Remarks regarding some Algebras of Logic

Mariana Floricica Călin, Cristina Flaut, Dana Piciu

Abstract. Algebras of Logic deal with some algebraic structures, often bounded
lattices, considered as models of certain logics, including logic as a domain
of order theory. There are well known their importance and applications in
social life to advance useful concepts, as for example computer algebra.

In this paper we reffer in specially to BL-algebras and we present properties
of finite rings or rings with a finite number of ideals in their connections
with BL-rings.

Keywords: Algebras of Logic, BL-algebras, BL-rings

AMS Classification: 03G10, 03G25, 06A06, 06D05,

08C05, 06F35.

0\. Introduction

Algebras of Logic are explicit algebraic systems that show the basic
mathematical structure of Logic. These algebras deal with some algebraic
structures, often bounded lattices, considered as models of certain logics,
including logic as a domain of order theory. There are well known their
importance and applications in social life to advance useful concepts, as for
example computer algebra, also called symbolic computation or algebraic
computation, which it is a scientific area that refers to the study and
development of algorithms or/and software used for computing mathematical
expressions and manipulating other mathematical objects, as well as in the
study of truth values from social analyses. In the following, we will
exemplify some of the applications of Algebras of Logic.

Basic Logic (BL, for short) was introduced by Hajek in [H; 98] to formalize
the many-valued semantics induced on the real interval $[0,1]$ by a continuous
t-norm. Basic Logic generalizes the three most used logics in the theory of
fuzzy sets: Łukasiewicz logic, Gődel logic and Product logic. BL-algebras are
Lindenbaum-Tarski algebras for Basic Logic.

In this paper we refer especially to BL-algebras and BL-rings.

BL-rings are commutative unitary rings whose lattice of ideals can be equipped
with a structure of BL-algebra. We present properties of finite rings or rings
with a finite number of ideals and their connections with BL-rings. We give
some examples in this purpose and we present in Conclusions directions of
research to find classifications of such rings.

In this paper, all considered rings are commutative and unitary rings.

1\. Preliminaries

Let $R$ be a commutative unitary ring. The set $Id\left(R\right)$ denotes the
set of all ideals of the ring $R$. Let $I,J\in Id\left(R\right)$. The
following sets are also ideals in $R:$

$I+J=<I\cup J>=\\{i+j,i\in I,j\in J\\}\text{, }$ $I\otimes
J=\\{\underset{k=1}{\overset{n}{\sum}}i_{k}j_{k},\text{ }i_{k}\in I,j_{k}\in
J\\}\text{, }$ $\left(I:J\right)=\\{x\in R,x\cdot J\subseteq I\\}\text{,}$
$Ann\left(I\right)=\left(\mathbf{0}:I\right)\text{, where
}\mathbf{0}=<0>,\text{ }$

called sum, product, quotient and annihilator of the ideal $I$.

Definition 1.1. ([AM; 69]) We consider $I,J\in Id\left(R\right)$. The ideals
$I$ and $J$ are called coprime if $I+J=R$, that means there are $i\in I$,
$j\in J$ such that $i+j=1$.

Remark 1.2. If $R_{1}$ and $R_{2}$ are commutaive rings and $I\,_{1}$ is an
ideal in $R_{1}$ and $I_{2}$ is an ideal in $R_{2},$ then $I_{1}\times I_{2}$
is an ideal in the ring $R=R_{1}\times R_{2}$. Moreover, each ideal in the
ring $R$ is on the form $I\times J$, where $I$ is an ideal in $R_{1}$ and $J$
is an ideal in $R_{2}$.

Proposition 1.3. ([AM; 69], The Chinese Remainder Theorem for Rings) If $R$ is
a commutative ring and $I,J$ two ideals in $R$ such that $I+J=R$, we have
$I\otimes J=I\cap J$. Then

$\frac{R}{I\otimes J}=\frac{R}{I\cap J}\simeq\frac{R}{I}\times\frac{R}{J}.$

$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proposition 1.4. 1) If $R_{i},i\in\\{1,2,...,s\\}$ are commutative unitary
rings and $I_{i}$ is an ideal in the ring $R_{i}$, if we consider
$R=R_{1}\times...\times R_{s}$ and $I=I_{1}\times...\times I_{s}$ an ideal in
the ring $R,$ then

$\frac{R}{I}\simeq\frac{R_{1}}{I_{1}}\times...\times\frac{R_{s}}{I_{s}}\text{.
}$

2) For a commutative ring $R$, we have $R\left[X\right]\times
R\left[X\right]\simeq(R\times R)\left[X\right]$.

3) If $\left(m,n\right)=1$, then
$\mathbb{Z}_{m}\left[X\right]\times\mathbb{Z}_{n}\left[X\right]\simeq(\mathbb{Z}_{m}\times\mathbb{Z}_{n})\left[X\right]=\mathbb{Z}_{mn}\left[X\right]$.

4)

$\frac{\mathbb{Z}\left[X\right]}{\left(n\right)}\simeq(\mathbb{Z}/\left(n\right))[X].$

5) If $I=I_{1}\times I_{2}$ is an ideal in the ring $R\left[X\right]\times
R\left[X\right]$, then

$(R\left[X\right]\times
R\left[X\right])/I\simeq\frac{R\left[X\right]}{I_{1}}\times\frac{R\left[X\right]}{I_{2}}.$

Proof. 1) It is clear from the Fundamenthal Theorem of Isomorphism,
considering the surjective morphism $\phi:$
$\frac{R}{I}\rightarrow\frac{R_{1}}{I_{1}}\times...\times\frac{R_{s}}{I_{s}},\phi\left(x_{1},...,x_{s}\right)=\left(\widehat{x_{1}},...,\widehat{x_{s}}\right)$.

2) Let $f:R\left[X\right]\times R\left[X\right]\simeq(R\times
R)\left[X\right],f\left(p,q\right)=Q[X],$ where
$p=p_{0}+p_{1}X+...+p_{m}X^{m}$, $q=q_{0}+q_{1}X+...+q_{m}X^{m}$, with $q_{m}$
or $p_{m}\,\ $possible zero and
$Q[X]=\left(p_{0},q_{0}\right)+\left(p_{1},q_{1}\right)X+...+\left(p_{m},q_{m}\right)X^{m}$.
It is clear that $f$ is a bijective morphism.

3) Indeed, let $p\left(X\right)\in\mathbb{Z}\left[X\right]$ be a polynomial
and $f:$
$\mathbb{Z}_{mn}\left[X\right]\rightarrow\mathbb{Z}_{m}\left[X\right]\times\mathbb{Z}_{n}\left[X\right],$

$f\left(\overline{p\left(X\right)}\right)=\left(\widehat{p_{m}\left(X\right)},\widetilde{p_{n}\left(X\right)}\right)$,
where $\overline{p\left(X\right)}$ is the polynomial $p\left(X\right)$ with
the coefficients reduced mod $mn$, $\widehat{p_{m}\left(X\right)}~{}$ is the
polynomial $p\left(X\right)$ with the coefficients reduced mod $m$ and
$\widetilde{p_{n}\left(X\right)}$ is the polynomial $p\left(X\right)$ with the
coefficients reduced mod $n$. We have,
$f\left(\overline{p_{1}}+\overline{p_{2}}\right)=f\left(\overline{p_{1}}\right)+f\left(\overline{p_{2}}\right)$
and
$f\left(\overline{p_{1}}\overline{p_{2}}\right)=f\left(\overline{p_{1}}\right)f\left(\overline{p_{2}}\right)$.

4) We have $\pi:\mathbb{Z\rightarrow Z}/\left(n\right)$, canonical projection
and
$i:\mathbb{Z}/\left(n\right)\rightarrow\mathbb{Z}/\left(n\right)\left[X\right]$
canonical injection. Therefore, from universal propertie of polinomial rings,
we have the morphism ring
$f:\mathbb{Z}\left[X\right]\rightarrow\mathbb{Z}/\left(n\right)\left[X\right]$
with ker$f=n\mathbb{Z}\left[X\right]=\left(n\right)=\\{$ the set of
polynomials with coefficients multiple of $n\\}$. It is clear that $f$ is
surjective. Therefore, from Isomorphism Fundamental Theorem, we have that
$\frac{\mathbb{Z}\left[X\right]}{\left(n\right)}\simeq(\mathbb{Z}/\left(n\right))[X]$.

5) It is clear from 1). $\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Definition 1.5. ([WD; 39]) A (commutative) residuated lattice  is an algebra
$(L,\wedge,\vee,\odot,\rightarrow,0,1)$ such that:

(1) $\ (L,\wedge,\vee,0,1)$ is a bounded lattice;

(2) $\ \ (L,\odot,1)$ is a commutative ordered monoid;

(3) $\ z\leq x\rightarrow y$ iff $x\odot z\leq y,$ for all $x,y,z\in L.$

The property (3) is called __ residuation, where $\leq$ is the partial order
of the lattice $\ (L,\wedge,\vee,0,1).$

In a residuated lattice is defined an additional operation: for $x\in L,$ we
denote $x^{\ast}=x\rightarrow 0.$

If we preserve these notations, for a commutative and unitary ring we have
that $(Id(R),\cap,+,\otimes\rightarrow,0=\\{0\\},1=R)$ is a residuated lattice
in which the order relation is $\subseteq$, $I\rightarrow J=(J:I)$ and $I\odot
J=I\otimes J,$ for every $I,J\in Id(R)$, see [TT;22]

In a residuated lattice $(L,\wedge,\vee,\odot,\rightarrow,0,1)$ we consider
the following identities:

$(prel)\qquad(x\rightarrow y)\vee(y\rightarrow x)=1\qquad\text{
({prelinearity)}};$ $(div)\qquad x\odot(x\rightarrow y)=x\wedge y\qquad\text{
({divisibility)}}.$

Definition 1.6. ([T; 99]) 1) A residuated lattice $L$ is called a BL-algebra
if $L$ verifies $(prel)$ and $(div)$ conditions.

