the regime of weak
lensing with $\kappa\ll 1$ and $\gamma\ll 1$ is confined to the vicinity of
the origin of the diagram in Figure 15.

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# Bifurcation of solitary waves

Vladimir Kozlov1

###### Abstract.

We consider solitary water waves on the vorticity flow in a two-dimensional
channel of finite depth. The main object of study is a branch of solitary
waves starting from a laminar flow and then approaching an extreme wave. In
[9] it was proved that there always exists a bifurcation point on such
branches and the first bifurcation occurs at a simple eigenvalue. Here we
describe the structure of global bifurcating branches starting from this
bifurcation point.

##  1\. Introduction

We consider steady water waves in a two-dimensional channel bounded below by a
flat, rigid bottom and above by a free surface that does not touch the bottom.
The surface tension is neglected and the water motion can be rotational. In
appropriate Cartesian coordinates $(X,Y)$, the bottom coincides with the
$X$–axis and gravity acts in the negative $Y$–direction. We choose the frame
of reference so that the velocity field is time-independent as well as the
free-surface profile which is supposed to be the graph of $Y=\xi(X)$,
$x\in\mathbb{R}$, where $\xi$ is a positive and continuous unknown function.
Thus

$D=D_{\xi}=\\{X\in\mathbb{R},0<Y<\xi(X)\\},\;\;S=S_{\xi}=\\{X\in\mathbb{R},\;Y=\xi(X)\\}$

is the water domain and the free surface respectively. We will use the stream
function $\Psi$, which is connected with the velocity vector $({\bf u},{\bf
v})$ as $\Psi_{Y}=c-{\bf u}$ and ${\bf v}=\Psi_{X}$, where $c$ is a constant
wave speed.

We assume that $\xi$ is a positive, even function. Since the surface tension
is neglected, $\Psi$ and $\xi$ after a certain scaling satisfy the following
free-boundary problem (see, for example, [10]):

$\displaystyle\Delta\Psi+\omega(\Psi)=0\;\;\mbox{in $D_{\xi}$},$
$\displaystyle\frac{1}{2}|\nabla\Psi|^{2}+\xi=R\;\;\mbox{on $S_{\xi}$},$
$\displaystyle\Psi=1\;\;\mbox{on $S_{\xi}$},$
$\displaystyle\Psi=0\;\;\mbox{for $Y=0$},$ (1.1)

where $\omega\in C^{1,\alpha}$, $\alpha\in(0,1)$, is a vorticity function and
$R$ is the Bernoulli constant. We assume that $\Psi$ is even in $X$ and

$\Psi_{Y}>0\;\;\mbox{on $\overline{D_{\xi}}$},$ (1.2)

which means that the flow is unidirectional.

The main our subject of study is the solitary waves. The solitary wave
solution satisfies

$\;\;\xi(X)\to d,\;\;\Psi(X,Y)\to U(Y)\;\;\mbox{as $X\to\pm\infty$}$ (1.3)

uniformly in $Y$. Here $(U,d)$ is a uniform stream solution with the Bernoulli
constant $R=d+U^{\prime}(d)^{2}/2$. Moreover the Froude number

$F=\Big{(}\int_{0}^{d}\frac{dY}{U^{\prime}(Y)^{2}}\Big{)}^{-2}$ (1.4)

for this uniform stream solution must be $>1$. In this case the corresponding
solitary wave is non-trivial, i.e. satisfies $\xi(X)>d$ and it is symmetric,
after a certain translation, and monotonic decreasing for positive $X$. In
what follows we assume that such translation is already done and we consider
even solitary waves.

If the Froude number $F$ is equal to $1$ or equivalently $R=R_{c}$, then the
only solution of (1) is the uniform stream solution $(U_{*}(Y),d_{*})$, where
$d_{*}=d(\theta_{c})$ (see Sect. 1.1 for details).

The Frechet derivative for the problem is evaluated for example in [15], [7]
in the periodic case. The same evaluation can be used to obtain the Frechet
derivative in the case, when the solution is a solitary wave. The
corresponding eigenvalue problem for the Frechet derivative has the form

$\displaystyle\Delta w+\omega^{\prime}(\Psi)w+\mu w=0\;\;\mbox{in $D_{\xi}$},$
$\displaystyle\partial_{n}w-\rho w=0\;\;\mbox{on $S_{\xi}$},$ $\displaystyle
w=0\;\;\mbox{for $Y=0$},$ (1.5)

where $n$ is the unite outward normal to $Y=\xi(X)$ and

$\rho=\rho(X)=\frac{(1+\Psi_{X}\Psi_{XY}+\Psi_{Y}\Psi_{YY})}{\Psi_{Y}(\Psi_{X}^{2}+\Psi_{Y}^{2})^{1/2}}\Big{|}_{Y=\xi(X)}.$
(1.6)

The function $w$ in (1) is supposed also to be even.

Let us introduce several function spaces. For $\alpha\in(0,1)$ and
$k=0,1,\ldots$, the space $C^{k,\alpha}(D)$ consists of bounded functions in
$D$ such that the norms $C^{k,\alpha}(D_{a,a+1})$ are uniformly bounded with
respect to $a\in\mathbb{R}$. Here

$D_{a,a+1}=\\{(X,Y)\in\overline{D},\;:\,a\leq X\leq a+1\\}.$

The space $C^{k,\alpha}_{0}(D)$ consists of function in $C^{k,\alpha}(D)$
vanishing for $Y=0$. The space $C^{k,\alpha}_{0,e}(D)$ consists of even
functions from $C^{k,\alpha}_{0}(D)$. Similarly we define the space
$C^{k,\alpha}_{e}(\mathbb{R})$) consisting of even functions in
$C^{k,\alpha}(\mathbb{R})$.

We will consider a branch of solitary waves depending on a parameter $t>0$,
i.e.

$\Psi=\Psi(X,Y;t),\;\;\xi=\xi(X,t),\;\;R=R(t),\;\;t\in(0,\infty),$ (1.7)

where $R$ is the Bernoulli constant.

We assume that the above branch of solitary waves satisfies the conditions of
the following theorem, see [9] and [23].

######  Theorem 1.1.

Fix a Hölder exponent $\alpha\in(0,1/2]$. There exists a continuous curve
(1.7) of solitary wave solutions to (1) with the regularity

$(\Psi(t),\xi(t),R(t)))\in C^{2,\alpha}(D(t))\times
C^{2,\alpha}({\mathbb{R}})\times(R_{c},\infty),$ (1.8)

where $D(t):=D_{\xi(t)}$. The solution curve satisfies the following property
(critical laminar flow, i.e. $R=R_{c}$)

$\lim_{t\to 0}(\Psi(t),\xi(t),R(t))=(U_{*},d_{*},R_{c}).$ (1.9)

Furthermore, there exists a sequence $\\{t_{j}\\}_{j=1}^{\infty}$ such that
$t_{j}\to\infty$ as $j\to\infty$ and $(X_{j},Y_{j})\in D(t_{j})$ such that the
following is valid

$(\Psi(t_{j}))_{Y}(X_{j},Y_{j})\to 0\;\;\mbox{as
$j\to\infty$}\;\;\;\mbox{(Stagnation)}.$ (1.10)

######  Remark 1.2.

(i) The convergence in (1.9) is understood in the spaces

$C^{2,\alpha}(D(t))\times C^{2,\alpha}({\mathbb{R}})\times\mathbb{R}$

(ii) The dependence on $t\in(0,\infty)$ in (1.8) is analytic in the variables
$(q,p)$ using in partial hodograph transformation, see [24], Sect.6.

(iv) In paper [23] the Froude number $F(t)$ is used instead of $R(t)$. Since
the dependence $R$ on $F$ is monotone and analytic (see [9]), we use here the
function $R(t)$.

In the next proposition we present a clarification of the stagnation condition
(1.10), where the constants $R_{0}$, $\theta_{0}$ and the function
$\Omega(\tau)$ are defined in the next section.

######  Proposition 1.3.

The stagnation condition (1.10) is equivalent to the following options

(i) If $R_{0}=\infty$ or $R_{0}<\infty$ and $\Omega(1)>0$ then there exists a
sequence $\\{t_{j}\\}_{j\geq 1}$, $t_{j}\to\infty$ as $j\to\infty$ such that

$\lim_{j\to\infty}|\xi^{\prime}(t_{j})(0)|=\infty$ (1.11)

or

$\lim_{j\to\infty}R(t_{j})=R\;\;\mbox{and}\;\;\lim_{j\to\infty}\xi(t_{j})(0)=R.$
(1.12)

(ii) If $R_{0}<\infty$ and $\theta_{0}=0$ then besides the options in (i)
there is one more option:

$\lim_{j\to\infty}R(t_{j})=R\;\;\mbox{and}\;\;\lim_{j\to\infty}\sup_{X\in\mathbb{R}}\Psi_{Y}(t_{j})(X_{j},Y_{j})=0,$
(1.13)

where $Y_{j}\to 0$ as $j\to\infty$.

