# A Short Introduction to the Analysis on Manifolds with Conical Singularities

Elmar Schrohe  Elmar Schrohe, Leibniz University Hannover, Institute of
Analysis
Welfengarten 1, 30167 Hannover, Germany<EMAIL_ADDRESS>
###### Abstract.

These notes recall central elements of the cone calculus. The focus lies on
conically degenerate differential operators and the Laplace-Beltrami operator
with respect to a conically degenerate metric as a prototypical example. The
topics include manifolds with conical singularities, the Mellin transform,
cone Sobolev spaces, and the notion of ellipticity in terms of the
invertibility of the principal pseudodifferential symbol and the principal
Mellin symbol. The notes end with a sketch the full cone calculus.

These are lecture notes for a 3 hour course during the summer school Modern
Problems in PDEs and Applications at Ghent University, Belgium, August 23 –
September 2, 2023.

###### Key words and phrases:

Conical singularities, conically degenerate operators, cone Sobolev spaces,
cone calculus

###### 2020 Mathematics Subject Classification:

35S05, 58J40, 46E35, 47L15

##  1\. Introduction

The following example highlights some of the features that come up in
connection with conical singularities. We consider a closed angular domain $B$
in the plane ${\mathbb{R}}^{2}$ with coordinates $x$ and $y$. We assume that
the origin is the vertex of the domain and one of the bounding half lines is
the non-negative $x$-axis while the other intersects the $x$-axis under an
angle $0<{\alpha}<2\pi$.

We are interested in solutions to the homogeneous Dirichlet problem

$\displaystyle\Delta u=0$ $\displaystyle\text{ in }B^{\circ}$ (1.1)
$\displaystyle u=0$ $\displaystyle\text{ on }\partial B\setminus\\{0\\}.$
(1.2)

Naively, one might expect the problem to only have the trivial solution
$u\equiv 0$. We will see, however, that something unexpected can happen.

We introduce polar coordinates $(t,{\theta})$ at the origin, so that $B$ can
be written

$B=\\{(t\cos{\theta},t\sin{\theta})):t\geq 0,0\leq{\theta}\leq{\alpha}\\}.$

We let $v(t,{\theta})=u(t\cos{\theta},t\sin{\theta})$. Equations (1.1) and
(1.2) become

$\displaystyle((t\partial_{t})^{2}+\partial^{2}_{\theta})v(t,{\theta})=0$
$\displaystyle\text{in }(0,\infty)\times(0,{\alpha}),$ (1.3) $\displaystyle
v(t,{\theta})=0$ $\displaystyle\text{for }t>0,{\theta}=0,{\alpha}.$ (1.4)

By ${\mathcal{M}}$ we denote the Mellin transform, which we apply with respect
to $t$:

${\mathcal{M}}v(z,{\theta})=\int_{0}^{\infty}t^{z}v(t,{\theta})\frac{dt}{t},\
z\in{\mathbb{C}}.$

Ignoring for the moment problems of convergence of the integral we see that
$z{\mathcal{M}}v(z)={\mathcal{M}}((-t\partial_{t})v)(z).$ Then Equations (1.3)
and (1.4) become

$\displaystyle(z^{2}+\partial_{\theta}^{2}){\mathcal{M}}v(z,{\theta})=0,$
$\displaystyle z\in{\mathbb{C}},{\theta}\in(0,{\alpha})$ (1.5)
$\displaystyle{\mathcal{M}}v(z,{\theta})=0$ $\displaystyle\text{for
}{\theta}=0,{\alpha}.$ (1.6)

Now we note that the ODE

$\displaystyle(z^{2}+\partial_{\theta}^{2})w({\theta})=0,$ $\displaystyle
z\in{\mathbb{C}},{\theta}\in(0,{\alpha})$ $\displaystyle w({\theta})=0$
$\displaystyle\text{for }{\theta}=0,{\alpha}.$

has nonzero solutions for $z=j\pi/\alpha$, namely
$\sin(j\pi{\theta}/{\alpha})$ for $j\in\mathbb{Z}\setminus\\{0\\}$, and the
functions

$t^{j\pi/\alpha}\sin(j\pi{\theta}/{\alpha})$

actually solve the problem (1.1), (1.2). Note that $t^{j\pi/{\alpha}}$ gets
more and more singular as $j\to-\infty$.

Figure 1: Example of a singular solution

The first unbounded solution is
$u_{1}(t,{\theta})=t^{-\pi/{\alpha}}\sin(\pi{\theta}{\alpha})$. It is in
$L^{2}$ near $t=0$ if and only if ${\alpha}>\pi$, i.e. the angle is obtuse.
The presence of such singular solutions may create problems in numerical
computations, so it is important to know them and to take them into account.

##  2\. Manifolds with Conical Singularities

######  Definition 2.1.

A manifold with conical singularities $B$ is a topological space which is a
smooth manifold outside a finite number of points while, close to each of
these points, it has the structure of a cone whose cross-section is a smooth
$n$-dimensional manifold $X$, i.e., there is a neighborhood $U$ of this point
in $B$ with

$U\cong X\times[0,1)/X\times\\{0\\}.$

Moreover, $B$ comes with a smooth structure that preserves the cones.

For more details see [20, Section 1] or [6, Section 3]. It makes the analysis
much easier to blow up the tip and to work on the blown-up space which is a
smooth manifold ${\mathbb{B}}$ of dimension $n+1$ with boundary
$\partial{\mathbb{B}}=X$. We know that every manifold ${\mathbb{B}}$ with
boundary has a ‘collar neighborhood’ of the boundary, i.e. a neighborhood of
$\partial{\mathbb{B}}$ of the form $[0,1)\times\partial{\mathbb{B}}$. In this
neighborhood we choose local coordinates $(t,x)$ in $[0,1)\times X$, so that
$t$ ‘measures’ the distance to the boundary.

Figure 2. A manifold with a conical singularity $B$ and its blow-up
${\mathbb{B}}$

We model the geometry of $B$ on ${\mathbb{B}}$ by endowing ${\mathbb{B}}$ with
a Riemannian metric $g$ that degenerates near the boundary, namely

$\displaystyle g=dt^{2}+t^{2}h(t),$ (2.1)

where $t\mapsto h(t)$ is a smooth family of non-degenerate Riemannian metrics
on the cross-section $X$. One speaks of a straight cone, if $h$ is independent
of $t$ and of a warped cone, if $h$ varies with $t$. Recall that (2.1) is
another way of representing the metric $g$ in local coordinates $(t,x)$ as the
bilinear form given by the matrix

$\displaystyle g=\begin{pmatrix}1&0\ldots 0\\\ 0&\\\ \vdots&t^{2}h(t)\\\
0&\end{pmatrix}$ (2.2)

Notice that $g$ models precisely the conical situation: Distances to the
boundary remain unchanged (the $dt^{2}$ term) while - in the situation of a
straight cone - distances in $X$ are scaled by a factor $t$ as one approaches
$\partial{\mathbb{B}}$.

##  3\. Cone Differential Operators

Let $A$ be a differential operator of order $\mu$ on ${\mathbb{B}}$ with
smooth coefficients up to the boundary. In local coordinates near
$\partial{\mathbb{B}}$ it can be written in the form

$A=\sum_{j+|{\alpha}|\leq\mu}c_{j,{\alpha}}(t,x)D_{x}^{\alpha}D_{t}^{j}$

with smooth functions $c_{j,{\alpha}}$. Regrouping the coefficients and
exchanging $D_{t}=-i\partial_{t}$ by $\partial_{t}$ we may rewrite $A$ in that
neighborhood in the form

$\displaystyle
A=t^{-\mu}\sum_{j=0}^{\mu}a_{j}(t,x,D_{x})(-t\partial_{t})^{j},$ (3.1)

where $t\mapsto a_{j}(t,x,D_{x})$ is a smooth family of differential operators
of order $\leq\mu-j$ on $X$. We call operators that can be written close to
$\partial{\mathbb{B}}$ in the form (3.1) conically degenerate or cone
differential operators. Note that the class of operators of the form (3.1) is
much larger than the class of smooth differential operators on ${\mathbb{B}}$.

######  Example 3.1.

The Laplace-Beltrami operator $\Delta_{g}$ with respect to a metric $g$ that,
in local coordinates is given by a matrix $(g_{jk}(y))$ and variables
$y=(y_{1},\ldots,y_{m})$ is defined by the formula

$\displaystyle{\Delta}_{g}=\frac{1}{\sqrt{\det
g}}\sum_{j,k}\partial_{y_{j}}(g^{jk}\sqrt{\det g})\partial_{y_{k}}.$ (3.2)

where $(g^{jk})$ is the inverse matrix to $(g_{jk})$.

It is instructive to compute the Laplace-Beltrami operator with respect to the
metric $g$ in (2.2). Here, $\det g=t^{2n}\det h(t)$ and the inverse matrix to
(2.2) is

$\displaystyle(g^{jk})=\begin{pmatrix}1&0\ldots 0\\\ 0&\\\
\vdots&t^{-2}h^{-1}(t)\\\ 0&\end{pmatrix}.$ (3.3)

Denoting by ${\Delta}_{h(t)}$ the Laplace-Beltrami operator with respect to
the metric $h(t)$ on $X$ and letting $-F(t)=\frac{1}{2}t\partial_{t}(\det
h(t))/\det h(t)={\mathcal{O}}(t)$, we obtain

$\displaystyle{\Delta}_{g}=t^{-2}\left((-t\partial_{t})^{2}-(n-1-F(t))(-t\partial_{t})+\Delta_{h(t)}\right),$
(3.4)

Note that $F\equiv 0$ if the cone is straight, i.e. $h$ is independent of $t$.

We see that $\Delta_{g}$ is of the form (3.1) with $\mu=2$,
$a_{2}(t,x,D_{x})=1$, $a_{1}(t,x,D_{x})=n-1-F(t)$ and
$a_{0}(t,x,D_{x})={\Delta}_{h(t)}$.

##  4\. The Mellin transform

We denote by $\Gamma_{\beta}=\\{z\in{\mathbb{C}}:{\mathop{\rm
Re\,}}z={\beta}\\}$ the vertical line of all complex numbers of real part
${\beta}$.

######  Definition 4.1.

Let $u\in C^{\infty}_{c}({\mathbb{R}}_{>0})$ (or, more generally, $u\in
C^{\infty}_{c}({\mathbb{R}}_{>0},E)$ for some locally convex space $E$). The
Mellin transform ${\mathcal{M}}u$ of $u$ is defined by

${\mathcal{M}}u(z)=\int_{0}^{\infty}t^{z}u(t)\frac{dt}{t},\quad
z\in{\mathbb{C}}.$

The following properties are easy consequences of the definition.

######  Lemma 4.2.

Let $u\in C^{\infty}_{c}({\mathbb{R}}_{>0})$.

  1. (a)

${\mathcal{M}}u$ is an entire function, i.e. holomorphic on ${\mathbb{C}}$.

  2. (b)

${\mathcal{M}}(t^{\gamma}u)(z)=({\mathcal{M}}u)(z+{\gamma})$,
${\gamma}\in{\mathbb{R}}$.

  3. (c)

${\mathcal{M}}(\ln tu)(z)=\partial_{z}{\mathcal{M}}u(z)$.

  4. (d)

${\mathcal{M}}(-t\partial_{t}u)(z)=z{\mathcal{M}}u(z)$.

  5. (e)

${\mathcal{M}}$ extends to an isomorphism
${\mathcal{M}}:L^{2}({\mathbb{R}}_{>0})\to L^{2}(\Gamma_{1/2})$.

######  Lemma 4.3.

Let ${\omega}$ be a cut-off function on ${\mathbb{R}}_{>0}$, i.e.
$0\leq{\omega}\leq 1$, ${\omega}$ is non-increasing, ${\omega}(t)\equiv 1$
near zero, ${\omega}(t)\equiv 0$ for large $t$. Then

${\mathcal{M}}(t^{-p}\ln^{k}t{\omega}(t))(\cdot)$

is meromorphic on ${\mathbb{C}}$ with a single pole of order $k+1$ in $z=p$.
Outside any neighborhood of $p$, it is rapidly decreasing on each line
$\Gamma_{\beta}$, uniformly for ${\beta}$ in compact intervals.