2) A BL-chain __ is a totally ordered BL-algebra, i.e., a BL-algebra such that
its lattice order is total.

Definition 1.7. ([CHA; 58]) An MV-algebra is an algebra
$\left(L,\oplus,^{\ast},0\right)$ satisfying the following axioms:

(1) $\left(L,\oplus,0\right)$ is an abelian monoid;

(2) $(x^{\ast})^{\ast}=x;$

(3) $x\oplus 0^{\ast}=0^{\ast};$

(4) $\left(x^{\ast}\oplus y\right)^{\ast}\oplus y=$ $\left(y^{\ast}\oplus
x\right)^{\ast}\oplus x$, for all $x,y\in L.\vskip 6.0pt plus 2.0pt minus
2.0pt$

Remark 1.8. If in a BL- algebra $L$ we have $x^{\ast\ast}=x,$ for every $x\in
L$, and, for $x,y\in L,$ we denote

$x\oplus y=(x^{\ast}\odot y^{\ast})^{\ast},$

then we obtain an MV-algebra structure $(L,\oplus,^{\ast},0).$ Conversely, if
$(L,\oplus,^{\ast},0)$ is an MV-algebra, then
$(L,\wedge,\vee,\odot,\rightarrow,0,1)$ becomes a BL-algebra, in which for
$x,y\in L$ we have$:$

$x\odot y=(x^{\ast}\oplus y^{\ast})^{\ast},$ $x\rightarrow y=x^{\ast}\oplus
y,1=0^{\ast},$ $x\vee y=(x\rightarrow y)\rightarrow y=(y\rightarrow
x)\rightarrow x\text{ and }x\wedge y=(x^{\ast}\vee y^{\ast})^{\ast},$

see [T; 99].

We recall that in [NL; 03], Di Nolla and Lettieri analyze the structure of
finite BL-algebras. They introduced the concept of BL-comets, a class of
finite BL-algebras which can be seen as a generalization of finite BL-chains.
Using BL-comets, any finite BL-algebra can be represent as a direct product of
BL-comets.

Definition 1.9. 1) ([NL; 03], Definition 21) Let $L$ be a BL-algebra. An
element $x\in L$ is called idempotent if $x\odot x=x$.

2) Let $L$ be a finite BL-algebra and $\mathcal{I}\left(L\right)$ the set of
idempotent elements in $L$. For $x\in\mathcal{I}\left(L\right)$, we denote
$\mathcal{C}\left(x\right)=\\{y\in\mathcal{I}\left(L\right)~{}$ such that $x$
and $y$ are comparable$\\}$. We defined the set
$\mathcal{D}\left(L\right)\subseteq\mathcal{I}\left(L\right)$ as follows:

$x\in\mathcal{D}\left(L\right)$ if and only if

i) $\mathcal{C}\left(x\right)=\mathcal{I}\left(L\right);$

ii) The set $\\{y\in\mathcal{I}\left(L\right),$ $y\leq x\\}$ is a chain.

It is clear that $\mathcal{D}\left(L\right)\neq\emptyset$ since
$0\in\mathcal{D}\left(L\right).$

A finite BL-algebra $L$ is called a BL-comet if
max$\mathcal{D}\left(L\right)\neq 0.\vskip 6.0pt plus 2.0pt minus 2.0pt$

We recall that in a BL-comet $L,$ the element max$\mathcal{D}\left(L\right)$
is called the pivot of $L$ and it is denoted by $pivot(L).\vskip 6.0pt plus
2.0pt minus 2.0pt$

Proposition 1.10. ([NL; 03], Proposition 26) Let $L$ be a finite BL-algebra.
The following assertions are equivalent:

(i) $L$ is a BL-comet and $pivot(L)=1;$

(ii) $L$ is a BL-chain.

Using the characterization of Boolean elements in BL-algebras, we establish
the connections between BL-comets and BL(MV)-chains.

Proposition 1.11. Let $L$ be a BL-comet. Then $L$ is a BL-chain iff
$pivot(L)^{\ast\ast}=pivot(L).$

Proof. If $L$ is a BL-chain, using Proposition 1.10, $pivot(L)=1$ and
obviously, $1^{\ast\ast}=1.$

Conversely, if we suppose that $pivot(L)^{\ast\ast}=pivot(L),$ using the
characterizations of Boolean elements in BL-algebras, see [P; 07, Proposition
3.3], it is clear that $pivot(L)$ is a boolean element in $L,$ so,
$pivot(L)\vee pivot(L)^{\ast}=1.$ We deduce that $pivot(L)^{\ast}$ is also a
boolean element, so, an idempotent element in $L.$ Then $pivot(L)^{\ast}\in
C(pivot(x))$. We conclude that $pivot(L)$ and $pivot(L)^{\ast}$ are
comparable.

If $pivot(L)\leq pivot(L)^{\ast},$ then $1=pivot(L)\vee
pivot(L)^{\ast}=pivot(L)^{\ast},$ so, $pivot(L)=0,$ a contradiction.

If $pivot(L)^{\ast}\leq pivot(L),$ then $1=pivot(L)\vee
pivot(L)^{\ast}=pivot(L).$ Using Proposition 1.10, $L$ is a BL-chain.
$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

From Proposition 1.11 we deduce the following result:

Proposition 1.12. Let $L$ be a finite MV-algebra. The following assertions are
equivalent:

(i) $L$ is a BL-comet;

(ii) $L$ is an MV-chain.

2. Some remarks regarding commutative unitary rings

Definition 2.1. ([AM; 69]) Let $R$ be a commutative unitary ring.

1) The ideal $M$ of the ring $R$ is maximal if it is maximal, with respect of
the set inclusion, amongst all proper ideals of the ring $R$. That means,
there are no other ideals different from $R$ contained $M$. The ideal $J$ of
the ring $R$ is a minimal ideal if it is a nonzero ideal which contains no
other nonzero ideals.

2) A commutative local ring $R$ is a ring with a unique maximal ideal.

3) Let $P$ $\not=R$ be an ideal in the ring $R$. Let $a,b\in R$ such that
$ab\in P$. If we have $a\in P$ or $b\in P$, therefore $P$ is called a prime
ideal of $R$.

Remark 2.2. Let $R$ be a commutative unitary ring.

1) The ideal $M$ of the ring $R$ is maximal if and only if $R/M$ is a field;

2) The ideal $P$ of the ring $R$ is prime if and only if $R/P$ is an integral
domain.

From here, we have that a maximal ideal it is a prime ideal.

For other details and properties, the reader is referred to [AM; 69].

Definition 2.3. i) [HLN; 18] A commutative ring $R$ is called a multiplication
ring if for every ideals $I_{1},I_{2}$ of $R$ with $I_{1}\subseteq I_{2}$,
there is an ideal $I_{3}$ of $R$ such that $I_{1}=I_{2}\otimes I_{3}$.

ii) A commutative ring $R$ is called Noetherian ring if the condition of
ascending chain is satisfied, that means every increasing sequence of ideals
$I_{1}\subseteq I_{2}\subseteq...\subseteq I_{r}\subseteq...$ is stationary,
that means there is $q$ such that $I_{q}=I_{q+1}=...$.

iii) A commutative ring $R$ is called Artinian ring if the condition of
descending chain is satisfied, that means every decreasing sequence of ideals
$I_{1}\supseteq I_{2}\supseteq...\supseteq I_{r}\supseteq...$ is stationary,
that means there is $q$ such that $I_{q}=I_{q+1}=...$.

Remark 2.4.

i) ([AB; 19], Lemma 3.5) Let $R=\underset{j\in J}{\prod}R_{i}$ be a direct
product of rings. $R$ is a multiplication ring if and only if $R_{j}$ is a
multiplication ring for all $j\in J$.

ii) ([AB; 19], Lemma 3.6) Let $R$ be a multiplication ring and $I$ be an ideal
of $R$. Therefore, the quotient ring $R/I$ is a multiplication ring.

Remark 2.5. ( [A; 76]. Corollary 6.1) Let $R$ be a ring. The following
conditions are equivalent:

i) $R[X]$ is a multiplication ring;

ii) $R$ is a finite direct product of fields.

Remark 2.6. ([AF; 92] and [AM; 69])

1) If $R$ is a Noetherian ring, therefore the polynomial ring
$R\left[X\right]$ is Noetherian and the quotient ring $R/I$ is also a
Noetherian ring, for $I~{}$an ideal of $R$.

2) Any field and any principal ideal ring is a Noetherian ring.

3) Every ideal of the Noetherian ring $R$ is finitely generated.

4) An integral domain $R$ is Artinian ring if and only if $R$ is a field.

5) The ring $K\left[X\right]/\left(X^{t}\right)$ is Artinian ring, for $K$ a
field and $t$ a positive integer.

6) A commutative Noetherian ring $R$ is Artinian if and only if $R$ is a
product of local rings.

7) In an Artinian ring every prime ideal is maximal.

8) An Artinian ring is a finite direct product of Artinian local rings.

Proposition 2.7. Let $R$ be a commutative unitary ring which has exactly three
ideals $\\{0\\},I,R$. Therefore, we have $I^{2}=\\{0\\}$.

Proof. It is clear that $I$ is the only minimal ideal in $R$. Since
$I^{2}\subseteq I$, we have $I^{2}=\\{0\\}$ or $I^{2}=I$. In the same time,
$I$ is the only maximal ideal of $R$ and each element from $I$ is a non-unit.
Suppose that $I^{2}=I$. Let $x\in I$ such that $x\neq 0$. It is clear that
$I=<x>$ and $x^{2}\neq 0$. We consider the ideal $A=\\{z\in
R~{}/~{}xz=0\\}\,$.We prove that $A\neq\\{0\\}$. Since $x^{2}\neq 0$,
therefore there is an element $y\in R$ such that $x^{2}y=x$. It results
$x\left(xy-1\right)=0,$ with $xy-1\neq 0,$ since $x$ is a non unit element in
$R$. Then $xy-1\in A$ and $A\neq\\{0\\}$. We have that $A\neq R$ since
$x\notin A$ ( we have $x^{2}\neq 0$). From here, we obtain that $A=I,$ a
contradiction, therefore $I^{2}=\\{0\\}$.$\Box\vskip 6.0pt plus 2.0pt minus
2.0pt$

Proposition 2.8. Each nonzero element in a finite commutative unitary ring $R$
is a unit or a zero divisor.