This proposition is proved in [9] except of the case $R_{0}<\infty$ and
$\Omega(1)>1$. We prove this case in Sect. 3.4.

Denote by ${\mathcal{A}}(t)={\mathcal{A}}_{(\Psi,\xi)}$ the unbounded operator
in $L^{2}(D_{\xi})$ corresponding to the problem (1), i.e.
${\mathcal{A}}_{(\Psi,\xi)}=-\Delta-\omega^{\prime}(\Psi)$ with the domain

${\mathcal{D}}(t)={\mathcal{D}}({\mathcal{A}}_{(\Psi,\xi)})=\\{w\in
H^{2}(D_{\xi})\,:\,w(X,0)=0,\;\;\partial_{\nu}w-\rho
w=0\,\mbox{for}\;Y=\xi(X)\\}.$

For every $t>0$ we denote by

$(U(t),d(t),R(t))$ (1.14)

the limit uniform stream solution and the corresponding Bernoulli constant in
the asymptotics (1.3).

Consider the eigenvalue problem

$\displaystyle-v^{{}^{\prime\prime}}-\omega^{\prime}(U(t))v=\nu v\;\;\mbox{on
$(0,d(t))$},$ $\displaystyle
v(0)=0\;\;\mbox{and}\;\;v^{\prime}(d(t))-\rho_{0}(t)v(d(t))=0,$ (1.15)

where

$\rho_{0}(t)=\frac{1+U_{Y}U_{YY}}{U_{Y}^{2}}\Big{|}_{Y=d(t)}.$ (1.16)

As it is shown in [9], the eigenvalues of this problem are positive for all
$t>0$, since the Froude number corresponding to the uniform stream solution
(1.14) is larger than $1$. We denote by $\nu_{0}=\nu_{0}(t)$ the lowest
eigenvalue. This eigenvalue is simple and corresponding eigenfunction does not
change sign in $(0,d(t)]$. Using the just intruduce number $\nu_{0}$, we can
write the asymptotics of the solitory wave ar infinity:

$\xi(X;t)=d(t)+O(e^{-\sqrt{\nu_{0}}X}),\;\;\Psi(X,Y;t)=U(Y)++O(e^{-\sqrt{\nu_{0}}|X|})\;\;\mbox{as
$|X|\to\infty$}.$

######  Proposition 1.4.

(see [9]) For every $t>0$ the operator ${\mathcal{A}}$ has a continuous
spectrum $[\nu_{0}(t),\infty)$. The spectrum on the half line $(-\infty,a]$
consists of finite number of isolated eigenvalues of finite multiplicity for
any $a<\nu_{0}(t)$.

The following result is proved in [9]

######  Theorem 1.5.

(i) The lowest eigenvalue of the operator ${\mathcal{A}}(t)$ for $t>0$ is
always negative and simple with the eigenfunction which does not change sign
in $D(t)$. We denote it by $\mu_{0}(t)$.

(ii) For small positive $t$ the spectrum of the operator ${\mathcal{A}}(t)$ on
the half line $\mu\leq 0$ consists of the eigenvalue $\mu_{0}(t)$ only.

(iii) If for a certain $t_{1}>0$ and $t\in(0,t_{1})$ the operator
${\mathcal{A}}(t)$ has the only negative eigenvalue $\mu_{0}$ and
${\mathcal{A}}(t_{1})$ has the eigenvalue $\mu=0$ then this eigenvalue is
simple.

Denote by $\mu_{1}(t)$ the second eigenvalue of the operator
${\mathcal{A}}(t)$ on the half-line $(-\infty,\nu_{0})$ if it exists,
otherwise we put $\mu_{1}(t)=\nu_{0}(t)$. Assume that

$\displaystyle\mbox{there exists a point $t_{*}>0$ such that: (i)
$\mu_{1}(t)\geq 0$ for $t\in(0,t_{*})$},$ $\displaystyle\mbox{ $\mu_{1}(t)<0$
for small positive $t-t_{*}$}.$ (1.17)

According to Theorem 1.5(iii) the eigenvalue $\mu_{1}(t_{*})=0$ is simple.
This implies that the crossing number of the Frechet derivative is $1$ at
$t_{*}$. Since the eigenvalue $\mu_{1}(t)$ is analytic it can be represented
as

$\mu_{1}(t)=\sum_{j=m}^{\infty}\mu_{j}(t-t_{*})^{j},$ (1.18)

where $m$ is odd and $\mu_{m}<0$.

######  Proposition 1.6.

([9]) Let the condition (1) hold. Then $t_{*}$ is a bifurcation point and
bifurcating solution builds a connected continuum.

Let the option (1.12) be valid then the condition (1) is satisfied.

In the next proposition we give a condition when (1) is fulfilled

######  Proposition 1.7.

([9]) Let there exists a sequence $\\{t_{j}\\}_{j=1}^{\infty}$ such that the
curve (1.7) approach an extreme solitary wave with the angle111Actually the
condition on the angle can be removed, see the paper [17]. Since it is not
published yet we will keep this angle requirement. $120^{\circ}$ at the crest
along this sequence then there exists a point $t_{*}$ which satisfies the
condition (1).

The aim of this paper is a description of the bifurcating continuum in more
detail.

######  Theorem 1.8.

Let condition (1) and hence condition (1.18) hold. There exists at least one
and at most $m$ continuous curves

$(\widehat{\Psi}_{s},\widehat{\xi}_{s},t(s))\in C^{2,\alpha}(Q)\times
C^{2,\alpha}(\mathbb{R})\times(0,\infty),\;\;\;s\in[0,\infty),$

bifurcating from $(\Psi(t_{*}),\xi(t_{*}),R(t_{*}))$. They admits an analytic
re-parameterisation near each point $s\geq 0$, and $t(0)=t_{*}$,
$(\widehat{\Psi}_{0},\widehat{\xi}_{0})=(\Psi(t_{*}),\xi(t_{*})$. Moreover
each curve can be one of the following:

(i) $t(s)=t_{*}$ for all $s\geq 0$;

(ii) $\mu_{1}((t(s))=0$ for all $s\geq 0$;

(iii) $t^{\prime}(s)\neq 0$ and the Frechet derivative is invertible for small
positive $t-t_{*}$;

Furthermore the bifurcation curve satisfies on of the following properties:

(a) $\sup_{s>0,(X,Y)\in
D_{\widehat{\xi}_{s}}}|\widehat{\xi}_{s}^{\prime}(X)|=\infty$ (overhanging);

(b) $\inf_{s>0,(X,Y)\in
D_{\widehat{\xi}_{s}}}\partial_{Y}\widehat{\Psi}_{s}(X,Y)=0$ (stagnation);

(c) $t(s)\to\infty$ as $s\to\infty$;

(d) the branch is a periodic curve.

The stagnation property (b) can be clarified as follows

######  Proposition 1.9.

The stagnation condition b is equivalent to the options (i) and (ii) in
Proposition 1.9.

This paper is devoted to the proof of Theorem 1.8.

The first study of bifurcations on the branches of solitary water waves was
done by Plotnikov in [19] in the irrotational case. In the case of vortical
flows one cannot apply the same technique as in irrotational case since it is
based on the complex analysis approach. The important contributions allowing
to approach the bifurcation problems are papers [23], [24] and [9], where the
global branches of solitary waves were constructed, and the papers [15]
together with [7] and [8], where the bifurcation analysis for branches of
Stokes waves were presented.

###  1.1. Uniform stream solution and dispersion equation

The uniform stream solution $\Psi=U(Y)$ with the constant depth $\xi=d$
satisfies the problem

$\displaystyle U^{{}^{\prime\prime}}+\omega(U)=0\;\;\mbox{on $(0,d)$},$
$\displaystyle U(0)=0,\;\;U(d)=1,$
$\displaystyle\frac{1}{2}U^{\prime}(d)^{2}+d=R.$ (1.19)

To find solutions to this problem we introduce a parameter
$\theta=U^{\prime}(0)$. We assume that
$\theta>\theta_{0}:=\sqrt{2\max_{\tau\in[0,1]}\Omega(\tau)}$, where

$\Omega(\tau)=\int_{0}^{\tau}\omega(p)dp.$ (1.20)

Then the problem (1.1) has a solution $(U,d)$ with a strongly monotone
function $U$ for

$R={\mathcal{R}}(\theta):=\frac{1}{2}\theta^{2}+d(\theta)-\Omega(1).$ (1.21)

The solution is given by

$Y=\int_{0}^{U}\frac{d\tau}{\sqrt{\theta^{2}-2\Omega(\tau)}},\;\;d=d(\theta)=\int_{0}^{1}\frac{d\tau}{\sqrt{\theta^{2}-2\Omega(\tau)}}.$
(1.22)

The function $d$ is strongly decreasing.