### The weighted Mellin transform

According to Lemma 4.2(b) we have

$\displaystyle({\mathcal{M}}u)_{|\Gamma_{1/2-{\gamma}}}(z)={\mathcal{M}}(t^{-{\gamma}}u)(z+{\gamma}),\quad
u\in C_{c}^{\infty}({\mathbb{R}}_{>0}).$

Since
${\mathcal{M}}:L^{2}({\mathbb{R}_{>0}})\stackrel{{\scriptstyle\cong}}{{\to}}L^{2}(\Gamma_{1/2})$,
this yields an extension of ${\mathcal{M}}$ to an isomorphism

$\displaystyle{\mathcal{M}}_{\gamma}:t^{\gamma}L^{2}({\mathbb{R}_{>0}})\to
L^{2}(\Gamma_{1/2-{\gamma}}).$

The inverse is given by

$\displaystyle{\mathcal{M}}_{\gamma}^{-1}(t)=\frac{1}{2\pi
i}\int_{\Gamma_{1/2-{\gamma}}}t^{-z}h(z)\,dz.$

Using Lemma 4.2 and the inversion formula for the Melllin transform, we can
write the operator $A$ with the representation (3.1) in the form

$\displaystyle Au(t,x)=\frac{1}{2\pi
i}\int_{\Gamma_{\frac{1}{2}-\gamma}}t^{-z}\sum_{j=0}^{\mu}a_{j}(t,x,D_{x})z^{j}({\mathcal{M}}_{t\to
z}u)(z,x)\,dz,\;u\in C_{c}^{\infty}({\mathbb{B}}^{\circ}),$ (4.1)

for $\gamma\in{\mathbb{R}}$. In fact, an application of Cauchy’s integral
formula shows that the result is the same for all ${\gamma}$, since $u\in
C^{\infty}_{c}({\mathbb{B}}^{\circ})$, so that ${\mathcal{M}}u$ is holomorphic
in $z$ and rapidly decreasing along vertical lines $\Gamma_{{\beta}}$,
uniformly for ${\beta}$ in compact intervals.

##  5\. Cone Sobolev Spaces

The form (3.1) of a cone differential operator suggests that one should
consider these operators not on standard Sobolev spaces but rather on a scale
of spaces that

  * •

incorporates the weight $t^{-\mu}$ and, more generally, power weights near
$t=0$

  * •

measures smoothness not with respect to derivatives $\partial_{x_{j}}$,
$j=1,\ldots,n$, and $\partial_{t}$ but rather with respect to
$\partial_{x_{j}}$ and $t\partial_{t}$.

######  Definition 5.1.

For $s=k\in{\mathbb{N}}_{0},{\gamma}\in{\mathbb{R}}$, $1<p<\infty$ let

$\displaystyle{\mathcal{H}}^{s,{\gamma}}_{p}({\mathbb{B}})$ $\displaystyle=$
$\displaystyle\left\\{u\in H^{k}_{loc}({\mathbb{B}}^{\circ}):\right.$
$\displaystyle\left.t^{\frac{n+1}{2}-{\gamma}}\partial^{\alpha}_{x}(t\partial_{t})^{j}u(t,x){\omega}(t)\in
L^{p}\left({\mathbb{B}},\frac{dxdt}{t}\right),\forall j+|{\alpha}|\leq
k\right\\}.$

Here, ${\omega}$ is an arbitrary cut-off function in
$C^{\infty}_{c}(\overline{\mathbb{R}}_{>0})$.

This definition extends to $s\in{\mathbb{R}}$ by interpolation and duality.
Note that $s$ measures smoothness and ${\gamma}$ flatness near $t=0$.

######  Remark 5.2.

The weight and the measure in the definition of the spaces
${\mathcal{H}}^{s,{\gamma}}_{p}({\mathbb{B}})$ are motivated by the fact that
in this way, ${\mathcal{H}}^{0,0}_{2}({\mathbb{B}})=L^{2}({\mathbb{B}},g)$,
the $L^{2}$-space with respect to the metric $g$. In fact, as we saw,
$\sqrt{\det g}=t^{n}\sqrt{\det h(t)}$. Since $\sqrt{\det h(t)}$ is bounded and
bounded away from zero we see that, for $u$ supported in $[0,1)\times X$,

$\displaystyle u\in L^{2}({\mathbb{B}},g)$ $\displaystyle\Leftrightarrow$
$\displaystyle\int_{0}^{1}\int_{X}|u(t,x)|^{2}t^{n}\sqrt{\det
h(t)}\,dtdx<\infty$ $\displaystyle\Leftrightarrow$ $\displaystyle
t^{\frac{n+1}{2}}u(t,x)\in L^{2}\Big{(}\frac{dtdx}{t}\Big{)}\Leftrightarrow
u\in{\mathcal{H}}^{0,0}_{2}({\mathbb{B}}).$

Note that this equality does not extend to $p\not=2$. We instead have
$L^{p}({\mathbb{B}},g)={\mathcal{H}}^{0,{\gamma}_{p}}({\mathbb{B}})$,
${\gamma}_{p}=(n+1)(\frac{1}{2}-\frac{1}{p})$. The advantage of the above
definition is that, in this way, the spectrum of an operator $A$ is
independent of $p$, see [22].

######  Lemma 5.3.

  1. (a)

${\mathcal{H}}^{s,{\gamma}}_{p}({\mathbb{B}})^{\prime}={\mathcal{H}}^{-s,-{\gamma}}_{p^{\prime}}({\mathbb{B}})$
, where $1/p+1/p^{\prime}=1$.

  2. (b)

${\mathcal{H}}^{s,{\gamma}}_{p}({\mathbb{B}})\hookrightarrow{\mathcal{H}}^{s^{\prime},{\gamma}^{\prime}}_{p}({\mathbb{B}})$
if and only if $s\geq s^{\prime}$ and ${\gamma}\geq{\gamma}^{\prime}$.

  3. (c)

${\mathcal{H}}^{s,{\gamma}}_{p}({\mathbb{B}})\stackrel{{\scriptstyle\text{\rm
comp}}}{{\hookrightarrow}}{\mathcal{H}}^{s^{\prime},{\gamma}^{\prime}}_{p}({\mathbb{B}})$
if and only if $s>s^{\prime}$ and ${\gamma}>{\gamma}^{\prime}$.

Hence one has to gain smoothness and weight in order to obtain compactness.

For cone differential operators the following theorem is obvious.

######  Theorem 5.4.

Let $A$ be a conically degenerate operator of order $\mu$. Then

$A:{\mathcal{H}}^{s+\mu,{\gamma}+\mu}_{p}({\mathbb{B}})\longrightarrow{\mathcal{H}}^{s,{\gamma}}_{p}({\mathbb{B}})$

is bounded.

##  6\. Symbols

Recall that a differential operator on a smooth manifold $M$ has an
invariantly defined principal symbol on $T^{*}M$. Namely, if
$P=\sum_{|{\alpha}|\leq\mu}c_{\alpha}(y)D^{\alpha}_{y}$, then

${\sigma}_{\psi}^{\mu}(P)(x,\xi)=\sum_{|{\alpha}|=\mu}c_{\alpha}(y)\xi^{\alpha}.$

Choosing coordinates $(t,x,\tau,\xi)$ for $T^{*}([0,1)\times X)$ and recalling
that $D_{t}=-i\partial_{t}$, we obtain for a cone differential operator of the
form (3.1)

${\sigma}_{\psi}^{\mu}(A)(t,x,\tau,\xi)=t^{-\mu}\sum_{j=0}^{\mu}{\sigma}_{\psi}^{\mu-j}(a_{j})(t,x,\xi)(-it\tau)^{j}.$

We notice the degeneracy in $t$ as $t\to 0$ that is inherent to the
singularity and therefore should not play a role when analyzing objects in the
conical setting. We define the rescaled symbol as

$t^{\mu}{\sigma}^{\mu}_{\psi}(A)(t,x,\tau/t,\xi)=\sum_{j=0}^{\mu}{\sigma}_{\psi}^{\mu-j}(a_{j})(t,x,\xi)(-i\tau)^{j}.$

######  Example 6.1.

For the Laplace-Beltrami operator ${\Delta}_{g}$ Equation (3.4) implies that

${\sigma}_{\psi}^{2}({\Delta}_{g})(t,x,\tau,\xi)=t^{-2}((-it\tau)^{2}+{\sigma}_{\psi}^{2}({\Delta}_{h(t)})(x,\xi)).$

Its rescaled symbol therefore is

$t^{2}{\sigma}_{\psi}^{2}(\Delta_{g})(t,x,\tau/t,\xi)=\tau^{2}+{\sigma}^{2}_{\psi}({\Delta}_{h(t)})(x,\xi)=-\tau^{2}-\|\xi\|_{h^{*}(t)}^{2},$

where $\|\xi\|_{h^{*}(t)}^{2}=\sum h^{jk}(t,x)\xi_{j}\xi_{k}$ is the squared
norm of $\xi$ with respect to the metric $h^{*}(t)$ on the cotangent space
induced from the metric $h(t)$.

For conically degenerate operators a second symbol is important. The so-called
conormal symbol or principal Mellin symbol of a cone differential operator
$A$, denoted by ${\sigma}_{M}^{\mu}(A)$ is a function on ${\mathbb{C}}$ with
values in differential operators on $X$. For $A$ as in (3.1), we define

$\displaystyle({\sigma}_{M}^{\mu}(A))(z)=\sum_{j=0}^{\mu}a_{j}(0,x,D_{x})z^{j},\quad
z\in{\mathbb{C}}.$

Hence we obtain bounded maps

${\sigma}_{M}^{\mu}(A)(z):H^{s+\mu}(X)\to H^{s}(X),\quad z\in{\mathbb{C}}.$

######  Definition 6.2.

A cone differential operator $A$ with a representation as in (3.1) is called
elliptic with respect to the weight line $\Gamma_{\beta}$, if

  1. (i)

The principal pseudodifferential symbol ${\sigma}_{\psi}^{\mu}$ is invertible
in the interior of ${\mathbb{B}}$ and, near $\partial{\mathbb{B}}$, the
rescaled principal symbol is uniformly invertible up to $t=0$, i.e. its
inverse is ${\mathcal{O}}(|\xi|+|\tau|)^{-\mu})$ up to $t=0$.

  2. (ii)

${\sigma}_{M}^{\mu}(A)(z):H^{s+\mu}(X)\to H^{s}(X)$ is invertible for some
$s\in{\mathbb{R}}$ and all $z\in\Gamma_{\beta}$.

A standard result from the theory of pseudodifferential operators implies that
${\sigma}_{M}^{\mu}(A)(z)$ in (ii) is invertible for all $s\in{\mathbb{R}}$ if
it is invertible for some $s$.

######  Example 6.3.

For the Laplace-Beltrami operator

$\Delta_{g}=t^{-2}\left((-t\partial_{t})^{2}-(n-1-F(t))(-t\partial_{t})+{\Delta}_{h(t)}\right)$

the principal Mellin symbol is (recall that $F(t)={\mathcal{O}}(t)$)

${\sigma}^{2}_{M}({\Delta}_{g})(z)=z^{2}-(n-1)z+{\Delta}_{h(0)}.$

For which $z$ is it invertible? This is the case if and only if $z^{2}-(n-1)z$
is not in the spectrum of $-{\Delta}_{h(0)}$. Denoting by
$0={\lambda}_{0}>{\lambda}_{1}>\ldots$ the different eigenvalues of
${\Delta}_{h(0)}$ we see that ${\sigma}_{M}^{2}({\Delta}_{g})(z)$ is not
invertible, if, and only if

$z\in\Big{\\{}q_{j}^{\pm}=\frac{n-1}{2}\pm\sqrt{\Big{(}\frac{n-1}{2}\Big{)}^{2}-{\lambda}_{j}},j\in{\mathbb{N}}_{0}\Big{\\}}.$

In particular, ${\sigma}_{M}^{2}(\Delta_{g})(\cdot)$ is invertible on every
line $\Gamma_{{\beta}}$ with ${\beta}\not=q^{\pm}_{j}$,
$j\in{\mathbb{N}}_{0}$. As $X$ is a closed manifold, ${\lambda}_{0}=0$ and so
${\sigma}_{M}^{2}({\Delta}_{g})(z)$ is not invertible in $z=q_{0}^{-}=0$ and
$z=q_{0}^{+}=n-1$ (both coincide for $n=1$).

##  7\. The Cone Calculus

Let ${\omega}_{1},{\omega}_{2},{\omega}_{3}$ be cut-off functions with
${\omega}_{1}{\omega}_{2}={\omega}_{1}$,
${\omega}_{1}{\omega}_{3}={\omega}_{3}$ and ${\gamma}\in{\mathbb{R}}$.

An operator of order $\mu\in{\mathbb{R}}$ in the cone calculus with respect to
the line $\Gamma_{\frac{n+1}{2}-{\gamma}}$ is an operator
$A:C^{\infty}_{c}({\mathbb{B}}^{\circ})\to C^{\infty}({\mathbb{B}}^{\circ})$
that can be written in the form (cf.(4.1))

$\displaystyle A=t^{-\mu}{\omega}_{1}{\text{\rm
op}}_{M}^{\gamma}(h+h_{0}){\omega}_{2}+(1-{\omega}_{1})P(1-{\omega}_{3})+G,$
(7.1)

where

  1. (i)

$h=h(t,z)$ is smooth in $t\geq 0$ and holomorphic in $z\in{\mathbb{C}}$,
taking values in pseudodifferential operators of order $\mu$ on $X$ such that
$h(t,z)_{|z\in\Gamma_{{\beta}}}\in\Psi^{\mu}(X;\Gamma_{{\beta}})$ is a
parameter-dependent (w.r.t. $\Gamma_{\beta}$) pseudodifferential operator of
order $\mu$, uniformly for ${\beta}$ in compact intervals.