Proof. Let $R=\\{x_{1},x_{2},...,x_{n}\\}$ be a ring with $n$ elements and
$a\in R$ a non unit element. Therefore, the following elements
$ax_{1},ax_{2},...,ax_{n}$ are all different from $1$. It results that we can
have maximum $n-1$ distict values, then there are equal at least two of them.
We otain that $ax_{i}=ax_{j}$, for $i\neq j$, therefore
$a\left(x_{i}-x_{j}\right)=0$. Since $x_{i}\neq x_{j}$, it results that $a$ is
a zero divisor.$~{}\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Remark 2.9. (see [FK; 12])

1\. In a commutative ring $R$, the set of non-unit elements is an ideal if and
only if the ring $R$ is local.

2\. If $R$ is a commutative Noetherian ring and $I$ is an ideal in $R$
consists of zero-divisors, then its annihilator is a non zero ideal.

Proposition 2.10. Let $R$ be an integral domain with a finite number of
ideals. Then $R$ is a field.

Proof. Let $a\in R$ be a nonzero element. We have the ideals
$<a>,<a^{2}>,...,<a^{k}>,...$Since we have a finite number of ideals, it
results that we can find two equal ideals in this chain:
$<a^{k_{1}}>=<a^{k_{2}}>$. Therefore, there is an element $r\in R$ such that
$a^{k_{1}}r=a^{k_{2}}$. Assuming that $k_{1}\geq k_{2}$, it results that
$a^{k_{2}}\left(a^{k_{1}-h_{2}}r-1\right)=0$. Since $a\not=0$, we have that
$a^{k_{1}-h_{2}}r-1=0$, therefore $a$ is an invertible element. $\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

Proposition 2.11. Let $R$ be a commutative and unitary ring with a finite
number of ideals. Therefore, an ideal $I$ is maximal if and only if it is a
prime ideal.

Proof. Let $I$ be a maximal ideal. Therefore, $R/I$ is a field, that means an
integral domain, therefore $I$ is a prime ideal. Conversely, if $I$ is a prime
ideal, therefore $R/I$ is an integral domain with a finite number of ideals.
From the above proposition, $R/I$ is a field, then $I$ is a maximal ideal.
$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proposition 2.12. Let $R$ be a commutative and unitary ring with a finite
number of ideals. Let $n_{m}\left(R\right)$ be the number of maximal ideals in
$R$, $n_{p}\left(R\right)$ be the number of prime ideals in $R$ and
$n_{I}\left(R\right)$ be the number of all ideals in $R$. Therefore,
$n_{m}\left(R\right)=n_{p}\left(R\right)=\alpha$ and
$n_{I}\left(R\right)=\underset{j=1}{\overset{\alpha}{\prod}}\beta_{j},\beta_{j}$
positive integers, $\beta_{j}\geq 2$.

Proof. Relation $n_{m}\left(R\right)=n_{p}\left(R\right)=\alpha$ arises from
the above proposition. It is clear that $R$ is an Artinian ring, therefore it
is a product of local nonzero rings, $R=R_{1}\times...\times R_{\alpha}$.
Therefore, each ideal $J$ in $R$ is of the form $J=J_{1}\times...\times
J_{\alpha}$, with $J_{i}$ ideal in $R_{i}$. Since each ring $R_{i}$ has at
least two ideals, we obtain that
$n_{I}\left(R\right)=\underset{j=1}{\overset{\alpha}{\prod}}\beta_{j}$, where
$\beta_{j}\geq 2$ is the number of ideals in $R_{j}$. $\Box\vskip 6.0pt plus
2.0pt minus 2.0pt$

Remark 2.13. 1) First, we must remark that $n_{I}\left(R_{j}\right)$ can be
any positive integer $\beta_{j}\geq 2\,$. For example, the ring
$K\left[X\right]/\left(X^{\beta_{j}}\right)$ has $\beta_{j}$ ideals,
$\beta_{j}\geq 2$.

2) If $n_{I}\left(R\right)$ is finite, from Proposition 2.11, we have only the
following two possibilities:

-$R$ is an integral domain, therefore it is a field and, in this case, we have $n_{m}\left(R\right)=n_{p}\left(R\right)=1$ and $n_{I}\left(R\right)=2~{}$or

\- $R$ is not an integral domain and
$n_{m}\left(R\right)=n_{p}\left(R\right)\geq 1$ and $n_{I}\left(R\right)>2$.

3) From the above proposition it is clear that there are not commutative
unitary rings such that
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(3,3,5\right)$
or
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(2,2,5\right)$,
since $5$ is a prime number. To find such an example, we must search in non-
commutative rings. Therefore we have examples only in the case
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,5\right)$.
The same situation for $n_{I}\left(R\right)=7$. We have only the case
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,7\right)$
But, we can find examples of commutative unitary rings such that
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(2,2,4\right)$
or commutative unitary rings such that
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,4\right)$.
For example, for the first case, we have the ring
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ or $\mathbb{Z}_{6}$ and for the second,
we have the ring $\mathbb{Z}_{8}.$

3. Remarks regarding BL-rings

Let $R$ be a commutative unitary ring and $Id\left(R\right)$ be the set of all
ideals of the ring $R$. As we remarked above, we have that
$(Id(R),\cap,+,\otimes\rightarrow,0=\\{0\\},1=R)$ is a residuated lattice in
which the order relation is $\subseteq$ and $I\rightarrow J=(J:I),$ for every
$I,J\in Id(R)$. A commutative ring is a BL-ring if and only if $Id(R)$ is a
BL-algebra (see [HLN; 18], Corollary 2.3.). A Noetherian multiplicative ring
is an example of BL-ring (see [HLN; 18], Example 2.4, 2.).

If we consider $R_{1}$ and $R_{2}$ two BL-rings and their lattices of ideals
$Id\left(R_{1}\right)$ and $Id\left(R_{2}\right)$, therefore the lattice
$Id\left(R_{1}\times R_{2}\right)=Id\left(R_{1}\right)\times
Id\left(R_{2}\right)$ is a BL-algebra, therefore $R_{1}\times R_{2}$ is a BL-
ring. Indeed, for $Z,W\in Id\left(R_{1}\times
R_{2}\right),Z=\left(I_{1},J_{1}\right)$ and $W=\left(I_{2},J_{2}\right)$, we
define

$Z\rightarrow W=\left(I_{1}\rightarrow I_{2},J_{1}\rightarrow J_{2}\right),$

and

$Z\otimes W=\left(I_{1}\otimes J_{1},I_{2}\otimes J_{2}\right).$

In this way, we obtain a BL-algebra structure on $Id\left(R_{1}\times
R_{2}\right)$.

Proposition 3.1. There are no commutative unitary rings $R$ with three ideals
having $(Id(R),\cap,+,\otimes\rightarrow,0=\\{0\\},1=R)$ as a BL-algebra which
is not an MV-algebra.

Proof. From Proposition 2.7, it results that if $R$ is a commutative unitary
ring which has exactly three ideals $\\{0\\},I,R$, therefore we have
$I^{2}=\\{0\\}$. From here, we have $I\rightarrow 0=\left(0:I\right)=I$ and
the following implication and multiplication tables are obtained

$\begin{tabular}[]{l|lll}$\rightarrow$&$0$&$I$&$R$\\\
\hline\cr$0$&$R$&$R$&$R$\\\ $I$&$I$&$R$&$R$\\\ $R$&$0$&$I$&$R$\end{tabular}\ \
\ \begin{tabular}[]{l|lll}$\otimes$&$0$&$I$&$R$\\\ \hline\cr$0$&$0$&$0$&$0$\\\
$I$&$0$&$0$&$I$\\\ $R$&$0$&$I$&$R$\end{tabular}\ \ ,$ (3.1.)

which give us an MV-algebra structure. An example of such a ring is
$\left(\mathbb{Z}_{4},+,\cdot\right)$, with
$I=\left(\widehat{2}\right),0=\left(\widehat{0}\right)$. $\Box\vskip 6.0pt
plus 2.0pt minus 2.0pt$

Remark 3.2. From the above, a commutative unitary ring $R$ with three ideals
always has on the algebra $Id\left(R\right)$ an MV-algebra structure.
Therefore, to obtain rings with three ideals such that $Id\left(R\right)$ is a
BL-algebra that is not an MV-algebra, we must search among non-unitary rings
or among non-commutative rings or both, or to find such an algebra as a subset
or as a subalgebra of a infinite BL-algebra. The only BL-algebra of order $3$
which is not an MV-algebra has the following implication and multiplication
tables:

$\begin{tabular}[]{l|lll}$\rightarrow$&$0$&$I$&$R$\\\
\hline\cr$0$&$R$&$R$&$R$\\\ $I$&$0$&$R$&$R$\\\ $R$&$0$&$I$&$R$\end{tabular}\ \
\ \begin{tabular}[]{l|lll}$\otimes$&$0$&$I$&$R$\\\ \hline\cr$0$&$0$&$0$&$0$\\\
$I$&$0$&$I$&$I$\\\ $R$&$0$&$I$&$R$\end{tabular}.$ (3.2.)

Remark 3.3. 1) We recall that if $K$ is a field and $K\left[X\right]$ is the
polynomial ring, then for $f\in K\left[X\right]$, the quotient ring
$A=K\left[X\right]/\left(f\right)$ is a BL-ring, see [FP; 22].