If we consider (1.21) as the equation with respect to $\theta$ then it is
solvable if $R\geq R_{c}$, where

$R_{c}=\min_{\theta\geq\theta_{0}}{\mathcal{R}}(\theta),$ (1.23)

and it has two solutions if $R\in(R_{c},R_{0})$, where

$R_{0}={\mathcal{R}}(\theta_{0}).$ (1.24)

We denote by $\theta_{c}$ the point where the minimum in (1.23) is attained.

The following connection between the Bernoulli constant $R$ and the Froude
number $F$ is proved in [9].

######  Proposition 1.10.

The Bernoulli constant $R\in[R_{c},\infty)$ can be considered as a continuous
function of the Froude number $F\in[1,\infty)$. This function is strongly
increasing, analytical on $(1,\infty)$ and $R(1)=R_{c}$. Moreover

$R(F)=\frac{1}{2}F^{4/3}+O(1)\;\;\mbox{as $F\to\infty$}.$

In order to describe how $d_{0}$ and the corresponding stream solution depend
on the vorticity distribution the following three options were considered in
[11].

(i) $\max_{0\leq\tau\leq 1}\Omega(\tau)$ is attained either at an inner point
of $(0,1)$ or at one (or both) of the end points. In the latter case, either
$\omega(1)=0$ when $\Omega(1)>\Omega(\tau)$ for $\tau\in(0,1)$ or
$\omega(0)=0$ when $\Omega(0)>\Omega(\tau)$ for $\tau\in(0,1)$ (or both of
these conditions hold simultaneously).

(ii) $\Omega(0)>\Omega(\tau)$ for $\tau\in(0,1]$ and $\omega(0)<0$.

(iii) $\Omega(\tau<\Omega(1)$ for $\tau\in(0,1)$ and $\omega(1)>0$. Moreover,
if $\Omega(1)=0$, then $\omega(0)<0$ and $\omega(1)>0$ must hold
simultaneously.

Conditions (i)–(iii) define three disjoint sets of vorticity distributions
whose union is the whole set of distributions continuous on $(0,1)$. It occurs
that conditions (i) imply that $d_{0}=\infty$, whereas conditions (ii) and
(iii) yield that $d_{0}$ is finite as well as $R_{0}$. The last two conditions
have the following consequences for solutions of problem (1.5) when $s=s_{0}$:
$U^{\prime}(d_{0})\neq 0$ under conditions (iii), whereas (ii) implies that
$U^{\prime}(0)=0$ and $U^{\prime}(d_{0})\neq 0$. This shows the difference
between the assumptions in (i) and (ii) in Proposition 1.9.

###  1.2. Partial hodograph transform

In what follows we will study branches of solitary waves
$(\Psi(X,Y;t),\xi(X;t))$ started from the uniform stream solution at $t=0$.
The existence of such branches is established in [23].

We assume that

$\Psi_{Y}>0\;\;\mbox{in $\overline{D_{\xi}}$}$

and use the variables

$q=X,\;\;p=\Psi.$

Then

$q_{X}=1,\;\;q_{Y}=0,\;\;p_{X}=\Psi_{X},\;\;p_{Y}=\Psi_{Y},$

and

$\Psi_{X}=-\frac{h_{q}}{h_{p}},\;\;\Psi_{Y}=\frac{1}{h_{p}}.$ (1.25)

System (1) in the new variables takes the form

$\displaystyle\Big{(}\frac{1+h_{q}^{2}}{2h_{p}^{2}}+\Omega(p)\Big{)}_{p}-\Big{(}\frac{h_{q}}{h_{p}}\Big{)}_{q}=0\;\;\mbox{in
$Q$},$ $\displaystyle\frac{1+h_{q}^{2}}{2h_{p}^{2}}+h=R\;\;\mbox{for $p=1$},$
$\displaystyle h=0\;\;\mbox{for $p=0$}.$ (1.26)

Here

$Q=\\{(q,p)\,:\,q\in\mathbb{R}\,,\;\;p\in(0,1)\\}.$

The uniform stream solution corresponding to the solution $U$ of (1.1) is

$H(p)=\int_{0}^{p}\frac{d\tau}{\sqrt{\theta^{2}-2\Omega(\tau)}},\;\;\theta=U^{\prime}(0)=H_{p}^{-1}(0).$
(1.27)

One can check that

$H_{pp}-H_{p}^{3}\omega(p)=0$ (1.28)

or equivalently

$\Big{(}\frac{1}{2H_{p}^{2}}\Big{)}_{p}+\omega(p)=0.$ (1.29)

Moreover it satisfies the boundary conditions

$\frac{1}{2H_{p}^{2}(1)}+H(1)=R,\;\;H(0)=0.$ (1.30)

The Froude number in new variables can be written as

$\frac{1}{F^{2}}=\int_{0}^{1}H_{p}^{3}dp.$

Furthermore, the Frechet derivative (the linear approximation of the functions
${\mathcal{F}}$ and ${\mathcal{G}}$ at a solution $h$) is the following

$Aw=A(h)w=\Big{(}\frac{h_{q}w_{q}}{h_{p}^{2}}-\frac{(1+h_{q}^{2})w_{p}}{h_{p}^{3}}\Big{)}_{p}-\Big{(}\frac{w_{q}}{h_{p}}-\frac{h_{q}w_{p}}{h_{p}^{2}}\Big{)}_{q}$
(1.31)

and

${\mathcal{N}}w={\mathcal{N}}(h)w=(Nw-w)|_{p=1},$ (1.32)

where

$Nw=N(h)w=\Big{(}-\frac{h_{q}w_{q}}{h_{p}^{2}}+\frac{(1+h_{q}^{2})w_{p}}{h_{p}^{3}}\Big{)}\Big{|}_{p=1}.$
(1.33)

The eigenvalue problem for the Frechet derivative, which is important for the
analysis of bifurcations of the problem (2.18), is the following

$\displaystyle A(h)w=\mu w\;\;\mbox{in $Q$},$
$\displaystyle{\mathcal{N}}(h)w=0\;\;\mbox{for $p=1$},$ $\displaystyle
w=0\;\;\mbox{for $p=0$}.$ (1.34)

Then according to [23] there exists a branch of even solitery wave solutions
to (1.2)

$h=h(q,p;t):(0,\infty)\rightarrow
C^{2,\gamma}_{0,e}(\overline{Q}),\;\;R=R(t):(0,\infty)\rightarrow(R_{c},\infty),$
(1.35)

which has a real analytic reparametrization locally around each $t>0$.

An equivalent formulation of Theorem 1.8 in the variables $q$ and $p$ is given
in the following

######  Theorem 1.11.

Let condition (1) hold. There exists at least one and at most $m$ continuous
curves

$(\widehat{h}(s),t(s))\in
C^{2,\alpha}_{0,e}(Q)\times(0,\infty),\;\;\;s\in[0,\infty),$

bifurcating from $(h(t_{*}),R(t_{*})$. They admits an analytic re-
parameterisation near each point, $t(0)=t_{*}$ and
$(\widehat{h}(0),t(0))=(h(t_{*}),t_{*})$. Moreover each curve is one of the
following:

(i) $t(s)=t_{*}$ for all $s\geq 0$;

(ii) $\mu_{1}((t(s))=0$ for all $s\geq 0$;

(iii) $t^{\prime}(s)\neq 0$ and the Frechet derivative is invertible for $s>0$
except a set of discrete points with the only possible accumulation point at
$\infty$.

Furthermore the bifurcation curve satisfies one of the properties (a)-(d) in
Theorem 1.8.

##  2\. Abstract bifurcation analysis

Here we present some known results about bifurcations in Banach spaces, which
are taken from [5], [6] and [1].