  2. (ii)

$h_{0}=h_{0}(z)$ is meromorphic on ${\mathbb{C}}$ with values in
pseudodifferential operators of order $\mu$ on $X$. The coefficients in the
Laurent expansions in the poles are pseudodifferential operators of finite
rank, so in particular smoothing. No pole lies on
$\Gamma_{({n+1)}/2-{\gamma}}$ and
$h_{0|\Gamma_{{\beta}}}\in\Psi^{\mu}(X;\Gamma_{{\beta}})$ outside the poles,
unformly for ${\beta}$ in compact intervals.

  3. (iii)

$P$ is a pseudodifferential operator of order $\mu$ on ${\mathbb{B}}^{\circ}$.

  4. (iv)

$G:{\mathcal{H}}_{2}^{s,{\gamma}}({\mathbb{B}})\to{\mathcal{H}}_{2}^{\infty,\gamma-\mu+{\varepsilon}}({\mathbb{B}})$
and
$G^{*}:{\mathcal{H}}_{2}^{-s+\mu,-{\gamma}+\mu}({\mathbb{B}})\to{\mathcal{H}}_{2}^{\infty,-{\gamma}+{\varepsilon}}({\mathbb{B}})$
for all $s\in{\mathbb{R}}$ and some ${\varepsilon}>0$.

The operator ${\text{\rm
op}}_{M}^{\gamma}(h+h_{0}):C^{\infty}_{c}({\mathbb{B}}^{\circ})\to
C^{\infty}(\mathbb{B}^{\circ})$ is then defined by

${\text{\rm op}}_{M}^{\gamma}(h+h_{0})u(t,x)=\frac{1}{2\pi
i}\int_{\Gamma_{\frac{n+1}{2}-{\gamma}}}t^{-z}(h(t,z)+h_{0}(z)){\mathcal{M}}_{t\to
z}u(z,\cdot)dz.$

For more details see [20] and [21] or [26, Chapter 2].

The operators $G$ in (iv) form the residual class. They are infinitely
smoothing and gain a little weight, so they are compact elements of
${\mathcal{L}}({\mathcal{H}}_{p}^{s,{\gamma}}({\mathbb{B}}),{\mathcal{H}}_{p}^{s-\mu,\gamma-\mu}({\mathbb{B}}))$.

######  Theorem 7.1.

Let $A$ be as above. For every $s\in{\mathbb{R}}$ and $1<p<\infty$, $A$ in
(7.1) then extends to a bounded linear operator

$A:{\mathcal{H}}_{p}^{s,{\gamma}}({\mathbb{B}})\to{\mathcal{H}}_{p}^{s-\mu,{\gamma}-\mu}({\mathbb{B}}).$

######  Theorem 7.2.

The operators in the cone calculus form a graded $*$-algebra, i.e.

  * •

The composition $A_{1}A_{2}$ of a cone operator $A_{2}$ of order $\mu_{2}$
with respect to the line $\Gamma_{\frac{n+1}{2}-{\gamma}}$ with $A_{1}$ of
order $\mu_{1}$ w.r.t. $\Gamma_{\frac{n+1}{2}-{\gamma}+\mu_{2}}$ is an element
of order $\mu_{1}+\mu_{2}$ w.r.t. $\Gamma_{\frac{n+1}{2}-{\gamma}}$.

  * •

The adjoint with respect to an extension of the $L^{2}$ inner product of an
operator of order $\mu$ w.r.t. the line $\Gamma_{\frac{n+1}{2}-{\gamma}}$ is
an operator of order $\mu$ w.r.t. $\Gamma_{\frac{n+1}{2}+{\gamma}-\mu}$.

If $A$ is a cone operator of order $\mu$ with respect to the line
$\Gamma_{\frac{n+1}{2}-{\gamma}}$ and $A$ is invertible as an operator in
$\mathcal{L}({\mathcal{H}}_{p}^{s,{\gamma}}({\mathbb{B}}),{\mathcal{H}}_{p}^{s-\mu,{\gamma}-\mu}({\mathbb{B}}))$
for some choice of $s\in\mathbb{R}$ and $1<p<\infty$, then $A^{-1}$ is a cone
operator of order $-\mu$ w.r.t. $\Gamma_{\frac{n+1}{2}-{\gamma}+\mu}$. The
spectrum of $A$ is independent of $s$ and $p$.

### Ellipticity

The notions of ellipticity extend to the general cone calculus. In fact, to
the operator ${\text{\rm op}}_{M}^{\gamma}(h)$ we may associate locally a
conically degenerate pseudodifferential symbol of the form $p(t,x,t\tau,\xi)$,
where $p(t,x,\tau,\xi)$ is a standard symbol of order $\mu$, smooth up to
$t=0$, modulo symbols of order $-\infty$ in the interior.

For the operator (7.1) we therefore have a well-defined principal symbol in
the interior with conical degeneration as $t\to 0$. We can define the rescaled
symbol as before by omitting the factor $t^{-\mu}$ and replacing $t\tau$ by
$\tau$. As in Definition 6.2(a) ellipticity requires the invertibility of the
interior pseudodifferential symbol together with the uniform invertibility of
the rescaled symbol.

Following 6.2(b), we define the principal Mellin symbol
${\sigma}^{\mu}_{M}(A)(z)=h(0,z)+h_{0}(z)$. This is a pseudodifferential
operator of order $\mu$ on $X$. As in Definition 6.2, ellipticity with respect
to the line $\Gamma_{\frac{n+1}{2}-{\gamma}}$ requires additionally the
invertibility of ${\sigma}^{\mu}_{M}(A)(z):H^{s}(X)\to H^{s-\mu}(X)$ for some
(and then all) $s\in{\mathbb{R}}$ and $z$ on this line.

A central theorem then is:

######  Theorem 7.3.

Assume that the cone operator $A$ is elliptic with respect to the line
$\Gamma_{\frac{n+1}{2}-{\gamma}}$ Then there exists a parametrix $B$ of order
$-\mu$ in the cone calculus such that

$AB-I_{{\mathcal{H}}_{p}^{s-\mu,\gamma-\mu}({\mathbb{B}})}=G_{R}\text{ and
}BA-I_{{\mathcal{H}}_{p}^{s,{\gamma}}({\mathbb{B}})}=G_{L}$

are bounded operators
$G_{R}:{\mathcal{H}}_{p}^{s-\mu,{\gamma}-\mu}({\mathbb{B}})\to{\mathcal{H}}_{p}^{\infty,\gamma-\mu+{\varepsilon}}({\mathbb{B}})$
and
$G_{L}:{\mathcal{H}}_{p}^{s,{\gamma}}({\mathbb{B}})\to{\mathcal{H}}_{p}^{\infty,\gamma+{\varepsilon}}({\mathbb{B}})$
for $1<p<\infty$ and suitable $\epsilon>0$. In particular, $A$ then is a
Fredholm operator.

Moreover, we have

######  Theorem 7.4.

If $A$ is a cone operator of order $\mu$ with respect to the line
$\Gamma_{\frac{n+1}{2}-{\gamma}}$ and $A$ is invertible as an operator in
$\mathcal{L}({\mathcal{H}}_{p}^{s,{\gamma}}({\mathbb{B}}),{\mathcal{H}}_{p}^{s-\mu,{\gamma}-\mu}({\mathbb{B}}))$
for some choice of $s\in\mathbb{R}$ and $1<p<\infty$, then $A^{-1}$ is a cone
operator of order $-\mu$ w.r.t. $\Gamma_{\frac{n+1}{2}-{\gamma}+\mu}$. The
spectrum of $A$ is independent of $s$ and $p$.

### Back to the beginning

One can now see the reason for the appearance of the singular solutions: In
analogy to the case of pseudodifferential parametrices, $B$ is constructed
starting from a rough version $B_{0}$ defined by inverting the principal
pseudodifferential symbol and the principal Mellin symbol and subsequent
refinements. Here,

$B_{0}=\tilde{\omega}_{1}{\text{\rm
op}}_{M}^{\gamma}(g(z))\tilde{\omega}_{2}t^{\mu}+(1-\tilde{\omega}_{1})Q(1-\tilde{\omega}_{3})$

with $g(z)=(h(0,z)+h_{0}(z))^{-1}$, a parametrix $Q$ to $P$ and cut-off
functions $\tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{\omega}_{3}$,
satisfying $\tilde{\omega}_{1}\tilde{\omega}_{2}=\tilde{\omega}_{1}$,
$\tilde{\omega}_{1}\tilde{\omega}_{3}=\tilde{\omega}_{3}$. Suppose $\mu>0$. By
Lemma 4.2(b) we have for $u\in C^{\infty}_{c}({\mathbb{B}}^{\circ})$

$\displaystyle{\text{\rm op}}_{M}^{\gamma}(g)(t^{\mu}u)(t,x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{(n+1)/2-{\gamma}}}t^{-z}g(z){\mathcal{M}}(t^{\mu}u)(z,x)\,dz$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{(n+1)/2-{\gamma}}}t^{-z}g(z){\mathcal{M}}(u)(z+\mu,x)\,dz$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{(n+1)/2-{\gamma}+\mu}}t^{-w+\mu}g(w-\mu){\mathcal{M}}(u)(w,x)\,dw$
$\displaystyle=$ $\displaystyle t^{\mu}\text{\rm
op}_{M}^{\gamma-\mu}g(\cdot-\mu)u(t,x)$ $\displaystyle=$ $\displaystyle
t^{\mu}{\text{\rm op}}_{M}^{\gamma}(g(\cdot-\mu))u(t,x)+\frac{t^{\mu}}{2\pi
i}\int_{\mathcal{C}}t^{-w}g(w){\mathcal{M}}(u)(w,x)\,dw,$

where $\mathcal{C}$ is the ‘contour’ in ${\mathbb{C}}$ given by the vertical
lines ${\Gamma_{(n+1)/2-{\gamma}}}$ and ${\Gamma_{(n+1)/2-{\gamma}+\mu}}$. The
integral makes sense, since $g$ decreases rapidly at infinity in the strip,
and the integration furnishes the residues of the poles of $w\mapsto
t^{-w}g(w-\mu){\mathcal{M}}u(w,x)$ between the lines. We know that
${\mathcal{M}}u(w,x)$ is holomorphic in $w$ and the Laurent coefficients of
$g$ in a pole are finite rank (hence smoothing) pseudodifferential operators.
So a pole of $g$ in the point $p$ between the two lines of order
$k\in{\mathbb{N}}_{0}$ furnishes contributions of the form
$t^{-p}\ln^{j}(t)s_{j}(x)$, where the $s_{j}$ are smooth functions on $X$,
$j\leq k+1$. Subsequent refinements lead additionally to terms
$t^{-p+l}\ln^{j}(t)\tilde{s}_{j}(x)$, $\tilde{s}_{j}\in C^{\infty}(X)$,
$l\in{\mathbb{N}}$, as long as $p-l$ also lies between the lines. Note that
$t^{\mu}{\text{\rm op}}_{M}^{\gamma}(g(\cdot-\mu))u$ lies in
$\mathcal{H}^{s,\gamma+\mu}(\mathbb{B})$ while the singular terms belong to
$\mathcal{H}^{s,\gamma}(\mathbb{B})\setminus\mathcal{H}^{s,\gamma+\mu}(\mathbb{B})$
for any $s\in\mathbb{R}$.

##  8\. Historical Remarks

Kondrat’ev’s paper [4] from 1967 is generally considered as the starting point
of the analysis on manifolds with conical singularities. His approach was
further developed e.g. by V.G. Maz’ja and B.A. Plamenevskiĭ [8].
Independently, Jeff Cheeger in [2] studied the spectral geometry of spaces
with cone-like singularities. However, it was only in the 1980’s that
pseudodifferential calculi were systematically developed. Early works are [11]
by Richard Melrose and Gerardo Mendoza, [24] by B.-W. Schulze and the
monographs [13] by Plamenevskiĭ, [25] by Schulze and [12] by Melrose. Several
other singular calculi were developed in the sequel, in particular the edge
calculus by Schulze, see e.g. [14], and Mazzeo [7], the zero calculus and the
$\phi$-calculus by Mazzeo and Melrose [9], [10], and the cone calculus for
boundary value problems by Schulze and the author [20], [21]. Index theory on
singular spaces was studied by many authors, two examples are Brüning and
Seeley [1] and Lesch [5]. In collaboration with Sandro Coriasco, Nikolaos
Roidos and Jörg Seiler, the author has developed an approach to the analysis
of nonlinear evolution equations on manifolds with conical singularities via
maximal regularity techniques, starting with [3]. The Cahn-Hilliard equation
and the Allen-Cahn equation were treated in [15] and [16], the porous medium
equation in [17], [18] and [19].