2) Let $n=p_{1}p_{2}$, with $p_{1}$ and $p_{2}$ two prime distinct integers.
We have the following isomorphism

$R=\frac{\mathbb{Z}_{p_{1}p_{2}}\left[X\right]}{\left(p_{1}X^{2}+p_{2}\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(n,~{}p_{1}X^{2}+p_{2}\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(p_{2},~{}X^{2}\right)}\simeq\frac{\mathbb{Z}_{p_{2}}\left[X\right]}{\left(X^{2}\right)}.\vskip
6.0pt plus 2.0pt minus 2.0pt$

This ring is a BL-ring.

Example 3.4. In the following, we present some examples of BL-rings, more
precisely, examples of Noetherian multiplicative rings which are BL-rings.

1) We consider the ring
$R=\frac{\mathbb{Z}_{6}\left[X\right]}{\left(X^{2}\right)}$. From the above,
we have
$\frac{\mathbb{Z}_{6}\left[X\right]}{\left(X^{2}\right)}\simeq\frac{\mathbb{Z}_{2}\left[X\right]}{\left(X^{2}\right)}\times\frac{\mathbb{Z}_{3}\left[X\right]}{\left(X^{2}\right)}$.
The ring $\frac{\mathbb{Z}_{2}\left[X\right]}{\left(X^{2}\right)}$ has three
ideals $\left(0\right),\left(X\right)$ and
$\frac{\mathbb{Z}_{2}\left[X\right]}{\left(X^{2}\right)}$. The ring
$\frac{\mathbb{Z}_{3}\left[X\right]}{\left(X^{2}\right)}$ has also three
ideals $\left(0\right),\left(X\right)$ and
$\frac{\mathbb{Z}_{3}\left[X\right]}{\left(X^{2}\right)}$ Therefore, the ring
$\frac{\mathbb{Z}_{6}\left[X\right]}{\left(X^{2}\right)}$ is a BL-ring with
$9$ ideals.

2) From the above results, we have the following rings isomorphisms

$R_{1}=\frac{\mathbb{Z}_{6}\left[X\right]}{\left(2X^{2}+3\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(6,~{}2X^{2}+3\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(3,~{}X^{2}\right)}\simeq\frac{\mathbb{Z}_{3}\left[X\right]}{\left(X^{2}\right)}$

and

$R_{2}=\frac{\mathbb{Z}_{6}\left[X\right]}{\left(3X^{2}+2\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(6,~{}3X^{2}+2\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(2,~{}X^{2}\right)}\simeq\frac{\mathbb{Z}_{2}\left[X\right]}{\left(X^{2}\right)}\text{.}$

Indeed, since $X^{2}=\left(2X^{2}+3\right)\left(3X^{2}+2\right)$, we have that
$\left(3,~{}X^{2}\right)\subseteq\left(6,~{}2X^{2}+3\right)$. Conversely it is
also true, since $2X^{2}+3=2\cdot X^{2}+3$ and $6=3+3$. Therefore
$\left(6,~{}2X^{2}+3\right)=\left(3,~{}X^{2}\right)$. Similar, we have
$\left(6,~{}3X^{2}+2\right)=\left(2,~{}X^{2}\right)$. In the rings $R_{1}$ and
$R_{2},$ we have $X^{2}=0,$ since
$X^{2}=\left(2X^{2}+3\right)\left(3X^{2}+2\right)$. Therefore, in $R_{1}$, we
have $3=0$ and in $R_{2}\,\ $we have $2=0.$

3) From the above results, we have the following isomorphism of rings:

$\frac{\mathbb{Z}_{6}\left[X\right]}{\left(2X^{2}+4\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(6,~{}2X^{2}+4\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(2\right)\left(3,~{}X^{2}+2\right)}\simeq$
$\simeq\frac{\mathbb{Z}\left[X\right]}{\left(2\right)}\times\frac{\mathbb{Z}\left[X\right]}{\left(3,~{}X^{2}+2\right)}\simeq\mathbb{Z}_{2}\left[X\right]\times\frac{\mathbb{Z}_{3}\left[X\right]}{\left(X^{2}+2\right)}.$

Therefore, the ring
$\frac{\mathbb{Z}_{6}\left[X\right]}{\left(2X^{2}+4\right)}$ is an infinite
BL-ring.

In the same way, we have the following isomorphisms of rings:

$\frac{\mathbb{Z}_{30}\left[X\right]}{\left(6X^{2}+24\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(30,~{}6X^{2}+24\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(6\right)\left(5,~{}X^{2}+4\right)}\simeq$
$\simeq\frac{\mathbb{Z}\left[X\right]}{\left(6\right)}\times\frac{\mathbb{Z}\left[X\right]}{\left(5,~{}X^{2}+4\right)}\simeq\mathbb{Z}_{6}\left[X\right]\times\frac{\mathbb{Z}_{5}\left[X\right]}{\left(X^{2}+4\right)},$

or

$\frac{\mathbb{Z}_{30}\left[X\right]}{\left(6X^{2}\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(30,~{}6X^{2}\right)}\simeq\frac{\mathbb{Z}\left[X\right]}{\left(6\right)\left(5,~{}X^{2}\right)}\simeq$
$\simeq\frac{\mathbb{Z}\left[X\right]}{\left(6\right)}\times\frac{\mathbb{Z}\left[X\right]}{\left(5,~{}X^{2}\right)}\simeq\mathbb{Z}_{6}\left[X\right]\times\frac{\mathbb{Z}_{5}\left[X\right]}{\left(X^{2}\right)}.$

Therefore, the rings
$\mathbb{Z}_{6}\left[X\right]\times\frac{\mathbb{Z}_{5}\left[X\right]}{\left(X^{2}+4\right)}$
and
$\mathbb{Z}_{6}\left[X\right]\times\frac{\mathbb{Z}_{5}\left[X\right]}{\left(X^{2}\right)}$
are infinite BL-rings.

4) From the above results, we have the following isomorphism of rings:

$\frac{\mathbb{Z}_{6}\left[X\right]}{\left(X^{2}-X\right)}\simeq\frac{\mathbb{Z}_{6}\left[X\right]}{X\left(X-1\right)}\simeq\frac{\mathbb{Z}_{6}\left[X\right]}{\left(X\right)}\times\frac{\mathbb{Z}_{6}\left[X\right]}{\left(X-1\right)}\simeq$
$\simeq\mathbb{Z}_{6}\times\mathbb{Z}_{6}.$

Example 3.5. Let
$R=\mathbb{Z}_{6}\left[X\right]\times\frac{\mathbb{Z}_{5}\left[X\right]}{\left(X^{2}\right)}$,
be a BL-ring from the above examples. Therefore $Id\left(R\right)$ is an
infinite BL-algebra. In $R,$ we consider the following ideals
$\\{0\\},I=\left(3\right)\times R$ and $R,$ where
$\left(3\right)=\\{f\in\mathbb{Z}_{6}\left[X\right],f=a_{n}X^{n}+...+a_{1}X+a_{0},$
with $a_{0}=0$ or $a_{0}=3\\}$. We have that $I^{2}=I$ and
$Ann\left(I\right)=\\{0\\}$. Therefore, $B=\\{\\{0\\},I,R\\}$ is a finite BL-
subalgebra of the algebra $Id\left(R\right)$, having the implication and
multiplication tables given by the relation $\left(3.2.\right)$.

Remark 3.6. Let $R$ be a commutative unitary ring which has exactly four
ideals $\\{0\\},I,J,R$. Therefore, the lattice $Id\left(R\right)$ can be of
the form $A$ or $B$ from Figure 1.

Figure 1: Latices with four elements.

Case i. $I$ and $J$ are maximal ideals.

In this situation, we have $\\{0\\}\subset I\subset R$ and $\\{0\\}\subset
J\subset R$, with $I+J=R$. Therefore, $Id\left(R\right)$ is a lattice as in
Figure 1, A. We have that $I$ and $J$ are coprime. It is clear that $I\otimes
J=I\cap J=\\{0\\}$ and $I^{2}=I,J^{2}=J$. From here, it results that
$Ann\left(I\right)=J$ and $Ann\left(J\right)=I$. Also, $J\rightarrow
I=\left(I:J\right)=I$ and $I\rightarrow J=\left(J:I\right)=J$. Therefore, we
obtain an MV-algebra structure with the following operations:

$\rightarrow$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $R$ | $R$ | $R$ | $R$
$I$ | $J$ | $R$ | $J$ | $R$
$J$ | $I$ | $I$ | $R$ | $R$
$R$ | $0$ | $I$ | $J$ | $R$
$\otimes$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$
$I$ | $0$ | $I$ | $0$ | $I$
$J$ | $0$ | $0$ | $J$ | $J$
$R$ | $0$ | $I$ | $J$ | $R$
. (3.3.)

Case ii. The lattice of ideals is a chain, $Id\left(R\right)=\\{\\{0\\}\subset
I\subset J\subset R\\}$, as in Figure 1, B.

Since $\\{0\\}\subset I\subset J\subset R$, the ring is local. Also, we remark
that $I$ is the only minimal ideal. Let $x\in I,x\neq 0$. Therefore,
$0\not=<x>\subseteq I$. Since $I$ is minimal, we have $I=<x>,$ then $I$ is
finitely generated. We have the following subcases.

1) $I^{2}=\\{0\\},J^{2}=I,I\otimes J=\\{0\\}$. We have
$Ann\left(I\right)=\left(0:I\right)=J$ and
$Ann\left(J\right)=\left(0:J\right)=I$. Also, $J\rightarrow
I=\left(I:J\right)=J$. Therefore, we obtain an MV-algebra structure with the
following operations:

$\rightarrow$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $R$ | $R$ | $R$ | $R$
$I$ | $J$ | $R$ | $R$ | $R$
$J$ | $I$ | $J$ | $R$ | $R$
$R$ | $0$ | $I$ | $J$ | $R$
$\otimes$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$
$I$ | $0$ | $0$ | $0$ | $I$
$J$ | $0$ | $0$ | $I$ | $J$
$R$ | $0$ | $I$ | $J$ | $R$
. (3.4.)