Let ${\bf X}$, ${\bf Y}$ and ${\bf Z}$ be real Banach spaces, ${\bf
X}\subset{\bf Z}$ and ${\bf X}$ continuously embedded into ${\bf Z}$. Consider
a function

${\mathcal{F}}=(F,G)=(F(x,\lambda),G(x,\lambda))\;\;\mbox{
mapping}\;\;{\mathcal{U}}\rightarrow{\bf Z}\times{\bf Y},$

where ${\mathcal{U}}$ is an open set in ${\bf X}\times\mathbb{R}$ and $F$ and
$G$ are analytic functions in ${\mathcal{U}}$. We assume that
$(0,0)\in{\mathcal{U}}$ and

${\mathcal{F}}(0,\lambda)=0\;\;\mbox{for all $(0,\lambda)\in{\mathcal{U}}$}.$
(2.1)

We are interested in solutions $(x,\lambda)$ to the problem222This formulation
is different from that in [5] and [6] because of the additional Banach space
$Y$. The advantage of this setting is that it allows applications to boundary
value problems and the forthcoming analysis does not undergo any changes.

${\mathcal{F}}(x,\lambda)=0$ (2.2)

with $x\neq 0$.

Let $\delta>0$ and

$B_{\delta}=\\{x\in{\bf X}\,:\,||x||_{\bf
X}<\delta\\}\;\;\;\mbox{and}\;\;I_{\delta}=\\{\lambda\in\mathbb{R}\,:\,|\lambda|<\delta\\}.$

For sufficiently small $\delta$ the set $B_{\delta}\times I_{\delta}$ belongs
to ${\mathcal{U}}$ and the analytic functions $F$ and $G$ can be represented
as

$F(x,\lambda)=\sum_{j\geq 0,k\geq
0}\lambda^{j}F_{jk}(x)\;\;\mbox{and}\;\;G(x,\lambda)=\sum_{j\geq 0,k\geq
0}^{\infty}\lambda^{j}G_{jk}(x),$ (2.3)

where $F_{jk}:{\bf X}^{k}\rightarrow{\bf Z}$ and $G_{jk}:{\bf
X}^{k}\rightarrow{\bf Y}$ are $k$–linear, symmetric and continuous operator,
and the above series are convergent in $B_{\delta}\times I_{\delta}$. Due to
(2.1), $F_{j0}=0$ and $G_{j0}=0$ for all $j\geq 0$. We represent $F$ and $G$
as

${\mathcal{F}}(x,\lambda)=A(\lambda)x+\widehat{{\mathcal{F}}}(x,\lambda),$
(2.4)

where

$A(\lambda)=\sum_{j=0}^{\infty}\lambda^{j}A_{j}\,:\,X\to Z\times
Y,\;\;A_{j}x=(F_{j1}(x),G_{j1}(x))$ (2.5)

and

$\widehat{{\mathcal{F}}}(x,\lambda)=(\widehat{F}(x,\lambda),\widehat{G}(x,\lambda)),\;\;\widehat{F}(x,\lambda)=\sum_{j\geq
0,k\geq 2}\lambda^{j}F_{jk}(x),\;\;\widehat{G}(x,\lambda)=\sum_{j\geq 0,k\geq
2}^{\infty}\lambda^{j}G_{jk}(x).$

We assume that the Frechet derivatives

$(D_{x}F(x,\lambda)v,D_{x}G(x,\lambda)v)\,:\,{\bf X}\rightarrow{\bf
Z}\times{\bf Y}$

is a Fredholm operator of index zero for all $(x,\lambda)\in{\mathcal{U}}$
satisfying (2.2).

The Frechet derivative at $x=0$ is given by

$(D_{x}F(0,\lambda)v,D_{x}G(0,\lambda)v)=A(\lambda)v.$

We will use the following spectral problem in order to study the bifurcation
of (2.2) at $(x,\lambda)=(0,0)$:

$A(\lambda)v=\mu(\lambda)(v,0).$ (2.6)

We assume also that for $\lambda=0$ the problem (2.6) has the eigenvalue
$\mu=0$, which is algebraically simple. Therefore there is a function
$\mu(\lambda)$ of eigenvalues of (2.6) consisting of simple eigenvalues of
(2.6). This function is analytic in a neighborhood of $\lambda=0$ and there
are no other eigenvalues in this neighborhood. There is also an analytic
function $v(\lambda)$ consisting of eigenfunctions of the (2.6) corresponding
to the eigenvalue $\mu(\lambda)$. We write the function $\mu$ as

$\mu(\lambda)=\sum_{j=m}^{\infty}\mu_{j}\lambda^{j}.$ (2.7)

Our main assumption is

$m\;\;\mbox{is odd and}\;\;\mu_{m}\neq 0.$ (2.8)

It is reasonable to study small bifurcation curves of (2.2) first and then to
discuss their continuation to curves containing large bifurcating solutions.

###  2.1. Small bifurcation curves

Since $v(\lambda)$ is analytic function, it can be represented as

$v(\lambda)=\sum_{j=0}^{\infty}v_{j}\lambda^{j}.$

Furthermore, the spectral problem
$A(\lambda)^{*}(\tilde{z}.\tilde{y})=\tilde{\mu}\tilde{z}$ has the eigenvalue
$\tilde{\mu}=\mu(\lambda)$ which is also simple and we denote the
corresponding eigenfunction by
$\tilde{w}(\lambda)=(\tilde{z}(\lambda),\tilde{y}(\lambda))\in{\bf
Z}^{*}\times{\bf Y}^{*}$. They can be chosen such that the functions
$\tilde{z}(\lambda)$ and $\tilde{y}(\lambda)$ are analytic in $\lambda$ and,
due to algebraic simplicity of the eigenvalue $\mu(\lambda)$, satisfy

$\langle v(\lambda),\tilde{z}(\lambda)\rangle\neq 0\;\;\mbox{for small
$\lambda$}.$

The coefficients $v_{j}$ can be chosen to satisfy

$\langle v_{j},\tilde{z}(0)\rangle=0\;\;\mbox{for $j=1,\ldots$}.$

We put

$\widehat{v}(\lambda)=v(\lambda)\;\;\mbox{and}\;\;\widehat{w}(\lambda)=(\widehat{z},\widehat{y})=\frac{\tilde{w}(\lambda)}{\langle
v(\lambda),\tilde{z}(\lambda)\rangle}.$

Both these functions are still analytic and satisfy

$\langle\widehat{v}(\lambda),\widehat{z}(\lambda)\rangle=1\;\;\mbox{for small
$\lambda$}.$

In order to reduce the system (2.2) to an one dimensional equation one can use
the Lyapunov-Schmidt method. We define the following projections

$P_{\lambda}x=\langle
x,\widehat{z}(\lambda)\rangle\widehat{v}(\lambda),\;\;\widehat{P}_{\lambda}(z,y)=\langle(x,y),\widehat{w}(\lambda)\rangle(\widehat{v}(\lambda),0).$

Here and in what follows we identify the element $w\in{\bf X}$ with the
element $(w,0)\in{\bf Z}\times{\bf Y}$ in the definition of
$\widehat{P}_{\lambda}$. Clearly $P_{\lambda}x=\widehat{P}_{\lambda}(x,0)$.
Now we can write (2.2) as

$\displaystyle\widehat{P}_{\lambda}{\mathcal{F}}(v+w,\lambda)=0,\;\;x=P_{\lambda}x+(I-P_{\lambda})x=v+w,$
$\displaystyle(I-\widehat{P}_{\lambda}){\mathcal{F}}(v+w,\lambda)=0.$ (2.9)

Taking $v=t\widehat{v}(\lambda)$ and using representation (2.4), we get

$t\mu(\lambda)+\langle\widehat{{\mathcal{F}}}(t\widehat{v}+w,\lambda),\widehat{w}(\lambda)\rangle=0$
(2.10)

and

$(I-\widehat{P}_{\lambda})(A(\lambda)w+\widehat{{\mathcal{F}}}(t\widehat{v}+w,\lambda))=0.$
(2.11)

We write the second equation in the form

$(I-\widehat{P}_{\lambda})\big{(}A(\lambda)w+\widehat{{\mathcal{F}}}_{1}(t\widehat{v}+w,\lambda)+\widehat{{\mathcal{F}}}(t\widehat{v},\lambda)\big{)}=0,$
(2.12)

where

$\widehat{{\mathcal{F}}}_{1}(t\widehat{v},w,\lambda))=\widehat{{\mathcal{F}}}(t\widehat{v}+w,\lambda)-\widehat{{\mathcal{F}}}(t\widehat{v},\lambda).$
(2.13)

Since the operator

$A(\lambda):(I-P_{\lambda}){\bf X}\rightarrow(I-\widehat{P}_{\lambda})({\bf
Z}\times{\bf Y})$

is invertible by using the implicit function theorem one can find
$w=w(t,\lambda)$ which is analytically depends on $t$ and $\lambda$ and one
can verify that

$w(t,\lambda)=t^{2}\overline{w}(t,\lambda),$

where the function $\overline{w}(t,\lambda)$ is also analytic in $t$ and
$\lambda$. Now the equation (2.109 can be written as

$\mu(\lambda)+t^{-1}\langle\widehat{{\mathcal{F}}}(t\widehat{v}+t^{2}\overline{w}(t,\lambda),\lambda),\widehat{w}(\lambda)\rangle=0$
(2.14)

Thus we arrive at the equation

$\mu(\lambda)+t{\mathcal{T}}(t,\lambda)=0,$ (2.15)

where ${\mathcal{T}}$ is an analytic function.