## References

  * [1] J. Brüning, R. Seeley. An index theorem for first order regular singular operators. Amer. J. Math. 110 (1988), 659–714.
  * [2] J. Cheeger. On the spectral geometry of spaces with cone-like singularities. Proc. Nat. Acad. Sci. U.S.A. 76, (1979) 2103–2106.
  * [3] S. Coriasco, E. Schrohe, J. Seiler. Bounded imaginary powers of differential operators on manifolds with conical singularities. Math. Z. 244 (2003), 235–269.
  * [4] V. A. Kondrat’ev. Boundary value problems for elliptic equations in domains with conical or angular points. Tr. Mosk. Mat. Obs. 16 (1967), 209–292
  * [5] M. Lesch. Operators of Fuchs type, Conical Singularities, and Asymptotic Methods. Teubner-Texte Math. 136, Teubner-Verlag, 1997.
  * [6] P. Lopes, E. Schrohe. Spectral invariance of pseudodifferential boundary value problems on manifolds with conical singularities. J. Fourier Analysis Appl. 25 (2019), 1147–1202.
  * [7] R. Mazzeo. Elliptic theory of differential edge operators. I. Comm. PDE 16 (1991), 1615–1664.
  * [8] V.G. Maz’ja, B.A. Plamenevskiĭ. The behavior of the solutions of quasilinear elliptic boundary value problems in the neighborhood of a conical point. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 94–97.
  * [9] R, Mazzeo, R. Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260–310.
  * [10] R, Mazzeo, R. Melrose. Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998), 833–866.
  * [11] R. Melrose and G. Mendoza. Elliptic operators of totally characteristic type. Preprint MSRI 047-83, MSRI, Berkeley, CA (1983).
  * [12] R. Melrose. The Atiyah-Patodi-Singer Index Theorem. A.K. Peters, New York 1993.
  * [13] B.A. Plamenevskijĭ. Algebras of pseudodifferential operators. Mathematics and its Applications (Soviet Series) 43. Kluwer Academic Publishers Group, Dordrecht, 1989.
  * [14] S. Rempel, B.-W. Schulze. Branching of asymptotics for elliptic operators on manifolds with edges. Partial differential equations (Warsaw, 1984), Banach Center Publ. 19, PWN, Warsaw, (1987), 231–236.
  * [15] N. Roidos, E. Schrohe. The Cahn-Hilliard equation and the Allen-Cahn equation on manifolds with conical singularities. Comm. PDE 38 (2013), 925–943.
  * [16] N. Roidos, E. Schrohe. Bounded imaginary powers of cone differential operators on higher order Mellin-Sobolev spaces and applications to the Cahn-Hilliard equation. J. Differential Equations 257 (2014), 611–637.
  * [17] N. Roidos, E. Schrohe. Existence and maximal $L^{p}$-regularity of solutions for the porous medium equation on manifolds with conical singularities. Comm. PDE 41 (2016), 1441–1471.
  * [18] N. Roidos, E. Schrohe. Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities. Comm. PDE 43 (2018), 1456-1484.
  * [19] N. Roidos, E. Schrohe, J. Seiler. Bounded $H_{\infty}$-calculus for boundary value problems on manifolds with conical singularities. J. Differential Equations 297 (2021), 370–408.
  * [20] E. Schrohe, B.-W. Schulze. Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities. I. Pseudo-differential calculus and mathematical physics, 97–209, Math. Top., 5,. Akademie Verlag, Berlin, 1994.
  * [21] E. Schrohe, B.-W. Schulze. Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities. II. Boundary value problems, Schrödinger operators, deformation quantization, 70–205, Math. Top., 8, Akademie Verlag, Berlin, 1995.
  * [22] E. Schrohe, J. Seiler. Ellipticity and invertibility in the cone algebra on $L_{p}$-Sobolev spaces. Integral Equations Operator Theory 41 (2001), 93-114.
  * [23] E. Schrohe, J. Seiler. The resolvent of closed extensions of cone differential operators. Canad. J. Math. 57 (2005), 771–811.
  * [24] B.-W. Schulze. Opérateurs pseudo-différentiels et asymptotique sur des variétés à singularités. (French) Séminaire sur les équations aux dérivées partielles, 1985–1986, Exp. No. III, 16 pp., École Polytech., Palaiseau, 1986.
  * [25] B.-W. Schulze. Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam, 1991.
  * [26] B.-W. Schulze. Boundary value problems and singular pseudo-differential operators. Pure and Applied Mathematics (New York). John Wiley & Sons, Ltd., 1998.


††thanks: Corresponding author

# Boussinesq’s equations for (2+1)-dimensional surface gravity waves in an
ideal fluid model

Anna Karczewska<EMAIL_ADDRESS>Faculty of Mathematics,
Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a,
65-516 Zielona Góra, Poland  Piotr Rozmej<EMAIL_ADDRESS>Institute
of Physics, Faculty of Physics and Astronomy, University of Zielona Góra,
Szafrana 4a, 65-516 Zielona Góra, Poland

###### Abstract

We study the problem of gravity surface waves for the ideal fluid model in
(2+1)-dimensional case. We apply a systematic procedure for deriving the
Boussinesq equations for a prescribed relationship between the orders of four
expansion parameters, the amplitude parameter $\alpha$, the long-wavelength
parameter $\beta$, the transverse wavelength parameter $\gamma$, and the
bottom variation parameter $\delta$. We also take into account surface tension
effects when relevant. For all considered cases, the (2+1)-dimensional
Boussinesq equations can not be reduced to a single nonlinear wave equation
for surface elevation function. On the other hand, they can be reduced to a
single, highly nonlinear partial differential equation for an auxiliary
function $f(x,y,t)$ which determines the velocity potential but is not
directly observed quantity. The solution $f$ of this equation, if known,
determines the surface elevation function. We also show that limiting the
obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known
cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.

Shallow water waves, the Boussinesq equations, perturbation approach, uneven
bottom

###### pacs:

02.30.Jr, 05.45.-a, 47.35.Bb, 47.35.Fg

##  I Introduction

The Korteweg-de Vries equation KdV  is one of the most widely used nonlinear
wave equations. It was derived by the perturbation calculus method from a
model of an ideal fluid (nonviscous and incompressible) in which the motion is
irrotational in (1+1)-dimensions and the bottom of the fluid container is
flat. The simplest nonlinear wave equation in (2+1)-dimensions is the
Kadomtsev-Petviashvili (KP for short) equation KP . Both the KdV and KP
equations are integral and satisfy many conservation laws. Their analytical
solutions are also known.

Since nonlinear waves on the surface of seas and oceans are very important in
practice, there are many studies in the literature on (2+1)-dimensional
equations of the KdV or KP type, see, e.g., Wazwaz08 ; Wazwaz09 ; Peng10 ;
Wazwaz11 ; WZZQ14 ; Adem16 ; Zhang17 ; Batwa18 ; Wanga19 ; Lou20 . The
equations used in these studies are also integral, allowing their authors to
construct analytical solutions of various types (solitons, periodic solutions,
lumps, breathers).

The extension of the ideal fluid model to the (2+1)-dimensional case leads to
significantly more complicated equations. The only such attempt known to us is
the work Fok . Unfortunately, this particular paper is erroneous, as we
demonstrated in RK21 . To our best knowledge, there are no correct
(2+1)-dimensional studies that take into account the model of an ideal fluid
in full detail. The reason is that the resulting equations are non-integrable.

In this paper, we generalise the perturbative approach described in BS2013
and KRcnsns  for (1+1)-dimensional case. The authors of BS2013  showed that
for the flat bottom, the KdV, extended KdV, fifth-order KdV, and Gardner
equations can be derived for different relations between small parameters
determining nonlinearity $(\alpha)$ and dispersion $(\beta)$. In KRcnsns , we
generalised this approach for the case with an uneven bottom, assuming that
the bottom variations are much smaller than the fluid’s depth. Here, we follow
KRcnsns  attempting to derive analogous wave equations for (2+1)-dimensional
case. It turns out that while it is possible to derive a system of appropriate
(2+1)-dimensional Boussinesq equations, it is rather intractable to reduce
them to a wave equation on the surface of a fluid. However, it is possible to
obtain a single nonlinear partial differential equation of the wave type in
which the argument is an auxiliary function that determines the velocity
potential. The solution of such an equation, if known, determines the
(2+1)-dimensional function representing the time-dependent surface profile.

The paper is organised as follows. In section II we recall the model, define
small parameters $\alpha,\beta,\gamma,\delta$ and set up the set of Euler’s
equations in scaled dimensionless variables. In section III, the case when all
small parameters are of the same order is considered. For $\gamma=0$ and
(1+1)-dimensions, this case leads to the Korteweg-de Vries equation extended
for an uneven bottom, derived by us in KRcnsns , and when the bottom is flat
($\delta=0$) reduces to the usual Korteweg-de Vries equation. Section IV is
devoted to the case when $\alpha,\gamma$ are of the order of $\beta$, but
$\delta$ is of the order of $\beta^{2}$. This case, for $\gamma=0$ and
(1+1)-dimensions leads to the extended KdV equation generalised for an uneven
bottom, derived in KRcnsns , and when the bottom is flat ($\delta=0$) leads to
the extended KdV equation derived by Marchant and Smyth in MS90 . In section V
we consider the case when parameters $\alpha,\gamma,\delta$ are of the order
of $\beta^{2}$ and perturbation approach is extended to second order. Here,
limitation to (1+1)-dimensions leads to fifth-order KdV equation generalised
in KRcnsns  for an uneven bottom. Section VI deals with the case when $\alpha$
is the leading parameter, $\beta,\gamma,\delta$ are of the order or
$\alpha^{2}$, and derivations are performed up to second order. In this case,
limitation to (1+1)-dimensions leads to the Gardner equation extended for an
uneven bottom KRcnsns , and when $\delta=0$, that is, when the bottom is flat,
reduces to the usual Gardner equation. Section VII contains conclusions.

##  II Description of the model

Let us consider the inviscid and incompressible fluid model whose motion is
irrotational in a container with an impenetrable bottom. In dimensional
variables, the set of hydrodynamical equations has the following form

$\displaystyle\phi_{xx}+\phi_{yy}+\phi_{zz}$ $\displaystyle=$ $\displaystyle
0,\quad\mbox{in~{}the~{}whole~{}volume},$ (1)
$\displaystyle\phi_{z}-(\eta_{x}\phi_{x}+\eta_{y}\phi_{y}+\eta_{t})$
$\displaystyle=$ $\displaystyle 0,\quad\mbox{at~{}the~{}surface},$ (2)
$\displaystyle\phi_{t}+\frac{1}{2}(\phi_{x}^{2}+\phi_{y}^{2}+\phi_{z}^{2})+g\eta+\frac{p_{s}}{\varrho}$
$\displaystyle=$ $\displaystyle 0,\quad\mbox{at~{}the~{}surface},$ (3)
$\displaystyle\phi_{z}-h_{x}\phi_{x}-h_{y}\phi_{y}$ $\displaystyle=$
$\displaystyle 0,\quad\mbox{at~{}the~{}impenetrable~{}bottom}.$ (4)

Here $\phi(x,y,z,t)$ denotes the velocity potential, $\eta(x,y,t)$ denotes the
surface profile function, $g$ is the gravitational acceleration, $\varrho$ is
fluid’s density, and $p_{s}$ is additional pressure due to the surface tension
AN2010

$p_{s}=-T\,\nabla\cdot\left(\frac{\nabla\eta}{(1+|\nabla\eta|^{2})^{1/2}}\right)=T\,\frac{\left(1+\eta_{y}^{2}\right)\eta_{xx}+\left(1+\eta_{x}^{2}\right)\eta_{yy}-2\eta_{x}\eta_{y}\eta_{xy}}{\left(1+\eta_{x}^{2}+\eta_{y}^{2}\right)^{3/2}},$
(5)

where $T$ is fluid’s surface tension coefficient, and
$\nabla=(\partial_{x},\partial_{y})$ is two-dimensional gradient. The bottom
can be non-flat and is described by the function $h(x,y)$. Indexes denote
partial derivatives, i.e. $\phi_{xx}\equiv\frac{\partial^{2}\phi}{\partial
x^{2}}$, and so on.

The next step consists in introducing a standard scaling to dimensionless
variables (in general, it could be different in $x$-, $y$\- and $z$-direction)

$\displaystyle\tilde{x}=$ $\displaystyle
x/L,\quad\tilde{y}=y/l,\quad\tilde{z}=z/H,\quad\tilde{t}=t/(L/\sqrt{gH}),\quad\tilde{\eta}=\eta/A,\quad\tilde{\phi}=\phi/(L\frac{A}{H}\sqrt{gH}),$
(6)

where $A$ is the amplitude of surface distortions from equilibrium shape (flat
surface), $H$ is average fluid depth, $L$ is the average wavelength (in
$x$-direction), and $l$ is a wavelength in $y$-direction. In general, $l$
should be the same order as $L$, but not necessarily equal. Then the set
(1)-(4) takes in scaled variables the following form (here and next, we omit
the tilde signs)

$\displaystyle\beta\phi_{xx}+\gamma\phi_{yy}+\phi_{zz}$ $\displaystyle=$
$\displaystyle 0,$ (7)
$\displaystyle\eta_{t}+\alpha(\eta_{x}\phi_{x}+\frac{\gamma}{\beta}\eta_{y}\phi_{y})-\frac{1}{\beta}\phi_{z}$
$\displaystyle=$ $\displaystyle 0,\quad\mbox{for}\quad z=1+\alpha\eta,$ (8)
$\displaystyle\phi_{t}+\frac{1}{2}\alpha\left(\phi_{x}^{2}+\frac{\gamma}{\beta}\phi_{y}^{2}+\frac{1}{\beta}\phi_{z}^{2}\right)+\eta+\text{ST}$
$\displaystyle=$ $\displaystyle 0,\quad\mbox{for}\quad z=1+\alpha\eta,$ (9)
$\displaystyle\phi_{z}-\beta\delta h_{x}\phi_{x}-\gamma\delta h_{y}\phi_{y}$
$\displaystyle=$ $\displaystyle 0,\quad\mbox{for}\quad z=\delta h.$ (10)

Besides standard small parameters $\alpha=\frac{a}{H}$,
$\beta=\left(\frac{H}{L}\right)^{2}$ and
$\gamma=\left(\frac{H}{l}\right)^{2}$, which are sufficient for the flat
bottom case, we introduced another one defined as $\delta=\frac{a_{h}}{H}$,
where $a_{h}$ denotes the amplitude of bottom variations KRcnsns ; KRI . In
the perturbation approach, all these parameters, $\alpha,\beta,\gamma,\delta$,
are assumed to be small but not necessarily of the same order. ST is the term
originating from surface tension. The explicit form of this term and its
useful approximation up to second order is given below, in (11). For details,
see the Appendix A.