2) $I^{2}=\\{0\\},J^{2}=I,I\otimes J=I$. In this case we do not obtaine a
residuated lattice, since $\otimes$ is not associative. For example, $I\otimes
J^{2}=I^{2}=\\{0\\}$ and $\left(I\otimes J\right)\otimes J=I\otimes J=I$.

3) $I^{2}=\\{0\\},J^{2}=J,I\otimes J=I$. We have
$Ann\left(I\right)=\left(0:I\right)=I$ and
$Ann\left(J\right)=\left(0:J\right)=\\{0\\}$. Also, $J\rightarrow
I=\left(I:J\right)=I$. Therefore, for $Id\left(R\right)~{}$we obtain a BL-
algebra structure (which is not an MV-algebra) with the following implication
and multiplication tables

$\rightarrow$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $R$ | $R$ | $R$ | $R$
$I$ | $I$ | $R$ | $R$ | $R$
$J$ | $0$ | $I$ | $R$ | $R$
$R$ | $0$ | $I$ | $J$ | $R$
$\otimes$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$
$I$ | $0$ | $0$ | $I$ | $I$
$J$ | $0$ | $I$ | $J$ | $J$
$R$ | $0$ | $I$ | $J$ | $R$
. (3.5.)

4) $I^{2}=\\{0\\},J^{2}=J,I\otimes J=\\{0\\}$. We have
$Ann\left(I\right)=\left(0:I\right)=J$ and
$Ann\left(J\right)=\left(0:J\right)=I$. Also, $J\rightarrow
I=\left(I:J\right)=I$. Therefore, we have $J\otimes(J\rightarrow I)=J\otimes
I=\\{0\\}$ and $J\wedge I=I$, false. Condition $\left(div\right)$ is not
satisfied. It results that $Id\left(R\right)$ is not a BL-algebra.

5) $I^{2}=\\{0\\},J^{2}=\\{0\\},I\otimes J=\\{0\\}$. We have
$Ann\left(I\right)=\left(0:I\right)=J$ and
$Ann\left(J\right)=\left(0:J\right)=J$. Also, $J\rightarrow
I=\left(I:J\right)=J$. Therefore, we have $J\otimes(J\rightarrow I)=J\otimes
J=\\{0\\}$ and $J\wedge I=I$, false. Condition $\left(div\right)$ is not
satisfied. It results that $Id\left(R\right)$ is not a BL-algebra.

6) $I^{2}=\\{0\\},J^{2}=\\{0\\},I\otimes J=I$, it is not possible, since
$I=I\otimes J\subset J^{2}=\\{0\\}$.

7) $I^{2}=I,J^{2}=I,I\otimes J=I$. We have
$Ann\left(I\right)=\left(0:I\right)=0$ and
$Ann\left(J\right)=\left(0:J\right)=0$. Also, $J\rightarrow
I=\left(I:J\right)=J$. Therefore, for $Id\left(R\right)~{}$we obtain a BL-
algebra structure(which is not an MV-algebra) with the following implication
and multiplication tables:

$\rightarrow$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $R$ | $R$ | $R$ | $R$
$I$ | $0$ | $R$ | $R$ | $R$
$J$ | $0$ | $J$ | $R$ | $R$
$R$ | $0$ | $I$ | $J$ | $R$
$\otimes$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$
$I$ | $0$ | $I$ | $I$ | $I$
$J$ | $0$ | $I$ | $I$ | $J$
$R$ | $0$ | $I$ | $J$ | $R$
. (3.6.)

8) $I^{2}=I,J^{2}=I,I\otimes J=\\{0\\}$, it is not possible, since
$I=I^{2}\subset I\otimes J=\\{0\\}$.

9) $I^{2}=I,J^{2}=J,I\otimes J=I$. We have
$Ann\left(I\right)=\left(0:I\right)=0$ and
$Ann\left(J\right)=\left(0:J\right)=0$. Also, $J\rightarrow
I=\left(I:J\right)=I$. Therefore, for $Id\left(R\right)~{}$we obtain a BL-
algebra structure (which is not an MV-algebra) with the following implication
and multiplication tables:

$\rightarrow$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $R$ | $R$ | $R$ | $R$
$I$ | $0$ | $R$ | $R$ | $R$
$J$ | $0$ | $I$ | $R$ | $R$
$R$ | $0$ | $I$ | $J$ | $R$
$\otimes$ | $0$ | $I$ | $J$ | $R$
---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$
$I$ | $0$ | $I$ | $I$ | $I$
$J$ | $0$ | $I$ | $J$ | $J$
$R$ | $0$ | $I$ | $J$ | $R$
. (3.7.)

10) $I^{2}=I,J^{2}=J,I\otimes J=\\{0\\}$, it is not possible, since
$I=I^{2}\subset I\otimes J=\\{0\\}$.

11) $I^{2}=I,J^{2}=\\{0\\},I\otimes J=I$, it is not possible, since
$I=I\otimes J\subset J^{2}=\\{0\\}$.

12) $I^{2}=I,J^{2}=\\{0\\},I\otimes J=\\{0\\}$, it is not possible, since from
$I\subset J$ we obtain $I^{2}\subset J^{2}=\\{0\\}$, so $I=\\{0\\},$ a
contradiction.

Remark 3.7. Counting the BL-algebras of order $4$ from the above proposition,
we obtain $5$ BL-algebras of order $4$, two MV and three BL-chain. In this
way, we recover the results obtained in [FP; 22-2], Table 2\. From the above
remark, we see that all BL-algebras of order four can be obtained as a lattice
of ideals of a commutative unitary ring $R$. For MV-algebra given by the
relation $\left(3.3\right)$, there is an example even of a ring which this MV-
algebras are isomorphic to, namely $R=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, and
for MV-algebra given by the relation $\left(3.4\right)$ an example can be the
ring $\mathbb{Z}_{8\text{ \ }}$(see [FP; 22-1], Table 2). It is interesting to
search and find, if there are, examples of commutative unitary rings $R$ with
$Id\left(R\right)$ a BL-algebra given by relations
$\left(3.5\right),\left(3.6\right),\left(3.7\right)$.

Remark 3.8. a) Other examples of BL-rings are presented in the following. From
([HLN; 18], Example 2.4 and Remark 2.5) we know that a Noetherian
multiplication ring is a BL-ring. The ring $\mathbb{\,Z}$ is a Noetherian
multiplication ring ( it is principal). Therefore, from the above properties
$(\mathbb{Z}_{n},+,\cdot)$ is a Noetherian multiplication ring.

b) We consider the ring $(\mathbb{Z}_{n},+,\cdot)$ with
$n=p_{1}p_{2}...p_{r},p_{1},p_{2},...,p_{r}$ being distinct prime numbers,
$r\geq 2$. The ring $R=\mathbb{Z}_{n}[X]/\left(f\right),$ with $f$ a
polynomial mod $n$ of degree $q\geq 2$, is a Noetherian multiplication ring,
therefore a BL-ring. From here, we have that the algebra of ideals,
$Id\left(R\right)$, are BL-algebra which is not a BL-chain ($\mathbb{Z}_{n}$
is a direct product of the fields $\mathbb{Z}_{p_{i}}$).

c) If $n=p^{r},$ $r\geq 2$, there is an MV-algebra isomorphic to
$\mathbb{Z}_{n}$ ([FP; 22-1]).

d) From ([AB; 19], Corollary 1.3), we know that an Artinian ring is a
multiplication ring if and only if it is a finite product of Artinian local
principal ideal rings. Moreover, if $r=1,$the ring
$R=\mathbb{Z}_{p}[X]/\left(f\right)$ is Artinian (is finite), local and
principal ring, therefore it is a BL-ring.

e) If we take the ring $\mathbb{Z}_{p}^{r}=$
$\underset{r-time}{\underbrace{\mathbb{Z}_{p}\times...\times\mathbb{Z}_{p}}}$,
we have that the ring $R=\mathbb{Z}_{p}^{r}[X]/\left(f\right)$ is a Noetherian
multiplication ring, therefore a BL-ring.

Remark 3.9. Let $R$ be a commutative unitary ring which has exactly five
ideals. Therefore, the lattice $Id\left(R\right)$ can be of the form $A$-$E$
from Figure 2 and Figure 3. Since $5$ is a prime number, with the above
notations, we have only commutative unitay rings $R$ of the form
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,5\right)$
with $Id\left(R\right)$ a BL-algebra, that means $Id\left(R\right)$ is a chain
(Figure E) or has Hasse diagram as in Figure C. Also, we remark that a BL-
algebra cannot have the form as in Figure C since, in this case $(a\rightarrow
b)\vee(b\rightarrow a)=c\neq 1,$ so (prel) condition is not satisfied. We
conclude that $Id\left(R\right)$ is a chain.

In [NL; 03], Corollary 28, the authors proved that each finite BL-algebras are
isomorphic to a direct product of BL-comets. A BL-comet of order $5$ has Hasse
diagram as in Figures 3, D or E. If we consider a BL-comet of order 5 as in
Figure 3, D, generated by a commutative ring $R,$ then $Id\left(R\right)$ has
two maximal ideals, impossible. We conclude that, BL-comets $Id\left(R\right)$
generated by commutative rings $R$ with five ideals are chains. As in Remark
3.6, we can obtain all $8$ BL-chain of order $5$, one MV and seven BL-chain as
$Id\left(R\right)$ of a ring with $5$ ideals with
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,5\right)$.
In this way, we recover the results obtained in [FP; 22-2], Table 2.

Figure 2: Latices with five elements.

Figure 3: Latices with five elements.