The main local bifurcation result for the equation (2.15), and hence for the
equation (2.2), is presented in the next theorem, which proof can be found in
[5], Theorem I.16.4.

######  Theorem 2.1.

Let $\mu(\lambda)$ satisfy (2.7) and (2.8). Then bifurcating solutions to
(2.2) consist of two groups of the same number of curves bifurcating at
$(x,\lambda)=(0,0)$, and this number is at least one and at most $m$. One
group consists of

$\displaystyle
x(s)=s\widehat{v}(\lambda)+\psi(s\widehat{v}(\lambda),\lambda(s))=s\widehat{v}(\lambda)+o(|s|),$
$\displaystyle\lambda(s)=s^{\gamma}\sum_{k=0}^{\infty}a_{k}s^{k/p\gamma_{2}}\;\;\mbox{for
$0\leq s<\delta$}$ (2.16)

and in another group the formulae for $x$ is the same but

$\lambda(s)=(-s)^{\tilde{\gamma}}\sum_{k=0}^{\infty}\tilde{a}_{k}(-s)^{k/\tilde{p}\tilde{\gamma}_{2}}\;\;\mbox{for
$-\delta<s\leq 0$}.$ (2.17)

Here $\gamma=\gamma_{1}/\gamma_{2}$,
$\tilde{\gamma}=\tilde{\gamma}_{1}/\tilde{\gamma}_{2}$ and $\gamma_{1}$,
$\gamma_{2}$, $\tilde{\gamma}_{1}$, $\tilde{\gamma}_{2}$ and $p$, $\tilde{p}$
are positive integers. The function $\psi$ is analytic near the point $(0,0)$
and satisfies

$\langle\psi,\widehat{z}_{0}^{\prime}\rangle=0.$

We include here possible vertical bifurcation $\lambda(s)\equiv 0$. If
$\lambda(s)$ is not the vertical bifurcation then $a_{0}\neq 0$,
$\tilde{a}_{0}\neq 0$.

It follows from [5], [6] the next

######  Proposition 2.2.

Let $M$ be the number of branches in each group of branches in the above
theorem. We numerate the branches (2.1) in increasing order if $M>1$, namely
$\lambda_{1}(s)<\cdots<\lambda_{M}(s)$ for $s\in(0,\delta)$. Then for each
branch $(\lambda_{j},x_{j})$ there exist a unique branch
$(\lambda^{(-)}_{j},x^{(-)}_{j})$ from the second group (2.17) in the above
theorem such that the function

$g_{j}(s)=(\lambda_{j}(s),x_{j}(s))\;\;\mbox{for $s\geq 0$
and}\;\;g_{j}(s)=(\lambda^{(-)}_{j}(s),x^{(-)}_{j}(s))\;\;\mbox{for $s<0$}$
(2.18)

admits an analytic injective reparametrization. If $\lambda_{j}$ has the form
(2.1) then the parametrization can be chosen as $t=s^{1/p\gamma_{2}}$.

Moreover the following alternatives are valid for the above branches

(i) If $\lambda_{j}$ is not identically zero then $\lambda^{\prime}(s)\neq 0$
for small $s\neq 0$ and

$D_{x}(F,G)(x_{j}(s),\lambda_{j}(s))$ is invertible for small $s\neq 0$,

(ii) $\lambda^{\prime}(s)\neq 0$ for small $s\neq 0$ and $\mu_{1}(s)=0$ for
all small $s$.

(iii) If $\lambda_{j}\equiv 0$ then the corresponding solution is given by

$\lambda=0\;\;\;x(s)=s\widehat{v}_{0}+\psi(s\tilde{v}_{0},0).$

######  Remark 2.3.

The conditions (2.7) and (2.8) can be relaxed as follows. We can assume that
the function in the left hand side of the bifurcation equation (2.15) is not
identically zero. In this case Theorem 2.1 and Proposition 2.2 are still true
except that the number of branches can be zero, but if we have a branch for
small $s>0$ then there is also a branch for small negative $s$ and the branch
defined now for small $|s|$ admits an analytic re-parametrization, see [5],
Sect. 1.15, 1.16.

###  2.2. Global bifurcation, continuation of local bifurcation curve.

We denote by ${\mathcal{B}}^{+}$ the branch of small bifurcating solutions
constructed in the previous section, see (2.18). We denote one of them by
$(\kappa(s),\Lambda(s))$, which is given for $0<s<\epsilon$. By Proposition
2.2 there exists at least one such curve and it can be extended in a
neighborhood of $0$ and the extended curve admits an analytic
reparametrization. But in what follows we will use only the branch defined for
$s\in(0,\epsilon)$.

We assume that

(A) ${\mathcal{F}}(0,\lambda)=0$ for all $(0,\lambda)\in{\mathcal{U}}$.

(B) $D_{x}{\mathcal{F}}[(x,\lambda)]$ is a Fredholm operator of index zero
when ${\mathcal{F}}(x,\lambda)=0$, $(x,\lambda)\in{\mathcal{U}}$.

(C) There is an interval $I\in\mathbb{R}$, in our case $I=(0,\epsilon)$ but
having in mind a more general application of the next extension theorem we
give a more general presentation, and an analytic function
$(x,\lambda)=(\kappa(s),\Lambda(s))$, $s\in I$,
$(\kappa(s),\Lambda(s))\in{\mathcal{U}}$, such that
${\mathcal{F}}(\kappa(s),\Lambda(s)))=0$ and $\Lambda^{\prime}(s)\neq 0$ for
$\lambda\in I$. Furthermore the operator
$D_{x}{\mathcal{F}}[(\kappa(s),\Lambda(s))]$ is invertible for $s\in I$.

Introduce

$\displaystyle{\mathcal{S}}=\\{(x,\lambda)\in{\mathcal{U}}\,:\,{\mathcal{F}}(x,\lambda)=0\\},$
$\displaystyle{\mathcal{B}}^{+}=\\{(\kappa(s),\Lambda(s))\,:\,s\in I\\}.$
(2.19)

######  Theorem 2.4.

Suppose (A)-(C) hold and all bounded closed sets of ${\mathcal{S}}$ are
compact in ${\bf X}\times\mathbb{R}$. Then there exists a continuous curve-
${\mathcal{B}}$, which extends ${\mathcal{B}}^{+}$ as follows

(a)
${\mathcal{B}}=\\{(\kappa(s),\Lambda(s))\,:\,s\in(0,\infty)\\}\subset{\mathcal{U}}$,
where $(\kappa,\Lambda):(0,\infty)\rightarrow{\bf X}\times\mathbb{R}$ is
continuous.

(b) ${\mathcal{B}}^{+}\subset{\mathcal{B}}\subset{\mathcal{S}}$.

(c) The set $\\{s>0\,:\,N(\partial_{x}F[(\kappa(s),\Lambda(s))]\\}\neq{O}$ has
no accumulation points.

(d) At each point, ${\mathcal{B}}$ has a local analytic re-parameterization in
the following sense. In the right neighbourhood of $s=0$, ${\mathcal{B}}$ and
${\mathcal{B}}^{+}$ coincide. For each $s^{*}\in(0,\infty)$ there exists
$\rho^{*}:(-1,1)\to\mathbb{R}$ which is continuous, injective, and

$\rho^{*}(0)=s^{*},\;\;t\rightarrow(\kappa(\rho^{*}(t),\Lambda(\rho^{*}(t)),\;t\in(-1,1),\;\;\mbox{is
analytic}.$

Furthermore $\Lambda$ is injective on a right neighbourhood of $0$ and for
$s^{*}>0$ there exist $\epsilon^{*}$ such that $\Lambda$ is injective on
$(s^{*},s^{*}+\epsilon^{*})$ and $(s^{*}-\epsilon^{*},s^{*})$.

(e) One of the following occurs.

(i) $||(\kappa(s),\Lambda(s))||\rightarrow\infty$ as $s\rightarrow\infty$.

(ii) $(\Lambda(s),\kappa(s))$ approaches the boundary of $U$ as $s$ tends to
$\infty$.

(iii) ${\mathcal{B}}$ is a close loop, i.e. for some $T>0$

${\mathcal{B}}=\\{(\kappa(s),\Lambda(s))\,:\,0<s\leq T\\}.$

We can assume here that $T$ is minimal of such periods.