ST
$\displaystyle=-\tau\,\frac{\beta\left(1+\alpha^{2}\beta\eta_{y}^{2}\right)\eta_{xx}+\gamma\left(1+\alpha^{2}\beta\eta_{x}^{2}\right)\eta_{yy}-2\alpha^{2}\beta\gamma\eta_{x}\eta_{y}\eta_{xy}}{\left(1+\alpha^{2}\beta\eta_{x}^{2}+\alpha^{2}\gamma\eta_{y}^{2}\right)^{3/2}}=-\tau\,(\beta\eta_{xx}+\gamma\eta_{yy}+O(\beta^{4})).$
(11)

The Bond number is defined as $\tau=\frac{T}{\varrho gH^{2}}$. For shallow-
water waves, when $H\gtrsim$ 1 m, ST term can be safely neglected, since
$\tau<10^{-7}$, but it becomes important for waves in thin fluid layers. In
sections III and IV we neglect surface tension terms, in sections V and VI we
take them into account.

The standard perturbation approach to the system of Euler’s equations (7)-(10)
consists of the following steps. First, the velocity potential is sought in
the form of power series in the vertical coordinate

$\phi(x,y,z,t)=\sum_{m=0}^{\infty}z^{m}\,\phi^{(m)}(x,y,t),$ (12)

where $\phi^{(m)}(x,y,t)$ are yet unknown functions. The Laplace equation (7)
determines $\phi$ in the form which involves only two unknown functions with
the lowest $m$-indexes, $f(x,y,t):=\phi^{(0)}(x,y,t)$ and
$F(x,y,t):=\phi^{(1)}(x,y,t)$ and their space derivatives. Hence,

$\displaystyle\phi(x,y,z,t)$
$\displaystyle=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(2m)!}z^{2m}\,(\beta\partial_{xx}+\gamma\partial_{yy})^{m}f(x,y,t)$
(13)
$\displaystyle+\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(2m+1)!}z^{2m+1}\,(\beta\partial_{xx}+\gamma\partial_{yy})^{m}F(x,y,t).$

The explicit form of this velocity potential reads as

$\displaystyle\phi=$ $\displaystyle\hskip 12.91663ptf-\frac{1}{2}z^{2}(\beta
f_{2x}+\gamma f_{2y})+\frac{1}{24}z^{4}(\beta^{2}f_{4x}+2\beta\gamma
f_{2x2y}+\gamma^{2}f_{4y})+\cdots$ $\displaystyle+zF-\frac{1}{6}z^{3}(\beta
F_{2x}+\gamma F_{2y})+\frac{1}{120}z^{5}(\beta^{2}F_{4x}+2\beta\gamma
F_{2x2y}+\gamma^{2}F_{4y})+\cdots$ (14)

Next, the boundary condition at the bottom (10) is utilized. For the flat
bottom case, it implies $F=0$, simplifying substantially next steps. In
particular, $F=0$ makes it possible to derive the Boussinesq equations up to
arbitrary order. For an uneven bottom, the equation (10) determines a
differential equation relating $F$ to $f$. This differential equation can be
resolved to obtain $F(f,f_{x},f_{xx},h,h_{x})$ but this solution can be
obtained only up to some particular order in leading small parameter. Then,
the velocity potential is substituted into kinematic and dynamic boundary
conditions at the unknown surface (8)-(9). Retaining only terms up to a given
order, one obtains the Boussinesq system of two equations for unknown
functions $\eta,f$ valid only up to a given order in small parameters. The
resulting equations, however, depend substantially on the ordering of small
parameters.

In 2013, Burde and Sergyeyev BS2013  demonstrated that for the case of
(1+1)-dimensional and the flat bottom, the KdV, the extended KdV, fifth-order
KdV, and Gardner equations can be derived from the same set of Euler’s
equations (7)-(10). Different final equations result from the different
ordering of small parameters and consistent perturbation approach up to first
or second order in small parameters.

In 2020, we extended their results to cases with an uneven bottom in KRcnsns ,
but still in (1+1)-dimensional theory. We showed that the terms originating
from the bottom have the same universal form for all these four nonlinear
equations. However, the validity of the obtained generalised wave equations is
limited to the cases when the bottom functions are piecewise linear. On the
other hand, the corresponding sets of the Boussinesq equations are valid for
the arbitrary form of the bottom functions.

In the present paper there are four small parameters,
$\alpha,\beta,\gamma,\delta$. In order to make calculations easier, we will
follow the idea from BS2013 ; KRcnsns , relating all small parameters to a
single one, called _leading parametr_. This method allows us for easier
control of the order of different terms, but the final forms of the obtained
equations are presented in original parameters $\alpha,\beta,\gamma,\delta$.
We discuss several cases, which are listed in Table 1. The table does not
contain all possible second-order cases, but only those that lead to well-
known KdV-type and Gardner equations when reduced to (1+1)-dimensions.

Table 1: Different ordering of small parameters considered in the paper. Case |  $\alpha$ |  $\beta\hskip 17.22217pt$ |  $\gamma$ |  $\delta$ | (1+1)-dim, $\delta=0$
---|---|---|---|---|---
1 | $O(\beta)$ | leading | $O(\beta)$ | $O(\beta)$ | KdV
2 | $O(\beta)$ | leading | $O(\beta)$ | $O(\beta^{2})$ | extended KdV
3 | $O(\beta^{2})$ | leading | $O(\beta^{2})$ | $O(\beta^{2})$ | 5th-ord KdV
3 | leading | $O(\alpha^{2})$ | $O(\alpha^{2})$ | $O(\alpha^{2})$ | Gardner

##  III Case 1: $\bm{\alpha}=O(\bm{\beta})$, $\bm{\gamma}=O(\bm{\beta})$,
$\bm{\delta}=O(\bm{\beta})$

Let us begin with the assumption that all small parameters are of the same
order. For easier control of the orders of different terms let us denote

$\alpha=A\,\beta,\quad\gamma=G\,\beta,\quad\delta=D\,\beta,$

where $A,G,D$ are arbitrary but close to 1.

Insertion the velocity potential (II) into the boundary condition at the
bottom (10) imposes the following relation

$\displaystyle F$
$\displaystyle=\beta^{2}D\left(h_{x}f_{x}+Gh_{y}f_{y}+h(f_{xx}+Gf_{yy})\right)$
(15) $\displaystyle\hskip
8.61108pt+\beta^{3}D^{2}\left(hh_{x}F_{x}+Ghh_{y}F_{y}+\frac{1}{2}h^{2}(F_{xx}+GF_{yy})\right)+O(\beta^{5}).$

Equation (15), when only terms up to second order in small parameters are
retained, that is

$F=\beta^{2}D\left(h_{x}f_{x}+Gh_{y}f_{y}+h(f_{xx}+Gf_{yy})\right)$ (16)

allows us to express $F$ by $f$ and reduce the set of unknown functions to
only $\eta$ and $f$. In higher order approxiation equation (15) cannot be
resolved for $F$. Then, retaining only terms up to second order, one obtains
the velocity potential, valid up to second-order approximation, expressed in
terms of a single unknown function $f(x,y,t)$ as

$\displaystyle\phi=f$
$\displaystyle-\frac{1}{2}\beta\,z^{2}(f_{xx}+Gf_{yy})+\frac{1}{24}\beta^{2}\,z^{4}(f_{xxxx}+2Gf_{xxyy}+G^{2}f_{yyyy})$
(17)
$\displaystyle+\beta^{2}z\,D\left(h_{x}f_{x}+Gh_{y}f_{y}+h(f_{xx}+Gf_{yy})\right).$

Now, we substitute the velocity potential (17) into (8) and (9), keeping terms
only up to first order. Due to the term $\frac{1}{\beta}\phi_{z}$ in (8) and
only second order valid expression for the potential (17) we can obtain valid
Boussinesq’s equations only in first order in $\alpha,\beta,\gamma,\delta$.

From (8) and (9) we get respectively

$\displaystyle\eta_{t}+f_{xx}+\frac{\gamma}{\beta}f_{yy}$
$\displaystyle+\alpha\left(\left(\eta
f_{x}\right)_{x}+\frac{\gamma}{\beta}\left(\eta
f_{y}\right)_{y}\right)-\frac{1}{6}\,\beta\left(f_{4x}+2\frac{\gamma}{\beta}f_{2x2y}+\left(\frac{\gamma}{\beta}\right)^{2}f_{4y}\right)$
$\displaystyle-\delta\left(\left(hf_{x}\right)_{x}+\frac{\gamma}{\beta}\left(hf_{y}\right)_{y}\right)=0\qquad\mbox{and}$
(18) $\displaystyle\eta+f_{t}$
$\displaystyle+\frac{1}{2}\alpha\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}\,\beta\left(f_{xxt}+\frac{\gamma}{\beta}f_{yyt}\right)=0.$
(19)

Equations (III)-(19) are the first order Boussinesq equations for
(2+1)-dimensional shallow water problem with an uneven bottom.

Let us check whether the reduction of the equations (III)-(19) to
(1+1)-dimensions leads to correct results. When we reduce these equations to
(1+1)-dimensional (assuming translational invariance in $y$ direction, that
is, setting all $y$-derivatives equal zero), then we arrive at

$\displaystyle\eta_{t}+f_{xx}$ $\displaystyle+\alpha\left(\eta
f_{x}\right)_{x}-\frac{1}{6}\,\beta f_{4x}-\delta\left(hf_{x}\right)_{x}=0,$
(20) $\displaystyle\eta+f_{t}$ $\displaystyle+\frac{1}{2}\alpha
f_{x}^{2}-\frac{1}{2}\,\beta f_{xxt}=0.$ (21)

Denoting $w=f_{x}$ and taking $x$-derivative of (21) one gets this set of
Boussinesq’s equation in the following form

$\displaystyle\eta_{t}+w_{x}$ $\displaystyle+\alpha\left(\eta
w\right)_{x}-\frac{1}{6}\,\beta w_{xxx}-\delta\left(hw\right)_{x}=0,$ (22)
$\displaystyle w_{t}+\eta_{x}$ $\displaystyle+\alpha ww_{x}-\frac{1}{2}\,\beta
w_{xxt}=0,$ (23)

identical to the equations (17)-(18) derived by us in KRcnsns . In KRcnsns ,
we also demonstrated that these two equations could be made compatible and
reduced to the KdV equation generalised for uneven bottom (KRcnsns, , Eq.
(29)), when the bottom profile $h(x)$ is a piecewise linear function. This
equation has the following form

$\eta_{t}+\eta_{x}+\frac{3}{2}\alpha\eta\eta_{x}+\frac{1}{6}\eta_{xxx}-\frac{1}{4}\delta(2h\eta_{x}+h_{x}\eta)=0.$
(24)

For the flat bottom, $\delta=0$, the equation (24) reduces to the usual
Korteweg-de Vries equation.

In Fok , the authors presented the derivation of two new (2+1)-dimensional
third- and fifth-order nonlinear evolution equations when
$\alpha,\beta,\gamma$ are of the same order and the bottom is flat
($\delta=0$). However, as we proved in RK21 , all results shown in Fok  are
false since the derivation is inconsistent and violates the fundamental
property of the velocity potential. When the method used by the authors is
applied consistently, the problem is reduced to well known KdV equation. For
details, see RK21 .