Remark 3.10. a) With the above notations, we do not have examples of
commutative unitary rings $R$ of the form
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(2,2,6\right)$
with $Id\left(R\right)$ a BL-algebra which is not an MV-algebra. Indeed, if
such a ring exists, it is an Artinian ring, therefore it is a product of
Artinian local nonzero rings, $R=R_{1}\times R_{2}$. One of these rings,
$R_{1},$ has two ideals, and the other one, $R_{2},$ has three ideals.
Therefore $Id\left(R_{1}\right)$ and $Id\left(R_{2}\right)$ must be BL-
algebras and at least one should not be an MV-algebra.. But, the only BL-
algebra with 2 elements, $Id(Z_{2}),$ is an MV-algebra and from Proposition
3.1, we have that there are not commutative unitary rings with three ideals
such that $Id\left(R_{2}\right)$ is a BL-algebra which are not an MV-algebra.
In this case we have only finite commutative unitay rings $R$ of the form
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,6\right)$
with $Id\left(R\right)$ a BL-algebra which is not an MV-algebra.

b) We can have only commutative unitay $R$ rings of the form
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,7\right)$
with $Id\left(R\right)$ a BL-algebra, since $7$ is a prime number, that means
$Id\left(R\right)$ is a chain.

b) If $n_{I}\left(R\right)=8$, we can have commutative unitary rings of the
form
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(2,2,8\right)\,\
$or
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(3,3,8\right)$,
with $Id\left(R\right)$ a BL-algebra.

d) For $n_{I}\left(R\right)=9,$ we do not have commutative unitary rings of
the form
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(2,2,9\right)$,
with $Id\left(R\right)$ a BL-algebra which is not an MV-algebra, since there
are not commutative unitary rings $R$ with three ideals such that
$Id\left(R\right)$ is a BL-algebra which is not an MV-algebra. But we can have
rings $R$ of the form
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,9\right)$,
that means $Id\left(R\right)$ is a chain. For MV-algebras of order $9$, there
are examples of rings whit which this MV-algebras is isomorphic to, namely
$\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ or $\mathbb{Z}_{9}$(see [FP; 22-1], Table
2).

e) For $n_{I}\left(R\right)=10,$ we can have
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(2,2,10\right)$
or
$\left(n_{m}\left(R\right),n_{p}\left(R\right),n_{I}\left(R\right)\right)=\left(1,1,10\right)$
such that $Id\left(R\right)$ is a BL-algebra which is not an MV-algebra.

Example 3.11. Let $R$ be a BL ring with $Id\left(R\right)$ infinite and $I$ be
an idempotent ideal in $R,I^{2}=I$. From Example 3.5, we know that such a ring
exists and $\\{0,I,R\\}$ forms a BL-subalgebra of order $3$ with the
multiplication tables given by the relation $\left(3.2\right)$. We consider
the ring $R\times R$, therefore

$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle\\{\left(0,0\right),\left(0,I\right),\left(0,R\right),\left(I,0\right),\left(I,I\right),\left(I,R\right),\left(R,0\right),\left(R,I\right),\left(R,R\right)\\}$
$\displaystyle=$ $\displaystyle\\{O,A,B,C,D,E,F,G,Z\\}$

is a BL-subalgebra of order $9$ (see Figure 4). We have the following
implication and multiplication tables:

$\rightarrow$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $Z$
---|---|---|---|---|---|---|---|---|---
$O$ | $Z$ | $Z$ | $Z$ | $Z$ | $Z$ | $Z$ | $Z$ | $Z$ | $Z$
$A$ | $F$ | $Z$ | $Z$ | $F$ | $Z$ | $Z$ | $F$ | $Z$ | $Z$
$B$ | $F$ | $G$ | $Z$ | $F$ | $G$ | $Z$ | $F$ | $G$ | $Z$
$C$ | $B$ | $B$ | $B$ | $\mathbf{Z}$ | $\mathbf{Z}$ | $\mathbf{Z}$ | $\mathbf{Z}$ | $\mathbf{Z}$ | $\mathbf{Z}$
$D$ | $O$ | $B$ | $B$ | $\mathbf{F}$ | $\mathbf{Z}$ | $\mathbf{Z}$ | $\mathbf{F}$ | $\mathbf{Z}$ | $\mathbf{Z}$
$E$ | $O$ | $A$ | $B$ | $\mathbf{F}$ | $\mathbf{G}$ | $\mathbf{Z}$ | $\mathbf{F}$ | $\mathbf{G}$ | $\mathbf{Z}$
$F$ | $B$ | $B$ | $B$ | $\mathbf{E}$ | $\mathbf{E}$ | $\mathbf{E}$ | $\mathbf{Z}$ | $\mathbf{Z}$ | $\mathbf{Z}$
$G$ | $O$ | $B$ | $B$ | $\mathbf{C}$ | $\mathbf{E}$ | $\mathbf{E}$ | $\mathbf{F}$ | $\mathbf{Z}$ | $\mathbf{Z}$
$Z$ | $O$ | $A$ | $B$ | $\mathbf{C}$ | $\mathbf{D}$ | $\mathbf{E}$ | $\mathbf{F}$ | $\mathbf{G}$ | $\mathbf{Z}$
$\otimes$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $Z$
---|---|---|---|---|---|---|---|---|---
$O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$
$A$ | $O$ | $A$ | $A$ | $O$ | $A$ | $A$ | $O$ | $A$ | $A$
$B$ | $O$ | $A$ | $B$ | $O$ | $A$ | $B$ | $O$ | $A$ | $B$
$C$ | $O$ | $O$ | $O$ | $\mathbf{C}$ | $\mathbf{C}$ | $\mathbf{C}$ | $\mathbf{C}$ | $\mathbf{C}$ | $\mathbf{C}$
$D$ | $O$ | $A$ | $A$ | $\mathbf{C}$ | $\mathbf{D}$ | $\mathbf{D}$ | $\mathbf{C}$ | $\mathbf{D}$ | $\mathbf{D}$
$E$ | $O$ | $A$ | $B$ | $\mathbf{C}$ | $\mathbf{D}$ | $\mathbf{E}$ | $\mathbf{C}$ | $\mathbf{D}$ | $\mathbf{E}$
$F$ | $O$ | $O$ | $O$ | $\mathbf{C}$ | $\mathbf{C}$ | $\mathbf{C}$ | $\mathbf{F}$ | $\mathbf{F}$ | $\mathbf{F}$
$G$ | $O$ | $A$ | $A$ | $\mathbf{C}$ | $\mathbf{D}$ | $\mathbf{D}$ | $\mathbf{F}$ | $\mathbf{G}$ | $\mathbf{G}$
$Z$ | $O$ | $A$ | $B$ | $\mathbf{C}$ | $\mathbf{D}$ | $\mathbf{E}$ | $\mathbf{F}$ | $\mathbf{G}$ | $\mathbf{Z}$
(3.8.)

Figure 4: Latices with nine elements.

From here, we remark that $\\{C,D,E,F,G,Z\\}$ is a BL-algebra of order $6~{}$(
see bold parts from relation $\left(3.8\right)$), extracted from the algebra
$\mathcal{B}$, $\\{O,D,E,G,Z\\}$ is a BL-subalgebra of order $5$ (it is a BL-
comet, from the below tables) and $\\{F,G,Z\\}$ is a BL-algebra of order
$3~{}$(as in relation $\left(3.7\right)$), extracted from the algebra
$\mathcal{B}$ (see Figure 5).

$\rightarrow$ | $O$ | $D$ | $E$ | $G$ | $Z$
---|---|---|---|---|---
$O$ | $Z$ | $Z$ | $Z$ | $Z$ | $Z$
$D$ | $O$ | $Z$ | $Z$ | $Z$ | $Z$
$E$ | $O$ | $G$ | $Z$ | $G$ | $Z$
$G$ | $O$ | $E$ | $E$ | $Z$ | $Z$
$Z$ | $O$ | $D$ | $E$ | $G$ | $Z$
$\otimes$ | $O$ | $D$ | $E$ | $G$ | $Z$
---|---|---|---|---|---
$O$ | $O$ | $O$ | $O$ | $O$ | $O$
$D$ | $O$ | $D$ | $D$ | $D$ | $D$
$E$ | $O$ | $D$ | $E$ | $D$ | $E$
$G$ | $O$ | $D$ | $D$ | $G$ | $G$
$Z$ | $O$ | $D$ | $E$ | $G$ | $Z$
.

Figure 5: Latices with six elements.

Conclusions. In this paper, we presented some rings with a finite number of
ideals and their connections with BL-rings. BL-rings are commutative unitary
rings whose lattice of ideals can be equipped with a structure of BL-algebra.
We proved that for a given number of ideals, some of these rings $R$ such that
$Id\left(R\right)$ is a BL-algebra can exist. As a further research we intend
to find more examples of such type of rings $R$ with $Id\left(R\right)$ a
finite BL-algebra.

References

[AB; 19] Alsuraiheed, T., Bavula, V. V., Characterization of multiplication
commutative rings with finitely many minimal prime ideals, Communications in
Algebra, 47(11))(2019), 4533-4540.

[A; 76] Anderson D. D., Multiplication ideals, multiplication rings, and the
ring $R(X)$, Can. J. Math., Vol. 28(4)(1976), 760-768.

[AF; 92] Anderson, F. W., Fuller, K., (1992), Rings and categories of modules,
Graduate Texts in Mathematics, 13(1992), 2 ed., Springer-Verlag, New York.

[AM; 69] Atiyah, M. F., MacDonald, I. G., Introduction to Commutative Algebra,
Addison-Wesley Publishing Company, London, 1969.

[CHA; 58] Chang, C.C., Algebraic analysis of many-valued logic, Trans. Amer.
Math. Soc. 88(1958), 467-490.

[FK; 12] Filipowicz, M., Kepczyk, M., A note on zero-divisors of commutative
rings, Arab J Math, 1(2012), 191–194.