(f) If, for some $s_{1}\neq s_{2}$

$(\kappa(s_{1}),\Lambda(s_{1}))=(\kappa(s_{2}),\Lambda(s_{2}))\;\;\mbox{where}\;\;N(\partial_{x}F[(\kappa(s_{1}),\Lambda(s_{1}))]=0,$

then (e),((iii) occurs and $|s_{1}-s_{2}|$ is an integer multiple of $T$, In
particular $(\kappa,\Lambda)\,:\,\mathbb{R}\rightarrow{\mathcal{S}}$ is
locally injective.

######  Remark 2.5.

The compactness property can be replaced by the following property of
${\mathcal{U}}$.

Let ${\mathcal{K}}_{{\bf n}}$, ${\bf n}\in\mathbb{R}^{N}$, be a sequence of
compact sets in ${\mathcal{U}}$ such that if ${\bf n}\leq{\bf m}$ then
${\mathcal{K}}_{{\bf n}}\subset{\mathcal{K}}_{{\bf m}}$. If
${\mathcal{U}}=\bigcup{\mathcal{K}}_{\bf n}$ then the compactness condition in
the theorem can be removed and the condition (e,ii) must be replaced by (ii’)
the curve $(\Lambda(s),\kappa(s))$, $s\in[0,\infty)$ does not belong to
${\mathcal{K}}_{{\bf n}}$ for any ${\bf n}$.

##  3\. Application

###  3.1. Bifurcation equation for solitary waves

Let $C^{k,\alpha}(Q)$ and $C^{k,\alpha}(\mathbb{R})$ be usual Hölder spaces in
$\overline{Q}$ and $\mathbb{R}$ respectively and let $C^{k,\alpha}_{0,e}(Q)$
be the subspace of $C^{k,\alpha}(Q)$ consisting of even functions $h$
vanishing for $p=0$. The space $C^{k,\alpha}_{e}(\mathbb{R})$ consists of even
functions in $C^{k,\alpha}(\mathbb{R})$. We put

${\bf X}=C^{2,\alpha}_{0,e}(Q),\;\;{\bf Z}=C^{0,\alpha}_{e}(Q),\;\;{\bf
Y}=C^{2,\alpha}(\mathbb{R}).$

and

${\mathcal{U}}_{\delta}=\\{(w,t)\,:\,w\in{\bf X},\,0\leq
t\leq\delta^{-1}\,:\,\delta<(h(t)+w)_{p}<\delta^{-1},(h(t)+w)_{q}<\delta^{-1}\\}.$

where $(h(t),R(t))$ is the branch (1.35). Let also

${\mathcal{U}}=\bigcup_{\delta}{\mathcal{U}}_{\delta}$ (3.1)

and

${\mathcal{S}}=\\{(w,t)\in{\bf X}\times(0,\infty)\,:\,\mbox{$(h(t)+w,R(t))$--a
solitary wave solution to (\ref{J4a})}\\}.$ (3.2)

Assuming that the property (1) is satisfied, we put $\lambda=t-t_{*}$, in
order to keep the same notation as in the previous sections, and introduce the
functions

$F(w,\lambda)=\tilde{F}(h(t)+w,R(t)),\;\;\;\tilde{F}(\tilde{h},R)=\Big{(}\frac{1+\tilde{h}_{q}^{2}}{2\tilde{h}_{p}^{2}}+\Omega(p)\Big{)}_{p}-\Big{(}\frac{\tilde{h}_{q}}{\tilde{h}_{p}}\Big{)}_{q}$

and

$G(w,\lambda)=\tilde{G}(h(t)+w,R(t)),\;\;\;\tilde{G}(\tilde{h},R)=\Big{(}\frac{1+\tilde{h}_{q}^{2}}{2\tilde{h}_{p}^{2}}+\tilde{h}\Big{)}\Big{|}_{p=1}-R.$

Here $(h(t),R(t))$ is the branch (1.35). Then

${\mathcal{F}}=(F,G)\,:\,{\mathcal{U}}\rightarrow{\bf Z}\times{\bf Y}.$ (3.3)

Using this definition, we see that

$F(0,\lambda)=0\;\;\;\mbox{and}\;\;\;G(0,\lambda)=0.$ (3.4)

Our aim is to study bifurcations of the problem

${\mathcal{F}}(w,\lambda)=(F(w,\lambda),G(w,\lambda))=0$ (3.5)

at $\lambda=0$ and $w=0$.

The Frechet derivative of the pair $(F,G)$ is the following

$Av=A(w,\lambda)v=\Big{(}\frac{\lambda^{2}(h+w)_{q}v_{q}}{(h+w)_{p}^{2}}-\frac{(1+(h+w)_{q}^{2})v_{p}}{(h+w)_{p}^{3}}\Big{)}_{p}-\Big{(}\frac{v_{q}}{(h+w)_{p}}-\frac{(h+w)_{q}v_{p}}{(h+w)_{p}^{2}}\Big{)}_{q}$
(3.6)

and

${\mathcal{N}}v={\mathcal{N}}(w,\lambda)v=(Nv-v)|_{p=1},$ (3.7)

where

$Nv=N(w,\lambda)v=\Big{(}-\frac{(h+w)_{q}v_{q}}{(h+w)_{p}^{2}}+\frac{(1+(h+w)_{q}^{2})v_{p}}{(h+w)_{p}^{3}}\Big{)}\Big{|}_{p=1}.$
(3.8)

The eigenvalue problem for the Frechet derivative, which is important for the
analysis of bifurcations of the problem (3.5), is the following

$\displaystyle Av=\mu v\;\;\mbox{in $Q$},$
$\displaystyle{\mathcal{N}}v=0\;\;\mbox{for $p=1$},$ $\displaystyle
v=0\;\;\mbox{for $p=0$}.$ (3.9)

Here $Q$ is the same strip as in Sect. 1.2.

By (1) $\mu_{1}(t_{*})=0$ and the eigenvalue $\mu_{1}(t)$ admits the
representation 1.18. When $\mu_{1}$ is simple we denote by
$\widehat{v}=\widehat{v}(t)$ the corresponding eigenfunction, which is
normalized by

$\int_{Q}\widehat{v}^{2}dqdp=1.$

The the projection operator, introduced in Sect. 2.1, can be written as

$Pw=\int_{Q}w\widehat{v}dqdp\,\widehat{v},\;\;\widehat{P}(x,y)=\Big{(}\int_{Q}x\widehat{v}dqdp+\int_{\mathbb{R}}y\widehat{v}(q,0)dq\Big{)}(\widehat{v},0)$
(3.10)

Formulation of the bifurcation problem in the sets ${\mathcal{U}}$ and
${\mathcal{S}}$ is convenient for application of the abstract bifurcation
analysis developed in the previous section.

Let us introduce

$\widehat{\mathcal{U}}_{\delta}=\\{(\widehat{h},R)\,:\,\widehat{h}\in{\bf
X}\,:\,\delta\leq\widehat{h}_{p}\leq\delta^{-1},|h_{q}|\leq\delta^{-1},R\geq
R_{c}+\delta\\},$ (3.11)
$\widehat{\mathcal{U}}=\bigcup_{\delta}\widehat{{\mathcal{U}}}_{\delta}$

and

$\widehat{{\mathcal{S}}}=\\{(\widehat{h},R)\in{\bf
X}\times(R_{c},\infty)\,:\,\mbox{$(\widehat{h},R)$--a solitary wave solution
to (\ref{J4a})}\\}.$ (3.12)

The above sets are convenient for study of global bifurcation branches. If
$(w,t)\in{\mathcal{U}}$ then $(\widehat{h}(t),R(t))\in\widehat{\mathcal{U}}$,
where $\widehat{h}(t)=h(t)+w$.

###  3.2. Lemmas

Introduce the constant

$\omega_{1}=\max_{\tau\in[0,1]}|\omega(\tau)|+\max_{\tau\in[0,1]}|\omega^{\prime}(\tau)|.$

The proof of the following lemma can be found in [12], Proposition 2.

######  Lemma 3.1.

Let

$R_{c}<R_{1}<\infty,\;\;\Psi_{Y}\geq 0\;\;\mbox{in $D_{\xi}$
and}\;\;|\xi^{\prime}(X)|\leq M\;\;\mbox{for $X\in\mathbb{R}$}.$ (3.13)

Then there exists $C=C(R_{1},M,\omega_{1})$ (this constant does not depend on
other parameters) such that if $(\Psi,\xi)\in C^{1}(\overline{D_{\xi}})\times
W^{1,\infty}(\mathbb{R})$ solves the problem (1) with $R\in(0,R_{1}]$ then

$|\nabla\Psi(X,Y)|\leq C.$ (3.14)

The next assertion follows from the above lemma and the local estimates
presented and proved in Sect. 2, Chapter 10, [18]. We use the notation
$B_{\rho}(X,Y)$ for the ball of radius $\rho$ and with the center at $(X,Y)$.