The Boussinesq equations (III)-(19) look very complicated. It is hard to
imagine how one can get a single equation for wave profile from them. On the
other hand, one can reduce this set to a single equation by inserting
$\eta=-\left[f_{t}+\frac{1}{2}\alpha\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}\,\beta\left(f_{xxt}+\frac{\gamma}{\beta}f_{yyt}\right)\right]$
determined from (19) into (III) and retaining only terms up to first order.
The result is

$\displaystyle f_{xx}+\frac{\gamma}{\beta}f_{yy}-f_{tt}$
$\displaystyle-\alpha\left[f_{t}\left(f_{xx}+\frac{\gamma}{\beta}f_{yy}\right)+\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\\!\right)_{t}\right]$
$\displaystyle-\beta\left[\frac{1}{6}\left(f_{4x}+2\frac{\gamma}{\beta}f_{2x2y}+\left(\frac{\gamma}{\beta}\right)^{2}f_{4y}\right)-\frac{1}{2}\left(f_{xxtt}+\frac{\gamma}{\beta}f_{yytt}\right)\right]$
(25)
$\displaystyle-\delta\left[\left(hf_{x}\right)_{x}+\frac{\gamma}{\beta}\left(hf_{y}\right)_{y}\right]=0.$

Equation (III) can be simplified utilizing zeroth-order solutions. In zeroth-
order we have from (III)-(19)

$\eta_{t}+f_{xx}+\frac{\gamma}{\beta}f_{yy}=0\qquad\quad\mbox{and}\qquad\quad\eta+f_{t}=0.$
(26)

So, $f_{t}=-\eta~{}$ and $f_{tt}=-\eta_{t}=f_{xx}+\frac{\gamma}{\beta}f_{yy}$.
Then we have

$f_{xxtt}=\frac{\partial^{2}}{\partial
x^{2}}\left(f_{xx}+\frac{\gamma}{\beta}f_{yy}\right)=f_{4x}+\frac{\gamma}{\beta}f_{xxyy},\quad
f_{yytt}=\frac{\partial^{2}}{\partial
y^{2}}\left(f_{xx}+\frac{\gamma}{\beta}f_{yy}\right)=f_{xxyy}+\frac{\gamma}{\beta}f_{4y},$

and the first-order term

$-\frac{1}{2}\left(f_{xxtt}+\frac{\gamma}{\beta}f_{yytt}\right)=-\frac{1}{2}\left(f_{4x}+2\frac{\gamma}{\beta}f_{2x2y}+\left(\frac{\gamma}{\beta}\right)^{2}f_{4y}\right)$

receives the same form like the term with the factor $\frac{1}{6}$ in front.
Then equation (III) receives the simpler form

$\displaystyle f_{xx}+\frac{\gamma}{\beta}f_{yy}-f_{tt}$
$\displaystyle-\alpha\left[f_{t}\left(f_{xx}+\frac{\gamma}{\beta}f_{yy}\right)+\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\\!\right)_{t}\right]$
$\displaystyle+\beta\left[\frac{1}{3}\left(f_{4x}+2\frac{\gamma}{\beta}f_{2x2y}+\left(\frac{\gamma}{\beta}\right)^{2}f_{4y}\right)\right]-\delta\left[\left(hf_{x}\right)_{x}+\frac{\gamma}{\beta}\left(hf_{y}\right)_{y}\right]=0.$
(27)

If the solution $f(x,y,t)$ to (III) is known, the equation (19) supplies the
surface profile function

$\eta(x,y,t)=-\left[f_{t}+\frac{1}{2}\alpha\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}\,\beta\left(f_{xxt}+\frac{\gamma}{\beta}f_{yyt}\right)\right].$
(28)

A preliminary idea how to look for solution to the equation (III) is presented
in the appendix B.

##  IV Case 2: $\bm{\alpha}=O(\bm{\beta})$, $\bm{\gamma}=O(\bm{\beta})$,
$\bm{\delta}=O(\bm{\beta}^{2})$

Denote now

$\alpha=A\,\beta,\quad\gamma=G\,\beta,\quad\delta=D\,\beta^{2}.$

In this case, the boundary condition at the bottom (10) imposes the following
relation

$\displaystyle F$
$\displaystyle=\beta^{3}D\left[(hf_{x})_{x}+G(hf_{y})_{y}\right]+\frac{1}{2}\beta^{5}D^{2}\left[(h^{2}F_{x})_{x}+G(h^{2}F_{y})_{y}\right]+O(\beta^{7}),$
(29)

which gives $F=\beta^{3}D\left[(hf_{x})_{x}+G(hf_{y})_{y}\right]$ valid up to
fourth order. Therefore, in this case, it is possible to obtain the Boussinesq
equations valid up to third order in small parameters.

From (8) we get (up to second order)

$\displaystyle\eta_{t}+f_{xx}+\frac{\gamma}{\beta}f_{yy}$
$\displaystyle+\alpha\left[(\eta f_{x})_{x}+\frac{\gamma}{\beta}(\eta
f_{y})_{y}\right]-\frac{1}{6}\beta\left[f_{4x}+2\frac{\gamma}{\beta}f_{2x2y}+\left(\frac{\gamma}{\beta}\right)^{2}f_{4y}\right]$
$\displaystyle-\frac{1}{2}\,\alpha\beta(\eta
f_{3x})_{x}-\frac{1}{2}\,\alpha\gamma\left[\eta_{x}f_{x2y}+2\eta
f_{2x2y}+\eta_{y}f_{2xy}\right]-\frac{1}{120}\,\beta^{2}f_{6x}$ (30)
$\displaystyle-\frac{1}{40}\,\beta\gamma\left[f_{4x2y}+\frac{\gamma}{\beta}\,f_{2x4y}+\frac{1}{3}\left(\frac{\gamma}{\beta}\right)^{2}\\!f_{6y}\right]-\delta\left[(hf_{x})_{x}+\frac{\gamma}{\beta}(hf_{y})_{y}\right]=0,$

whereas the result from (9), up to second order, is

$\displaystyle\eta+f_{t}$
$\displaystyle+\frac{1}{2}\alpha\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}\beta\left(f_{xxt}+\frac{\gamma}{\beta}f_{yyt}\right)+\alpha\beta\left[\frac{1}{2}(f_{xx}^{2}-f_{x}f_{3x})-\eta
f_{xxt}\right]$ $\displaystyle+\alpha\gamma\left[\eta
f_{yyt}+f_{xx}f_{yy}-\frac{1}{2}(f_{x}f_{xyy}+f_{y}f_{xxy})+\frac{\gamma}{\beta}(f_{yy}^{2}-f_{y}f_{3y})\right]$
(31)
$\displaystyle+\frac{1}{24}\beta^{2}\left[f_{4xt}+2\frac{\gamma}{\beta}f_{2x2yt}+\left(\frac{\gamma}{\beta}\right)^{2}\\!f_{4yt}\right]=0.$

Note that the term with $\delta$ in (IV), originating from the uneven bottom,
has identical form as such term in (III) although now it is of second order.
Let us rewrite (IV) in the form

$\displaystyle\eta\left(1\\!-\alpha\beta f_{xxt}-\\!\alpha\gamma
f_{yyt}\right)=-$
$\displaystyle\left\\{f_{t}+\frac{1}{2}\alpha\\!\left(\\!f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}\beta\\!\left(\\!f_{xxt}+\frac{\gamma}{\beta}f_{yyt}\right)+\alpha\beta\left[\frac{1}{2}(f_{xx}^{2}-f_{x}f_{3x})\right]\right.$
$\displaystyle+\alpha\gamma\left[f_{xx}f_{yy}-\frac{1}{2}(f_{x}f_{xyy}+f_{y}f_{xxy})+\frac{\gamma}{\beta}(f_{yy}^{2}-f_{y}f_{3y})\right]$
(32)
$\displaystyle\left.+\frac{1}{24}\beta^{2}\left[f_{4xt}+2\frac{\gamma}{\beta}f_{2x2yt}+\left(\frac{\gamma}{\beta}\right)^{2}\\!f_{4yt}\right]\right\\}.$

Multiplying both sides by $\left(1+\alpha\beta f_{xxt}+\alpha\gamma
f_{yyt}\right)$ and retaining only terms up to second order we obtain

$\displaystyle\eta=-$
$\displaystyle\left\\{f_{t}+\frac{1}{2}\alpha\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}\beta\left(\\!f_{xxt}+\frac{\gamma}{\beta}f_{yyt}\\!\right)+\alpha\beta\left[\frac{1}{2}(f_{xx}^{2}-f_{x}f_{3x})\right]\right.$
$\displaystyle+\alpha\gamma\left[f_{xx}f_{yy}-\frac{1}{2}(f_{x}f_{xyy}+f_{y}f_{xxy})+\frac{\gamma}{\beta}(f_{yy}^{2}-f_{y}f_{3y})\right]$
(33)
$\displaystyle\left.+\frac{1}{24}\beta^{2}\left[f_{4xt}+2\frac{\gamma}{\beta}f_{2x2yt}+\left(\frac{\gamma}{\beta}\right)^{2}\\!f_{4yt}\right]\right\\}+f_{t}\left(\alpha\beta
f_{xxt}+\alpha\gamma f_{yyt}\right).$

Substituting $\eta$ defined by (IV) into (IV) and keeping only the terms up to
second order allows us to obtain a nonlinear (2+1)-dimensional differential
equation for the function $f(x,y,t)$. This equation has the form (34). In (34)
we didn’t come back to original parameters $\alpha,\beta,\gamma,\delta$ for
easier recognition of first order and second order terms.

$\displaystyle
f_{xx}+Gf_{yy}-f_{tt}-AG(f_{tt}f_{yyt}+f_{t}f_{2y2t})-A(f_{tt}f_{2xt}+f_{t}f_{2x2t})$
$\displaystyle+\beta\bigg{[}-GA^{2}\left(f_{yyt}f_{x}f_{xt}+f_{t}f_{x}f_{x2yt}A^{2}+f_{t}f_{yyt}f_{xx}+f_{y}f_{xy}f_{2xt}+f_{t}f_{yy}f_{2xt}-f_{t}f_{y}f_{2xyt}+f_{t}f_{yy}\right)$
$\displaystyle\hskip
21.52771pt-G^{2}A^{2}\left(f_{y}f_{xy}f_{yyt}+f_{t}f_{yy}f_{yyt}+f_{t}f_{y}f_{3yt}\right)-A^{2}\left(f_{x}f_{xt}f_{2xt}-f_{t}f_{xx}f_{2xt}-f_{t}f_{x}f_{3xt}\right)$
$\displaystyle\hskip
21.52771pt-2GAf_{y}f_{xy}-2f_{x}f_{xt}-Af_{t}f_{xx}+\frac{1}{2}Gf_{2y2t}-\frac{1}{6}G^{2}f_{4y}+\frac{1}{2}f_{2x2t}-\frac{1}{3}Gf_{2x2y}-\frac{1}{6}f_{4x}\bigg{]}$
$\displaystyle+\beta^{2}\left[A^{2}G^{3}\left(f_{xy}f_{yyt}f_{3y}+f_{t}f_{3y}f_{3yt}+f_{t}f_{yyt}f_{4y}\right)+\frac{1}{120}G^{3}f_{6y}\right.$
$\displaystyle\hskip
21.52771pt+A^{2}G^{2}\left(-\frac{3}{2}f_{y}^{2}f_{yy}+\frac{1}{2}f_{yyt}f_{xt}f_{x2y}+\frac{1}{2}f_{t}f_{x2y}f_{x2yt}+\frac{1}{2}f_{xy}f_{3y}f_{2xt}+\frac{1}{2}f_{t}f_{4y}f_{2xt}\right.$
$\displaystyle\left.\hskip
21.52771pt+\frac{1}{2}f_{xy}f_{yyt}f_{2xy}+\frac{1}{2}f_{t}f_{3yt}f_{2xy}+\frac{1}{2}f_{t}f_{3y}f_{2xyt}+f_{t}f_{yyt}f_{2x2y}\right)$
$\displaystyle\hskip
21.52771pt+AG^{2}\left(-\frac{1}{2}f_{yy}f_{yyt}+f_{xy}f_{3y}+f_{y}f_{3yt}+\frac{1}{2}f_{t}f_{4y}\right)$
(34) $\displaystyle\hskip
21.52771pt+A^{2}G\left(-\frac{1}{2}f_{yy}f_{x}^{2}-\frac{1}{2}f_{y}^{2}f_{xx}+\frac{1}{2}f_{xt}f_{x2y}f_{2xt}+\frac{1}{2}f_{xy}f_{2xt}f_{2xy}+\frac{1}{2}f_{t}f_{2xy}f_{2xyt}+\frac{1}{2}f_{yyt}f_{xt}f_{3x}\right.$
$\displaystyle\left.\hskip
51.6665pt+\frac{1}{2}f_{t}f_{x2yt}f_{3x}+\frac{1}{2}f_{t}f_{x2y}f_{3xt}+\frac{1}{2}f_{t}f_{yyt}f_{4x}+f_{t}f_{2xt}f_{2x2y}-2f_{y}f_{x}f_{xy}\right)$
$\displaystyle\hskip
21.52771pt+G^{2}\left(-\frac{1}{24}f_{4y2t}+\frac{1}{40}f_{2x4y}\right)+DG\left(-h_{h}f_{y}-hf_{yy}\right)-D\left(hf_{x}+hf_{xx}\right)$
$\displaystyle\hskip
21.52771pt+AG\left(+f_{xt}f_{x2y}+f_{x}f_{x2yt}-\frac{1}{2}f_{yyt}f_{xx}-\frac{1}{2}f_{yy}f_{2xt}+f_{xy}f_{2xy}+f_{y}f_{2xyt}+f_{t}f_{2x2y}\right)$
$\displaystyle\hskip
21.52771pt+G\left(-\frac{1}{12}f_{2x2y2t}+\frac{1}{40}f_{4x2y}\right)+A\left(-\frac{1}{2}f_{xx}f_{2xt}+f_{xt}f_{3x}+f_{x}f_{3xt}+\frac{1}{2}f_{t}f_{4x}\right)$
$\displaystyle\left.\hskip
21.52771pt+A^{2}\left(\frac{1}{2}f_{xt}f_{2xt}f_{3x}+\frac{1}{2}f_{t}f_{3x}f_{3xt}+\frac{1}{2}f_{t}f_{2xt}f_{4x}\right)-\frac{1}{24}f_{4x2t}+\frac{1}{120}f_{6x}\right]=0.$

Equation (34) is the single nonlinear wave equation for auxiliary function $f$
determining the velocity potential, second order in small parameters for the
case $~{}\alpha=O(\beta),~{}\gamma=O(\beta),~{}\delta=O(\beta^{2})$.