[FP; 22-1] Flaut, C., Piciu, D., Connections between commutative rings and
some algebras of logic, Iranian Journal of Fuzzy Systems, 19(6)(2022), 93-110.

[FP; 22-2] Flaut, C., Piciu, D., Some Examples of BL-Algebras Using
Commutative Rings, Mathematics 10(2022), 4739, 1-15.

[HLN; 18] Heubo-Kwegna, O. A., Lele, C., Ndjeya, S., Nganou, J. B., BL-rings,
Logic Journal of the IGPL, 26(3)(2018), 290–299.

[H; 98] Hájek, P., Metamathematics of Fuzzy Logic, Trends in Logic-Studia
Logica Library 4, Dordrecht: Kluwer Academic Publishers 1998.

[NL; 03] Di Nola, A., Lettieri, A., Finite BL-algebras, Discrete Mathematics,
269(2003), 93-112.

[NL; 05] Di Nola, A., Lettieri, A., Finiteness based results in BL-algebras,
Soft Comput 9(2005), 889–896.

[P; 07] Piciu, D., Algebras of fuzzy logic, Ed. Universitaria, Craiova, 2007.

[TT;22] Tchoffo Foka, S. V., Tonga, M., Rings and residuated lattices whose
fuzzy ideals form a Boolean algebra, Soft Computing, 26 (2022) 535-539.

[T; 99] Turunen, E., Mathematics Behind Fuzzy Logic, Physica-Verlag, 1999.

[WD; 39] Ward, M., Dilworth, R.P., Residuated lattices, Trans. Am. Math. Soc.
45(1939), 335–354.

Mariana Floricica Călin

Faculty of Psychology and Educational Science,

Ovidius University of Constanţa, România,

Bd. Mamaia 124, 900527,

e-mail<EMAIL_ADDRESS>

Cristina Flaut

Faculty of Mathematics and Computer Science, Ovidius University,

Bd. Mamaia 124, 900527, Constanţa, România,

http://www.univ-ovidius.ro/math/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Dana Piciu

Faculty of Science, University of Craiova,

A.I. Cuza Street, 13, 200585, Craiova, Romania,

http://www.math.ucv.ro/dep_mate/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>


# Overhanging and touching waves
in constant vorticity flows

Vera Mikyoung Hur E-mail<EMAIL_ADDRESS>Department of Mathematics,
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA  Miles H. Wheeler E-mail<EMAIL_ADDRESS>Department
of Mathematical Sciences, University of Bath
Bath BA2 7AY, UK

###### Abstract

We show the existence of periodic traveling waves at the free surface of a two
dimensional, infinitely deep, and constant vorticity flow, under gravity,
whose profiles are overhanging, including one which intersects itself to
enclose a bubble of air. Numerical evidence has long suggested such
overhanging and touching waves, but a rigorous proof has been elusive.
Crapper’s celebrated capillary waves in an irrotational flow have recently
been shown to yield an exact solution to the problem for zero gravity, and our
proof uses the implicit function theorem to construct nearby solutions for
weak gravity.

##  1 Introduction

We consider periodic traveling waves at the free surface of an incompressible
inviscid fluid in two dimensions, under gravity, without the effects of
surface tension. When the flow is irrotational, the wave profile is
necessarily the graph of a single-valued function [23] (see also [6]). In
constant vorticity flows, by contrast, numerical investigations (see, for
instance, [22, 24, 15, 14]) have revealed profiles with multi-valued height
and even profiles which intersect themselves tangentially above the trough to
enclose a bubble of air. Constantin, Strauss and Vărvărucă [6] conjectured
that such overhanging and touching waves indeed exist. Here we give a proof of
this conjecture.

Crapper [9] discovered a remarkable family of exact solutions to the capillary
wave problem—that is, nonzero surface tension and zero gravity—in an
irrotational flow, whose profiles become more rounded as the amplitude
increases, opposite to gravity waves, so that overhanging profiles appear,
limited by a touching wave. See Figure 1. Akers, Ambrose and Wright [2] then
employed a perturbation method to construct nearby solutions for sufficiently
weak gravity, and in particular overhanging capillary-gravity waves. Córdoba,
Enciso and Grubic [8] took matters further and constructed a touching wave.
Recently, the authors [18] (see also [17]) showed that Crapper’s capillary
waves also give the profiles of periodic traveling waves in constant vorticity
flows, without the effects of gravity and surface tension. We follow a
perturbation argument, similar to [2, 8, 7] and others, to construct
overhanging and touching waves for nonzero gravity.

But it is the rotational effect which generates overhanging and touching
profiles for our problem, rather than the capillary effect [2, 8, 7]. Although
the unperturbed fluid surface is the same as Crapper’s wave, the fluid flow
beneath the surface is completely different, and so are the governing
equations. See [18] for more discussion, and also see [3] for a study of the
stability of the solutions with zero gravity.

For the capillary wave problem, Okamoto and Shōji [20, 21] produced closed-
form recurrence relations among the Fourier coefficients for the linearized
operator about Crapper’s wave, which enabled [2, 8, 7] and others to work out
their perturbation arguments. Unfortunately, such relations seem unwieldy for
nonzero constant vorticity and zero surface tension. Instead we reformulate
our problem for a holomorphic function in the unit disk, for which the zero-
gravity exact solution is given as a rational function (see (21)) and the
commutator, first introduced in [4, 5] for zero vorticity, and its linearized
operator can be evaluated by means of the calculus of residues (see (20) and
(27)). The novelty of our approach is that, to establish the invertibility of
the linearized operator, we relate it the problem of finding holomorphic
solutions to a complex ODE with meromorphic coefficients. In retrospect, this
technique seems quite natural, but so far we have been unable to find other
examples of its use in the literature. We believe that similar methods could
be applied to a much wider range of fluids problems which possess explicit
solutions given in terms of conformal mappings.

We begin in Section 2 by stating the problem and the results. In Section 3 we
reformulate the problem in conformal coordinates and, in turn, for a
holomorphic function in the unit disk. In Section 4 we employ the implicit
function theorem to prove our results. Section 5 discusses how one can
possibly take matters further to finite depth, point vortices, and hollow
vortices, among others. We pause to remark that Crapper’s waves also make
exact solutions for point vortices, rather than constant vorticity, without
the effects of gravity or surface tension [12]. Crowdy and his collaborators
[10, 26, 11] discovered exact solutions for non-rotating hollow vortices with
the effects of surface tension and showed that, interestingly, the same
conformal mapping makes exact solutions for rotating hollow vortices with
$N$-fold symmetry. Appendix A gives a summary of [18] and, importantly,
corrects errors in [18].

##  2 Preliminaries and the statement of the results

###  2.1 Stream function formulation

We consider a two dimensional, infinitely deep, and constant vorticity flow of
an incompressible inviscid fluid, under gravity, without the effects of
surface tension, and periodic traveling waves at the fluid surface. We assume
for simplicity that the fluid has the unit density. Suppose for definiteness
that in Cartesian coordinates, waves propagate in the $x$ direction and
gravity acts in the negative $y$ direction. In a frame of reference moving
with a constant velocity, suppose that the fluid flow is stationary and
occupies a region $D$ in the $(x,y)$ plane, bounded above by a free surface
$S$.

Let $\psi(x,y)$ denote a stream function so that $(\psi_{y},-\psi_{x})$ is the
velocity of the fluid, and $\psi$ satisfies

$\displaystyle\nabla^{2}\psi=-\omega$ $\displaystyle\text{in $D$},$ (1a)
$\displaystyle\psi=0$ $\displaystyle\text{on $S$},$ (1b)
$\displaystyle\tfrac{1}{2}|\nabla\psi|^{2}+gy=b$ $\displaystyle\text{on $S$},$
(1c) $\displaystyle\nabla\psi-(0,-\omega y-c)\to(0,0)\qquad$ as $y\to-\infty$
(1d)

for some $c>0$, the wave speed. Here $\omega$ denotes the vorticity and we
assume that it is constant in $D$. Note that $S$ is a free boundary, and (1b)
and (1c) are the kinematic and dynamic boundary conditions, where $g\geqslant
0$ is the gravitational constant, and $b\in\mathbb{R}$ the Bernoulli constant.
For zero vorticity, that is, $\omega=0$, (1d) expresses that there is no
motion of the fluid at the infinite bottom. Additionally we assume that $D$
and $\psi$ are $2\pi/k$ periodic in the $x$ direction for some wave number
$k>0$, and symmetric about the vertical lines below the crest and the trough.

Introducing dimensionless variables**$*$Rather than introducing new notation
for all the variables, we choose to write, for instance, $x\mapsto kx$. This
is to be read ‘$x$ is replaced by $kx$’, so that hereafter the symbol $x$ will
mean a dimensionless variable.

$x\mapsto kx,\qquad y\mapsto ky,\qquad\psi\mapsto(k/c)\psi,$

and dimensionless parameters

$\varOmega=\omega/ck,\qquad G=g/kc^{2},\qquad B=b/c^{2},$

we can rewrite (1) more conveniently as

$\displaystyle\nabla^{2}\psi=-\varOmega$ $\displaystyle\text{in $D$},$ (2)
$\displaystyle\psi=0$ $\displaystyle\text{on $S$},$
$\displaystyle\tfrac{1}{2}|\nabla\psi|^{2}+Gy=B$ $\displaystyle\text{on $S$},$
$\displaystyle\nabla\psi-(0,-\varOmega y-1)\to(0,0)\qquad$
$\displaystyle\text{as $y\to-\infty$}.$

Suppose that the fluid surface is given parametrically as

$S=\\{(x(\alpha),y(\alpha)):\alpha\in\mathbb{R}\\}.$ (3)

The periodicity and symmetry conditions become

$\displaystyle x(\alpha+2\pi)=x(\alpha)+2\pi$ $\displaystyle\text{and}\quad
y(\alpha+2\pi)=y(\alpha),$ (4) $\displaystyle x(-\alpha)=-x(\alpha)$
$\displaystyle\text{and}\quad y(-\alpha)=y(\alpha)$

for all $\alpha\in\mathbb{R}$, and

$\psi(-x,y)=\psi(x,y)=\psi(x+2\pi,y)\qquad\text{for all $(x,y)\in D$}.$

###  2.2 Exact solution for zero gravity

In what follows, we identify $\mathbb{R}^{2}$ with $\mathbb{C}$ whenever it is
convenient to do so and employ the notation $z=x+iy$.