######  Lemma 3.2.

Let the inequalities (3.13) be valid and let $(\Psi,\xi)\in
C^{2}(\overline{D_{\xi}})\times C^{2,\alpha}(\mathbb{R})$ be a solution to
(1). If

$\Psi_{Y}\geq\delta>0\;\mbox{in $B_{\rho}(X,Y)$}$ (3.15)

then there exists constants $C_{1}$ and $\alpha_{1}\in(0,1)$ depending also on
$M,\omega_{0},\delta,\rho$ such that

$||\Psi||_{C^{3,\alpha_{1}}(D_{\xi}\bigcap B_{\rho/2}((X,Y))}\leq
C_{1},\;\;\||\xi||_{C^{3,\alpha_{1}}(S_{\xi}\bigcap B_{\rho/2}((X,Y))}\leq
C_{1}.$

Next lemma is a version of the previous one in the variables $(q,p)$.

######  Lemma 3.3.

Let the inequalities (3.13) be valid and let be a solution $h\in
C^{2}(\overline{Q})$ to (1.2) subject to $\delta\leq H_{p}\leq\delta^{-1}$ in
$Q\bigcap B_{\rho}(q,p)$. Then there exist constants $C_{1}$ and $\alpha_{1}$
depending on $\delta$, $M$, $\omega$ and $R_{1}$, such that $h\in
C^{3,\alpha}(Q\bigcap B_{\rho}(q,p))$ and

$||h||_{C^{3,\alpha}(Q\bigcap B_{\rho}(q,p))}\leq C_{1}.$

######  Lemma 3.4.

Let $R>R_{c}$ and $(\Psi,\xi)\in C^{2,\alpha}(D_{\xi})\times
C^{2,\alpha}(\mathbb{R})$ satisfy .Then the following properties hold:

(i) If

$|\xi^{\prime}(X)|\leq M\;\;\mbox{for $X\in\mathbb{R}$}$ (3.16)

then

$\Psi_{Y}^{2}(X,\xi(X))\geq\frac{2a}{1+M^{2}},\;\;\mbox{where}\;\;a:=\sup_{X}(R-\xi(X)).$
(3.17)

(ii) If $R_{0}=\infty$ then

$\Psi_{Y}(X,0)\geq\delta_{1}:=\sqrt{s^{2}-s_{0}^{2}},$ (3.18)

where $s\in(s_{0},s_{c})$ is the root of the equation ${\mathcal{R}}(s)=R$.

(iii) Let $R_{0}<\infty$, $\Omega(1)>0$ and $\Psi(X,\xi(X))\geq\delta>0$. Then

$\Psi_{Y}(X,0)>s_{0}\;\;\mbox{for $X\in\mathbb{R}$}.$ (3.19)

###### Proof.

The items (i) and (ii) is proved in Sect. 3 in [9]. The proof of (iii) is
given in Sect. 3.4.

∎

The following property is proved in Proposition 1.3, [9].

######  Lemma 3.5.

Let the property (i) and one of the properties (ii) or (iii) be valid. Then
there exists a constant $c_{0}>0$ depending on $M$, $\delta$, $R$ such that

$\Psi_{Y}(X,Y)\geq c_{0}\;\;\mbox{for all $(X,Y)\in D_{\xi}$}.$ (3.20)

###### Proof.

By Proposition 3.1[14] there exists constants $\alpha\in(0,1)$ and $C>0$
depending on $M$, $\delta$ and $R$ such that

$||\Psi||_{C^{2,\alpha}(D_{\xi})}\leq C.$ (3.21)

Now the estimate (3.20) follows from (3.17), (3.18) and (3.21) by using the
Harnack principle. ∎

######  Lemma 3.6.

([9]) There exists a constant $\widehat{R}$ depending on $\omega$ such that
$R\leq\widehat{R}$ if the problem (1) has a solitary wave solution.

The following property is a consequence of Theorem 1, [12].

######  Proposition 3.7.

Let $(\Psi,\xi,R)$ be a solitary wave solution to (1) and let

$\sup|\xi^{\prime}(X)|\leq
M_{1}\;\;\mbox{and}\;\;||\Psi(X,Y)||_{C^{2,\alpha}(S_{\xi})}\leq M_{2}.$
(3.22)

Then there exists a positive $\epsilon$ depending on $M_{1}$ and $M_{2}$ such
that if $R\in(R_{c},R_{c}+\varepsilon)$ then $\Psi=\Psi(t)$, $\xi=\xi(t)$ and
$R=R(t)$ for a certain small $t$.

######  Proposition 3.8.

Let $\widehat{\mathcal{U}}_{\delta}$ and $\widehat{\mathcal{U}}$ be introduced
by (3.11) and (3.12). The set
$\overline{\widehat{{\mathcal{U}}}_{\delta}\bigcap\widehat{{\mathcal{S}}}}$ is
compact in $C^{2,\alpha}_{0}(Q)\times[R_{c},\infty)$.

###### Proof.

From the definition (3.11) it follows that

$\delta\leq\Psi_{Y}\leq\delta^{-1},\;\;|\Psi_{X}|\leq\delta^{-1}\;\;\mbox{and}\;\;|\xi^{\prime}(X)|\leq\delta^{-2}.$

By Lemma 3.6, $R\leq\widehat{R}(\omega)$. Therefore the inequalities (3.13)
and (3.15) hold with constants depending on $\delta$ and $\omega$. By Lemma
3.3 the set
$\overline{\widehat{{\mathcal{U}}}_{\delta}\bigcap\widehat{{\mathcal{S}}}}$ is
bounded.

Let $(h,R)$ be a solitary wave solution to (1.2). We represent it as

$h(q,p)=H(p)+v(q,p),$ (3.23)

where $h(q,p)\rightarrow H(p)$ as $p\to\pm\infty$ and $H$ is given by (1.27)
with ${\mathcal{R}}(\theta)=R$. According to Theorem 3.10, [23] the set of $v$
in representation (3.23) is compact. Since the set of $H$ in the same
representation is also compact we arrive at the required assertion.

∎

###  3.3. Proof of Theorems 1.11 and 1.8

Theorem 1.11 is a reformulation of Theorem 1.8 after application of partial
hodograph transform. Theorem 1.11 is more convenient for application of the
abstract bifurcation analysis and Theorem 1.8 is more convenient for
verification properties of global branches of bifurcating solutions (see
properties a)-d) there). Therefore we will prove both theorems simultaneously.

One can verify that the operator (3.3) satisfies the assumptions of Sect. 2.1
and 2.2 concerning Fredholm properties. The assumptions (2.7) and (2.8) follow
from (1). Furthermore, according to Theorem 2.1 there exist a branch

$(w,\lambda)=(\kappa(s),\Lambda(s)):(0,\epsilon)\rightarrow{\mathcal{U}},$

whose asymptotics is describe in that theorem. We denote this branch by
${\mathcal{B}}^{+}$. To obtain a global branch we intend to use Theorem 2.4.
The compactness required in that theorem, complemented by Remark 2.5, follows
from Proposition 3.8. It remains to verify the property (C) in the beginning
of Sect. 2.2 in order to apply Theorem 2.4. According to Proposition 2.2 we
consider three cases for the behavior of the branch ${\mathcal{B}}^{+}$. We
put

$\widehat{h}(q,p;s)=h(q,p;t(s))+w(q,p;t_{*}+t(s)).$ (3.24)

1\. Assume that the curve ${\mathcal{B}}^{+}$ satisfies the option (i) of
Proposition 2.2, which corresponds to (iii) in Theorem 2.4. Then the non-
degeneracy condition (C) for the branch ${\mathcal{B}}^{+}$ is satisfied and
we can apply Theorem 2.4 and get the existence of a curve of solitary
solutions ${\mathcal{B}}$, continuously parameterised by $s\in[0,\infty)$,
which extends ${\mathcal{B}}^{+}$. We denote by ${\mathcal{J}}^{+}$ and
${\mathcal{J}}$ the curves corresponding to ${\mathcal{B}}^{+}$ and
${\mathcal{B}}$ respectively, connected by (3.24). Both of them belongs to
$\widehat{\mathcal{U}}\bigcap\widehat{\mathcal{S}}$. According to Theorem 2.4
the assertions (a)-(d) imply all properties in Theorem 1.11 except of (a)–(d)
in Theorem 1.1. Let us turn to proving these properties. Assume that non of
the properties (a)-(d) in Theorem 1.8 holds. Then