In principle, if the solution $f(x,y,t)$ to (34) is known, the equation (IV)
supplies the surface profile function. The complexity of equations (IV) and
(34) may be the reason why there are so many small amplitude wrinkles and
ripples that are observed on the surface of seas and oceans. However, the
complexity of the equation (34) gives a little hope for finding the solution.

Let us check how the Boussinesq equations (IV)-(IV) look like when they are
reduced to (1+1)-dimensions. From (IV) we get (denoting $w=f_{x}$)

$\displaystyle\eta_{t}+w_{x}$ $\displaystyle+\alpha(\eta
w)_{x}-\frac{1}{6}\beta w_{3x}-\frac{1}{2}\,\alpha\beta(\eta
w_{2x})_{x}-\frac{1}{120}\,\beta^{2}w_{5x}-\delta(hw)_{x}=0,$ (35)

whereas from (IV) after differentiation over $x$ we obtain

$\displaystyle\eta_{x}+w_{t}$ $\displaystyle+\alpha ww_{x}-\frac{1}{2}\beta
w_{2xt}-\alpha\beta\left[\frac{1}{2}\left(w_{x}w_{2x}-ww_{3x}\right)-(\eta
w_{xt})_{x}\right]+\frac{1}{24}\beta^{2}w_{4xt}=0.$ (36)

The (1+1)-dimensional Boussinesq equations (35)-(36) are identical with
(KRcnsns, , Eqs. (37)-(38)), and for the case of flat bottom ($\delta=0$) are
identical with (BS2013, , Eqs. (11)-(12)). In the latter case these equations
lead to the extended KdV equation MS90 . So, the full (2+1)- dimensional
Boussinesq equations (IV)-(IV) seem to be correct.

##  V Case 3: $\bm{\alpha}=O(\bm{\beta}^{2})$,
$\bm{\gamma}=O(\bm{\beta}^{2})$, $\bm{\delta}=O(\bm{\beta}^{2})$

Denote now

$\alpha=A\,\beta^{2},\quad\gamma=G\,\beta^{2},\quad\delta=D\,\beta^{2}.$

Now, the boundary condition at the bottom (10) imposes the following relation

$\displaystyle F$
$\displaystyle=\beta^{3}D(hf_{x})_{x}+\beta^{4}DG(hf_{y})_{y}-\beta^{5}\frac{1}{2}D(h^{2}F_{x})_{x}+O(\beta^{6}),$
(37)

which gives $F=\beta^{3}D\left[(hf_{x})_{x}\right]$ and consequently the
velocity potential valid up to fourth order. Therefore, in this case it is
possible to obtain the Boussinesq equations valid up to third order in small
parameters. We will proceed up to second order which is enough complicated.
From (8) we obtain

$\displaystyle\eta_{t}+f_{xx}$ $\displaystyle-\frac{1}{6}\beta
f_{4x}+\frac{\gamma}{\beta}f_{yy}+\alpha(\eta
f_{x})_{x}-\frac{1}{120}\beta^{2}f_{6x}-\frac{1}{3}\gamma
f_{2x2y}-\delta(hf_{x})_{x}=0,$ (38)

whereas the result from (9) is

$\displaystyle\eta+f_{t}-\frac{1}{2}\beta f_{xxt}+\frac{1}{2}\alpha
f_{x}^{2}+\frac{1}{24}\beta^{2}f_{4xt}-\frac{1}{2}\gamma f_{yyt}=0.$ (39)

In (39), we didn’t take into account terms from surface tension (setting
$\tau=0$ in (9)). This omission is fully justified when water depth is on the
order of meters. Now, $\eta$ determined from (39) is substituted into (38).
Limiting this equation up to second order one obtains

$\displaystyle f_{xx}-f_{tt}$
$\displaystyle+\frac{\gamma}{\beta}f_{yy}+\beta\left(\frac{1}{2}f_{xxtt}-\frac{1}{6}f_{4x}\right)-\beta^{2}\left(\frac{1}{24}f_{4x2t}+\frac{1}{120}f_{6x}\right)$
(40)
$\displaystyle+\gamma\left(-\frac{1}{3}f_{xxyy}+\frac{1}{2}f_{yytt}\right)-\alpha(2f_{x}f_{xt}+f_{t}f_{xx})-\delta(hf_{x})_{x}=0.$

Equation (40) is the single nonlinear wave equation for auxiliary function $f$
determining the velocity potential, second order in small parameters for the
case $~{}\alpha=O(\beta^{2}),~{}\gamma=O(\beta^{2}),~{}\delta=O(\beta^{2})$.

For a check, let us reduce the Boussinesq equations (38)-(39) to
(1+1)-dimensional case setting $\gamma=0$. Then, after differentiating (39)
over $x$ and denoting $w=f_{x}$ we obtain the set

$\displaystyle\eta_{t}+w_{x}$ $\displaystyle-\frac{1}{6}\beta
w_{3x}+\alpha(\eta
w)_{x}-\frac{1}{120}\beta^{2}w_{5x}-\delta(hf_{x})_{x}=0\qquad\mbox{and}$ (41)
$\displaystyle w_{t}+\eta_{x}$ $\displaystyle-\frac{1}{2}\beta w_{xxt}+\alpha
ww_{x}+\frac{1}{24}\beta^{2}w_{4xt}=0.$ (42)

This set of equations, for $\delta=0$, is identical to the equations (29)-30)
in BS2013 , limited to second order. Eliminating by the standard method the
function $w$, one obtains from them the well-known fifth-order KdV equation
when $\delta=0$, i.e., for the flat bottom. When $\delta\neq 0$, the final
wave equation is the fifth-order KdV generalised for the uneven bottom
(KRcnsns, , Eq. (68)), in which $\tau=0$.

###  V.1 Case 5a: $\bm{\alpha}=O(\bm{\beta}^{2})$,
$\bm{\gamma}=O(\bm{\beta}^{2})$, $\bm{\delta}=O(\bm{\beta}^{2})$,
$\bm{\tau}\neq 0$

For thin fluid layers, surface tension may introduce significant changes in
the dynamics. Using surface tension term in the form (11) in dynamical
boundary condition at the surface (9), that is

$\displaystyle\phi_{t}+\frac{1}{2}\alpha\left(\phi_{x}^{2}+\frac{\gamma}{\beta}\phi_{y}^{2}+\frac{1}{\beta}\phi_{z}^{2}\right)+\eta-\tau\,(\beta\eta_{xx}+\gamma\eta_{yy})$
$\displaystyle=0,\quad\mbox{for}\quad z=1+\alpha\eta,$ (43)

we obtain instead (39) more complicated equation

$\displaystyle\eta+f_{t}-\beta\left(\frac{1}{2}f_{xxt}+\tau\eta_{xx}\right)+\frac{1}{2}\alpha
f_{x}^{2}+\frac{1}{24}\beta^{2}f_{4xt}-\gamma\left(\frac{1}{2}f_{yyt}+\tau\eta_{yy}\right)=0.$
(44)

In this case, because we can not express $\eta$ trough only $f$ and its
derivatives, we can not reduce the Boussinesq equations (38)-(44) to a single
equation for the function $f$. However, limiting to (1+1)-dimensions (as usual
setting $\gamma=0$, $w=f_{x}$ and differentiating (44) over $x$) we obtain

$\displaystyle\eta_{t}+w_{x}$ $\displaystyle-\frac{1}{6}\beta
w_{3x}+\alpha(\eta
w)_{x}-\frac{1}{120}\beta^{2}w_{5x}-\delta(hf_{x})_{x}=0\qquad\mbox{and}$ (45)
$\displaystyle w_{t}+\eta_{x}$
$\displaystyle-\beta\left(\frac{1}{2}w_{xxt}+\tau\eta_{xxx}\right)+\alpha
ww_{x}+\frac{1}{24}\beta^{2}w_{4xt}=0.$ (46)

These equations are identical to (KRcnsns, , Eqs. (61)-(62)), which lead to
the fifth-order KdV equation generalised for an uneven bottom (KRcnsns, , Eq.
(68)).

##  VI Case 4: $\bm{\beta}=O(\bm{\alpha}^{2})$,
$\bm{\gamma}=O(\bm{\alpha}^{2})$, $\bm{\delta}=O(\bm{\alpha}^{2})$

Denote now

$\beta=B\,\alpha^{2},\quad\gamma=G\,\alpha^{2},\quad\delta=D\,\alpha^{2}.$

Now, the boundary condition at the bottom (10) imposes the following relation

$\displaystyle F$
$\displaystyle=\alpha^{4}(BD(hf_{x})_{x}+DG(hf_{y})_{y})+\alpha^{6}\frac{1}{2}(BD^{2}(h^{2}F_{x})_{x}+GD^{2}(hf_{y})_{y})+O(\alpha^{8}),$
(47)

which gives $F=\alpha^{4}(BD(hf_{y})_{y}+DG(hf_{y})_{y})$ and consequently the
velocity potential valid up to fourth order in the leading parameter.
Therefore, in this case, it is possible to obtain the Boussinesq equations
valid up to third order in small parameters. We will proceed up to the second
order, which is complicated enough. Then we can safely neglect higher order
terms in surface tension term (11) and use the dynamic boundary condition in
the form (43). From kinematic boundary condition at the surface (8), we obtain

$\displaystyle\eta_{t}+f_{xx}+\frac{\gamma}{\beta}f_{yy}$
$\displaystyle+\alpha\left((\eta f_{x})_{x}+\frac{\gamma}{\beta}(\eta
f_{y})_{y}\right)-\frac{1}{6}\beta\left(f_{4x}+\frac{\gamma^{2}}{\beta^{2}}f_{4y}\right)$
$\displaystyle-\frac{1}{3}\gamma
f_{2x2y}-\delta\left((hf_{x})_{x}+\frac{\gamma}{\beta}(hf_{y})_{y}\right)=0,$
(48)

whereas, from the dynamic boundary condition at the surface (43), the result
is

$\displaystyle\eta+f_{t}+\frac{1}{2}\alpha\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}(\beta
f_{xxt}+\gamma f_{yyt})-\tau(\beta\eta_{xx}+\gamma\eta_{yy})=0.$ (49)

Equations (VI)-(49) are the Boussinesq equations for the case when
$\beta\approx\gamma\approx\delta=O(\alpha^{2})$.

When $\tau$ is of the order of 1, what occurs for thin fluid layers, equation
(49) can not be resolved to obtain $\eta=G(f,f_{x},f_{y},f_{{t}},\ldots)$,
that is, as a function of $f$ and its derivatives only. Therefore, in this
case, the Boussinesq set (VI)-(49) can not be reduced to a single equation for
$f$ function.

Limiting equations (VI)-(49) to (1+1)-dimnesions by setting all
$y$-derivatives equal to zero (alternatively setting $\gamma=0$) we obtain
from (VI)

$\eta_{t}+w_{x}+\alpha(\eta w)_{x}-\frac{1}{6}\beta w_{xxx}-\delta(hw)_{x}=0.$
(50)

From (49), after setting $\gamma=0$ and differentiating over $x$ we have

$\eta_{x}+w_{t}+\alpha ww_{x}-\frac{1}{2}\beta w_{xxt}-\tau\beta\eta_{xxx}=0.$
(51)

Equations (50)-(51) are identical with equations (85)-(86) from KRcnsns , when
the latter are limited to second order. Therefore, one can make them
compatible and derive a single wave equation for wave profile function
$\eta(x,t)$. Such procedure leads to the equation identical with the equation
(91) from KRcnsns , which is the Gardner equation generalised for an uneven
bottom. For $\delta=0$, that is for the flat bottom, equation (50) receives
the form (KRcnsns, , Eq. (92)), the usual Gardner equation.

The reduction of the equations (VI)-(49) to the equation for a single unknown
function $f$ is possible for cases when the depth of the fluid is of the order
of meters. Then $\tau$ is so small that terms
$-\tau(\beta\eta_{xx}+\gamma\eta_{yy})$ in (49) can be safely neglected. In
such case substituting
$\eta=-\left[f_{t}+\frac{1}{2}\left(f_{x}^{2}+\frac{\gamma}{\beta}f_{y}^{2}\right)-\frac{1}{2}(\beta
f_{xxt}+\gamma f_{yyt})\right]$ obtained from (49) with $\tau=0$ into (VI) we
obtain a single nonlinear wave equation for the function $f(x,y,t)$ in the
form

$\displaystyle f_{xx}+\frac{\gamma}{\beta}f_{yy}-f_{tt}$
$\displaystyle-\alpha\left[(2f_{x}f_{xt}+f_{t}f_{xx})+\frac{\gamma}{\beta}(2f_{y}f_{yt}+f_{t}f_{yy})\right]-\delta\left[(hf_{x})_{x}+(hf_{y})_{y}\right]$
$\displaystyle-\frac{1}{6}\beta\left[f_{4x}+\left(\frac{\gamma}{\beta}\right)^{2}f_{4y}\right]+\frac{1}{2}\beta
f_{xxtt}+\frac{1}{2}\gamma f_{yytt}-\frac{1}{3}\gamma f_{xxyy}$ (52)
$\displaystyle-\alpha^{2}\left[\frac{3}{2}\left(f_{x}^{2}f_{xx}+\left(\frac{\gamma}{\beta}\right)^{2}f_{y}^{2}f_{yy}\right)+\frac{1}{2}\frac{\gamma}{\beta}(f_{x}^{2}f_{yy}+f_{y}^{2}f_{xx})+4f_{x}f_{y}f_{xt})\right]=0.$

Equation (VI) is the single nonlinear wave equation for auxiliary function $f$
determining the velocity potential, second order in small parameters for the
case $~{}\beta=O(\alpha^{2}),~{}\gamma=O(\alpha^{2}),~{}\delta=O(\alpha^{2})$,
when surface tension effects are negligible $(\tau\ll 1)$.

##  VII Conclusions

In the model of an ideal fluid, we have derived several sets of Boussinesq’s
equations for (2+1)-dimensional nonlinear waves. When restricted to
(1+1)-dimensions, the cases discussed include the Korteweg-de Vries equation,
extended KdV, fifth-order KdV, and Gardner equations, both for a flat and an
uneven bottom. Even though these Boussinesq’s equations are only first or
second order in small parameters, they are too complicated in (2+1)-dimensions
to obtain single wave equations for surface profile function $\eta(x,y,t)$.

The Boussinesq equations, in each case, constitute a set of coupled partial
differential equations for two functions, $\eta(x,y,t)$ \- surface profile
function and the function $f(x,y,t)$ which determines the velocity potential
through the equation (13). For (1+1)-dimensions it is always possible to
eliminate $f(x,y,t)$ from the Bouusinesq set and arrive to a single wave
equation of the KdV-type for the wave profile function $\eta(x,y,t)$. It seems
to be impossible for (2+1)-dimensional cases due to the complexity of these
equations. However, for each (2+1)-dimensional case, it is possible to
eliminate the function $\eta(x,y,t)$ and obtain a single nonlinear wave
equation for the function $f(x,y,t)$. The disadvantage of this result is that
the function $f(x,y,t)$ is not a directly observed quantity. Only by knowing
the $f(x,y,t)$ solution can one construct a solution of a wave profile
$\eta(x,y,t)$.

Another possible approach is based on a consistent application of perturbation
theory. We start with zero-order solutions. We then use the first-order
Boussinesq equations to determine the first-order corrections, and if
necessary, we determine further corrections from the second-order Boussinesq
equations. In each of these steps, one has to solve several inhomogeneous wave
equations. This approach seems to be the most promising.

##  Appendix A Surface tension terms

In the original, dimension variables the term originating from additional
pressure due to the surface tension has the following form AN2010

$\text{ST}=-\frac{T}{\varrho}\;\nabla\cdot\left(\frac{\nabla\eta}{(1+|\nabla\eta|^{2})^{1/2}}\right)=-\frac{T}{\varrho}\frac{(1+\eta_{x}^{2})\eta_{yy}+(1+\eta_{y}^{2})\eta_{xx}-2\eta_{x}\eta_{y}\eta_{xy}}{(1+\eta_{x}^{2}+\eta_{y}^{2})^{3/2}}.$
(53)

In (53), $\nabla=(\partial_{x},\partial_{y})$ is two-dimensional gradient
operator, and $T$ is fluid’s surface tension coefficient. After transformation
(6) to non-dimensional scaled variables we obtain (tildas are again dropped)

ST $\displaystyle=-\frac{T}{\varrho
gH^{2}}\;\frac{\beta\left(1+\alpha^{2}\beta\eta_{y}^{2}\right)\eta_{xx}+\gamma\left(1+\alpha^{2}\beta\eta_{x}^{2}\right)\eta_{yy}-2\alpha^{2}\beta\gamma\eta_{x}\eta_{y}\eta_{xy}}{\left(1+\alpha^{2}\beta\eta_{x}^{2}+\alpha^{2}\gamma\eta_{y}^{2}\right)^{3/2}}$
$\displaystyle=-\tau\left[\beta\left(1+\alpha^{2}\beta\eta_{y}^{2}\right)\eta_{xx}+\gamma\left(1+\alpha^{2}\beta\eta_{x}^{2}\right)\eta_{yy}-2\alpha^{2}\beta\gamma\eta_{x}\eta_{y}\eta_{xy}\right]$
$\displaystyle\hskip
25.83325pt\times\left(1-\frac{3}{2}\alpha^{2}\beta\eta_{x}^{2}-\frac{3}{2}\alpha^{2}\gamma\eta_{y}^{2}+O(\beta^{6})\right)$
(54) $\displaystyle=-\tau(\beta\eta_{xx}+\gamma\eta_{yy}+O(\beta^{4})).$

In (A), we first expand the factor
$1/\left(1+\alpha^{2}\beta\eta_{x}^{2}+\alpha^{2}\gamma\eta_{y}^{2}\right)^{3/2}$
assuming that terms $\alpha^{2}\beta\eta_{x}^{2}$ and
$\alpha^{2}\gamma\eta_{y}^{2}$ are small.

Since we are interested in Boussinesq’s equations up to second order in small
parameters, it is enough to use the surface term in the form

$\text{ST}=-\tau\,(\beta\eta_{xx}+\gamma\eta_{yy}).$ (55)

##  Appendix B First step towards solving the equation (III)

We will try to find solution to equation (III) in two steps. First, assume
that the function $f^{(0)}$ fulfils the equation (III) reduced to zeroth
order, that is, the following holds

$f^{(0)}_{xx}+\frac{\gamma}{\beta}f^{(0)}_{yy}-f^{(0)}_{tt}=0.$ (56)

Next, postulate the solution to (III) in the form

$f=f^{(0)}+\alpha\,a(x,y,t)+\beta\,b(x,y,t)+\gamma\,g(x,y,t)+\delta\,d(x,y,t),$
(57)

where $a,b,g,d$ are first order correction functions. Inserting $f$ in the
form (57) into (III) and retaining terms up to first order in small parameters
yields

$\displaystyle f^{(0)}_{xx}+\frac{\gamma}{\beta}f^{(0)}_{yy}-f^{(0)}_{tt}$
$\displaystyle+\alpha\left(a_{xx}+\frac{\gamma}{\beta}a_{yy}-a_{tt}-2f^{(0)}_{x}f^{(0)}_{xt}-f^{(0)}_{t}f^{(0)}_{xx}-\frac{\gamma}{\beta}\left(2f^{(0)}_{y}f^{(0)}_{yt}+f^{(0)}_{t}f^{(0)}_{yy}\right)\right)$
$\displaystyle+\beta\left(b_{xx}+\frac{\gamma}{\beta}b_{yy}-b_{tt}+\frac{1}{2}f^{(0)}_{xxtt}-\frac{1}{6}f^{(0)}_{4x}\right)$
(58)
$\displaystyle+\delta\left(d_{xx}+\frac{\gamma}{\beta}d_{yy}-d_{tt}-\frac{\gamma}{\beta}\left(h\,f^{(0)}_{y}\right)_{y}-\left(h\,f^{(0)}_{x}\right)_{x}\right)$
$\displaystyle+\gamma\left(g_{xx}+\frac{\gamma}{\beta}g_{yy}-g_{tt}-\frac{\gamma}{\beta}f^{(0)}_{4y}+\frac{1}{2}f^{(0)}_{yytt}-2f^{(0)}_{xxyy}\right)=0.$

Since $f^{(0)}$ fulfils zeroth order equation (56) and small parameters can be
arbitrary, the condition (B) is equivalent to four inhomogeneous wave
equations for the correction functions

$\displaystyle a_{xx}+\frac{\gamma}{\beta}a_{yy}-a_{tt}$
$\displaystyle=2f^{(0)}_{x}f^{(0)}_{xt}+f^{(0)}_{t}f^{(0)}_{xx}+\frac{\gamma}{\beta}\left(2f^{(0)}_{y}f^{(0)}_{yt}+f^{(0)}_{t}f^{(0)}_{yy}\right),$
(59) $\displaystyle b_{xx}+\frac{\gamma}{\beta}b_{yy}-b_{tt}$
$\displaystyle=-\frac{1}{2}f^{(0)}_{xxtt}+\frac{1}{6}f^{(0)}_{4x},$ (60)
$\displaystyle g_{xx}+\frac{\gamma}{\beta}g_{yy}-g_{tt}$
$\displaystyle=\frac{\gamma}{\beta}f^{(0)}_{4y}-\frac{1}{2}f^{(0)}_{yytt}+2f^{(0)}_{xxyy},$
(61) $\displaystyle d_{xx}+\frac{\gamma}{\beta}d_{yy}-d_{tt}$
$\displaystyle=\frac{\gamma}{\beta}\left(h\,f^{(0)}_{y}\right)_{y}+\left(h\,f^{(0)}_{x}\right)_{x}.$
(62)

For flat bottom case, $~{}\delta=0$, equation (62) does not appear.

If solutions to equations (56), (59)-(62) are known, then the time dependent
surface profile is given by the equation (28).

## References

  * (1) Korteweg, D.J., de Vries, H: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philosophical Magazine 39, 422-443 (1895).
  * (2) Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Dokl Akad Nauk SSSR 192, 753 (1970). [Sov. Phys. Dokl. 15, 539 (1970)].
  * (3) Wazwaz, A.-M.: Single and multiple-soliton solutions for the (2+1)-dimensional KdV equation. Appl Math Comput 204 20-26 (2008).
  * (4) Wazwaz, A.-M.: Four (2+1)-dimensional integrable extensions of the KdV equation: Multiple-soliton and multiple singular soliton solutions. Applied Mathematics and Computation 215 1463–1476 (2009).
  * (5) Peng, Y.-Z.: A New (2+1)Dimensional KdV Equation and Its Localized Structures. Communications in Theoretical Physics 54 863-865 (2010).
  * (6) Wazwaz, A.-M.: A new (2+1)-dimensional Korteweg–de Vries equation and its extension to a new (3+1)-dimensional Kadomtsev–Petviashvili equation, Physica Scripta, 84 035010 (2011).
  * (7) Wang, Z., Zou, L., Zonh, Z., Qin, H.: A Family of Novel Exact Solutions to 2+1-Dimensional KdV Equation. Abstract and Applied Analysis 2014 764750 (2014).
  * (8) Adem, A.R.: A (2+1)-dimensional Korteweg–de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws. Int J Mod Phys B 30 1640001 (2016).
  * (9) Zhang, X., Chen, Y.: Deformation rogue wave to the (2+1)-dimensional KdV equation. Nonlinear Dynamics 90 755–763 (2017).
  * (10) Batwa, S., Ma, W.-X.: Lump Solutions to a (2+1)-Dimensional Fifth-Order KdV-Like Equation. Adv Math Phys 2018 2062398 (2018).
  * (11) Wanga, G., Karab, A.H.: A (2+1)-dimensional KdV equation and mKdV equation: Symmetries, group invariant solutions and conservation laws, Physics Letters A, bf 383 728–731 (2019)
  * (12) Lou, S.-Y.: A novel (2+1)-dimensional integrable KdV equation with peculiar solution structures. arXiv:2001.08571.
  * (13) Fokou, M., Kofane, T.C., Mohamadou, A., Yomba, E.: Two- dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension. Nonlinear Dynamics 91 1177–1189 (2018).
  * (14) Rozmej, P., Karczewska, A.: Comment on “Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]. Nonlinear Dynamics, 105 (2021). https://doi.org/10.1007/s11071-021-06716-5, Preprint https://arxiv.org/abs/2105.08519 .
  * (15) Burde, G.I., Sergyeyev, A.: Ordering of two small parameters in the shallow water wave problem. J Phys A Math Theor 46 075501 (2013).
  * (16) Karczewska, A., Rozmej, P.: Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry? Commun Nonlinear Sci Numer Simulat 82 105073 (2020).
  * (17) Akers, B., Nicholls, D.P.: Travelling waves in deep water with gravity and surface tension. SIAM J Appl Math 70 2373-2389, (2010).
  * (18) Karczewska, A., Rozmej, P., Infeld, E.: Shallow-water soliton dynamics beyond the Korteweg–de Vries equation. Physical Review E 90  012907 (2014).
  * (19) Marchant, T.R., Smyth, N.F.: _The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography_ , Journal of Fluid Mechanics, 221, 263-288 (1990).