$|\xi^{\prime}(X)|\leq M,\;\;\Psi_{Y}\geq\delta>0,\;\;t(s)\leq
T\;\;\mbox{along the curve ${\mathcal{J}}$}$

and the curve is not periodic. By Lemma 3.1 the inequality (3.14) is valid and
hence Lemma 3.2 implies that

$||\Psi||_{C^{3,\alpha_{1}}(D_{\xi})}\leq
C_{1},\;\;\||\xi||_{C^{3,\alpha_{1}}(S_{\xi})}\leq C_{1},$

where $C_{1}$ depends on the constants $M$, $\delta$ and $T$. By Proposition
3.7 there exist $\epsilon>0$ such that $R>R_{c}+\epsilon$. Otherwise the curve
must coincide with the curve (1.7) on a certain interval and hence everywhere,
which contradict to the construction of the bifurcating curve. All these
inequalities imply that the curve must belong to
$\widehat{\mathcal{U}}_{\delta}$ for certain $\delta>0$. This implies that the
closure of the curve is compact and since it is not periodic this contradicts
to the property (ii) in Theorem 2.4.

2\. Assume that the curve ${\mathcal{B}}^{+}$ satisfies the case (ii) in
Proposition 2.2 and hence the property (i) in Theorem 1.11. This means that
$\lambda\equiv 0$ or $t\equiv t_{*}$ on ${\mathcal{B}}^{+}$.

Introduce the following subspaces

$\widehat{\bf
U}=\\{(w,t)\in{\mathcal{U}}\,:\,P_{0}w=0\\}\;\;\mbox{and}\;\;\widehat{{\mathcal{Z}}}=\\{w\in{\bf
Z}\times{\bf Y}\,:\,\widehat{P}_{0}w=0=0\\},$

where the projector operators are given by (3.10) and used in the Lyapunov-
Schmidt method, see Sect. 2.1. Consider the bifurcation problem

${\mathcal{F}}_{0}(w,s):=(I-\widehat{P}_{0})((A(0)w,{\mathcal{N}}(0)w)+\widehat{\mathcal{F}}_{0}(s\widehat{v}_{0}+w,0))=0,$
(3.25)

which is the equation (2.11) with $\lambda=0$. Here ${\mathcal{F}}_{0}$ maps
${\bf U}$ into $\widehat{\mathcal{Z}}$. Then $w=\psi(s\tilde{v}_{0},0)$
satisfies this equation and the equation (2.10) which becomes

$\widehat{P}_{0}\widehat{{\mathcal{F}}}(s\widehat{v}+w,0)=0.$ (3.26)

We consider (3.25) as a bifurcation equation. The corresponding Frechet
derivative with respect to $w$ is invertible. So we can apply Theorem 2.4 to
the bifurcation problem (3.25), where

${\mathcal{B}}^{+}=\\{(w,s)\,:\,w=\psi(s\tilde{v}_{0},0),\;s\in(0,\epsilon)\\}.$

The extension branch $(w,s)$ we denote by ${\mathcal{B}}$. The relation (3.26)
is still valid for the whole curve ${\mathcal{B}}$ since this curve is
analytic and it is valid on ${\mathcal{B}}^{+}$. Thus the extension curve
satisfies both equations (3.25) and (3.26). The verification of properties
(a)-(d) in Theorem 1.11 can be done in the same way as in the previous case 1.

3\. Assume finally that the curve satisfies the case (iii) in Proposition 2.2,
i.e. $\lambda$ is not identically constant and $\mu_{1}(s)=0$ for small $s\neq
0$ and the bifurcating solution ${\mathcal{B}}^{+}$ is given by (2.1) and
(2.17). Consider the curve $(x(s),\Lambda(s))$ from Theorem 2.1. This curve is
analytic in an interval $(0,\epsilon)$ for a certain positive $\epsilon$. Let
${\mathcal{J}}^{+}$ be the curve obtained from ${\mathcal{B}}^{+}$ by (3.24).
Let us extend this curve to a global curve
${\mathcal{J}}\subset\widehat{\mathcal{S}}$ such that near every point in
${\mathcal{J}}$ it admits an analytic injective parameterisation. Due to local
analyticity of the curve, $\mu_{1}=0$ on this curve and $\lambda$ can be zero
only in a set consisting of isolated points of the curve. One of the end
points of ${\mathcal{J}}$ is $(h(t_{*}),t_{*})$. Consider the set
$\overline{\mathcal{J}}$. Assume that
$\overline{\mathcal{J}}\in{\mathcal{U}}_{\delta}$. Then this set is compact by
Proposition 3.8. By continuity the eigenvalue, $\mu_{1}$ is equal to $0$ in
all points in $\overline{\mathcal{J}}$. Let us take a point
$(\widehat{h}_{1},\lambda_{1})$ on the curve $\overline{\mathcal{J}}$. Let us
introduce the eigenfunction $\widehat{v}$ of the Frechet derivative at this
point. We can assume that $L^{2}$ norm of $\widehat{v}$ is $1$. Consider the
equation

${\mathcal{F}}(\widehat{h}_{1}+w,\lambda_{1}+\tau)=0$ (3.27)

in a neighborhood of this point. By introducing the projectors $P$ and
$\widehat{P}$ by (3.10), we can represent $w=s\widehat{v}+\tilde{v}$ where
$Pv=0$ and $s$ is a small number. Then the system (3.27) can be split as
follows

$\widehat{P}{\mathcal{F}}(\widehat{h}_{1}+s\widehat{v}+v,\lambda_{1}+\tau)=0$

and

$(I-\widehat{P}){\mathcal{F}}(\widehat{h}_{1}+s\widehat{v}+v,\lambda_{1}+\tau)=0.$

From the last system we can find $v=v(s,\tau)$ by using the implicit function
theorem, and write the first equation as

${\bf
F}(s,\tau):=P{\mathcal{F}}(\widehat{h}+s\widehat{v}+v(s,\tau),\lambda_{1}+\tau)=0.$

The function ${\bf F}(s,\tau)$ is analytic with respect $s$ and $\tau$ in a
neighborhood of $(s,\tau)=(0,0)$. Let us show that it is not identical zero.
Indeed, if is identical zero then the bifurcation equation (3.27) is satisfied
in a neighborhood of $(\widehat{h}_{1},\lambda_{1})$. Since there are point of
${\mathcal{J}}$ inside this neighborhood we obtain a contradiction with the
constraction of the curve ${\mathcal{J}}$. Therefore the function ${\bf
F}(s,\tau)$ is not identically zero. Then by Theorem 2.1 together with Remark
2.5 there exist local branches of local curves described there which admit a
local analytic injective parametrization. Therefore the curve ${\mathcal{J}}$
can be extended if $(\widehat{h}_{1},\lambda_{1})$ does not belong to
${\mathcal{J}}$ or it is a loop. Thus if ${\mathcal{J}}$ is the maximal
extension curve it must be a loop or it does not belong to any
$\widehat{\mathcal{U}}_{\delta}$ with $\delta>0$. So it remain to prove the
properties (a)-(d) in Theorem 1.11. This can be done in the same way as in the
case 1.

###  3.4. Proof of Lemma 3.4(iii) and Proposition 1.9

As it was mentioned before the only assertion which must be prove there is the
following If $R_{0}<\infty$ and $\Omega(1)>1$ then $\Psi_{Y}(X,0)>\theta_{0}$.

We can use the proof of Theorem 1’ (c’)-(d’), given in Sect.2.2.3, [13], with
some small changes. The first step consists of proving the inequality

$h(q,p)\geq H(p;\theta)\;\;\mbox{with some $\theta>\theta_{0}$},$ (3.28)

where

$H(p;\theta)=\int_{0}^{p}\frac{d\tau}{\sqrt{\theta^{2}-2\Omega(\tau)}}.$

The proof of this inequality requires only that $\Psi_{Y}\geq 0$ and
$\Psi_{Y}(X,\xi(X))\geq\delta$. The inequality (3.28) and the relations
$H(0;\theta)=h(q,0)=0$ implies $H_{p}(0;\theta)\leq h_{p}(q,0)$, which gives

$\Psi_{Y}(X,0)\geq\theta_{0}.$

This completes the proof Lemma 3.4(iii).

Reference to Lemma 3.5 gives the proof of the remaining part of Proposition
1.9 consisting of the case $R_{0}<\infty$ and $\Omega(1)>0$.

Data availability statement Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

##  4\. Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

##  5\. References

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