For zero gravity, that is, $G=0$, the authors [18] recently showed that

$z(\alpha;A):=\alpha-\frac{4iAe^{-i\alpha}}{1+Ae^{-i\alpha}},$ (5)

together with

$\varOmega(A):=\frac{1-A^{2}}{1-3A^{2}}\quad\text{and}\quad
B(A):=\frac{1}{2}\left(\frac{1+A^{2}}{1-3A^{2}}\right)^{2},$ (6)

make an exact solution to (2)–(4) for an appropriate stream function (see
(42)), depending on the real parameter $A$. But, unfortunately, there were
sign errors, among others, causing some equations in [18] to appear incorrect.
In Appendix A we detail how to correct these errors.

Surprisingly, the same fluid surface also makes Crapper’s exact solution to
the capillary wave problem (nonzero surface tension and zero gravity) in an
irrotational flow for an appropriate value of the surface tension coefficient,
depending on $A$ [9]. See also [18].

Figure 1: The profiles of (5) in the $(x,y)$ plane for four values of $A$. (a)
$A=0.8A_{\max}<\sqrt{2}-1$ (see (9)), and the profile is not overhanging. (b)
$A=0.97A_{\max}>\sqrt{2}-1$, and it is overhanging. (c) $A=A_{\max}$. The
profile intersects itself tangentially at one point of the trough line. (d)
$A=1.06A_{\max}$, and it intersects itself transversely at two points of the
trough line. The fluid surfaces in the panels (a)–(c) give rise to physical
solutions to (2)–(4) but (d) does not.

Figure 1 shows the profiles of (5) for four values of $A$ in the $(x,y)$ plane
in the range $x\in[-\pi,\pi]$. By symmetry, it suffices to take $A\geqslant
0$. When $A$ is small, the fluid surface is not overhanging, that is, $y$ can
be given as a function of $x$. See, for instance, Figure 1(a). When $A$ is
sufficiently large, on the other hand, the fluid surface intersects itself
transversely at two points of the trough line and, hence, the fluid flow
becomes multi-valued, giving rise to a physically ‘unrealistic’ solution to
(2)–(4). See, for instance, Figure 1(d). A straightforward calculation reveals
that the profile of (5) is not overhanging as long as

$A<\sqrt{2}-1=0.4142135623730950\ldots.$ (7)

Recall [9] that it does not intersect itself so long as

$A<A_{\max}=0.4546700164520109\dots,$ (8)

where

$A_{\max}:=\max_{\alpha\in[-\pi,\pi]}\left(\frac{2\sin\alpha}{\alpha}-\cos\alpha\right)-\sqrt{\left(\max_{\alpha\in[-\pi,\pi]}\left(\frac{2\sin\alpha}{\alpha}-\cos\alpha\right)\right)^{2}-1}.$
(9)

When $\sqrt{2}-1<A<A_{\max}$, the profile of (5) is overhanging but does not
intersect itself. See, for instance, Figure 1(b). When $A=A_{\max}$, it
intersects itself tangentially at one point of the trough line, enclosing a
bubble of air, namely, a touching wave. See Figure 1(c).

In what follows we restrict the attention to $A\in[0,1/2)$, which includes all
the physical solutions ($A\leqslant A_{\max}$) and some nonphysical solutions
($A>A_{\max}$). We remark that both the dimensionless vorticity parameter
$\varOmega(A)$ and the crest-to-trough vertical distance
$\operatorname{Im}(z(\pi;A)-z(0;A))$ are strictly increasing with
$A\in[0,1/2)$.

###  2.3 Statement of the results

The aim here is to construct solutions to (2)–(4) nearby (5) and (6) for small
but nonzero values of $G$, particularly, overhanging and touching waves.

Below we state our results.

######  Theorem 1 (Overhanging waves).

For each $A\in(\sqrt{2}-1,A_{\max})$ (see (9)) and $G$ sufficiently small,
(2)–(4) has a solution, where $\varOmega=\varOmega(A)$ and $B=B(A)$ are in
(6), whose fluid surface does not intersect itself but is overhanging, that
is, $y$ cannot be given as a function of $x$.

######  Theorem 2 (Touching waves).

For $G$ sufficiently small, there exists a solution to (2)–(4) for which $S$
intersects itself tangentially along the trough line, enclosing a bubble of
air.

In Theorem 2, $\varOmega=\varOmega(A)$ and $B=B(A)$ for $A\approx A_{\max}$.

We emphasize that Theorems 1 and 2 are the first to rigorously establish that
overhanging and touching profiles exist for surface gravity waves. There is
persuasive numerical evidence (see, for instance, [15, 14, 13] and references
therein) of their existence. Also there is a global bifurcation result [6]
which allows for overhanging profiles, although the result is incapable of
determining whether such profiles actually exist. For zero vorticity, that is,
$\omega=0$, by contrast, overhanging waves cannot exist. See, for instance,
[6] for more discussion.

The proof of Theorems 1 and 2 is based on the exact solution for $G=0$,
discussed in Section 2.2, and uses the implicit function theorem to construct
nearby solutions for small $G$. The same strategy has been implemented for the
existence of overhanging and touching capillary-gravity waves (see, for
instance, [2, 8, 7]), based instead on Crapper’s exact solution to the
capillary wave problem in an irrotational flow. Although the zero-gravity
fluid surface is the same, the physical problem here is completely different
from the capillary-gravity wave problem. Actually, the fluid flows are
completely different. See [18] for more discussion. The linearized operator of
the capillary wave problem about Crapper’s wave was treated in [20, 21],
examining closed-form recurrence relations among the Fourier coefficients. But
such an approach seems unwieldy for our problem. We develop an alternative
approach, relating the linearized operator to a complex ODE with meromorphic
coefficients, whose solvability can be studied by means of the calculus of
residues.

##  3 Reformulation

###  3.1 Reformulation via conformal mapping

We introduce

$z=z(\alpha+i\beta),$ (10)

which maps $\mathbb{R}\times(-\infty,0)$ to $D$ conformally,
$\mathbb{T}\times(-\infty,0)$ to $(\mathbb{T}\times\mathbb{R})\cap D$, and
satisfies

$z(\alpha+i\beta)-(\alpha+i\beta)\to 0\qquad\text{as $\beta\to-\infty$}.$

Suppose that (10) extends to map $\mathbb{R}\times(-\infty,0]$ to $D\cup S$
continuously. This allows us to reformulate (2) in ‘conformal coordinates’ as

$(1+\varOmega(y+y\mathcal{H}y_{\alpha}-\mathcal{H}(yy_{\alpha})))^{2}=(B-2Gy)((1+\mathcal{H}y_{\alpha})^{2}+y_{\alpha}^{2})\qquad\text{for
$\beta=0$}.$ (11)

Here $\mathcal{H}$ denotes the periodic Hilbert transform: for instance, for
$y\in L^{2}(\mathbb{T})$,

$\mathcal{H}y(\alpha)=\frac{1}{2\pi}\operatorname{PV}\int_{\mathbb{T}}y(\alpha^{\prime})\cot\left(\frac{\alpha-\alpha^{\prime}}{2}\right)\leavevmode\nobreak\
d\alpha^{\prime},$ (12)

where $\operatorname{PV}$ stands for Cauchy’s principal value integral.
Alternatively,

$\mathcal{H}e^{in\alpha}=-i\operatorname{sgn}(n)e^{in\alpha},\qquad
n\in\mathbb{Z}.$ (13)

See, for instance, [15, 14] for details. Here and in what follows, we regard
$y$ as a real-valued function of $\alpha\in\mathbb{R}$ whenever it is
convenient to do so. Throughout we use subscripts for partial derivatives and
primes for variables of integration. We pause to remark that
$z(\alpha)=\alpha+(\mathcal{H}+i)y(\alpha)$. In other words,
$S=\\{(\alpha+\mathcal{H}y(\alpha),y(\alpha)):\alpha\in\mathbb{R}\\}$.

For zero vorticity, that is, $\varOmega=0$, (11) becomes

$1=(B-2Gy)(y_{\alpha}^{2}+y_{\beta}^{2})\qquad\text{for $\beta=0$},$

and indeed (2) can be formulated as (local) elliptic boundary value problem
for $y$ in a fixed domain. For nonzero vorticity, on the other hand, the
nonlocal term $\mathcal{H}(yy_{\alpha})$ causes technical difficulties. But
the commutator formula [4, 5]

$(y\mathcal{H}y_{\alpha}-\mathcal{H}(yy_{\alpha}))(\alpha)=\frac{1}{8\pi}\int_{\mathbb{T}}(y(\alpha)-y(\alpha^{\prime}))^{2}\csc^{2}\left(\frac{\alpha-\alpha^{\prime}}{2}\right)\leavevmode\nobreak\
d\alpha^{\prime}$ (14)

turns out to be instrumental. See (20) and (27).

A (smooth) solution of (11) gives rise to a solution of (2), provided that

$\alpha\mapsto z(\alpha)$ is injective for all $\alpha\in\mathbb{R}$ (15) and
$\displaystyle\text{$z_{\alpha}(\alpha)\neq 0$ for all
$\alpha\in\mathbb{R}$}.$ (16)

We refer the reader to, for instance, [15, 14] for details.

Recall [15, 14] that (15) states that the fluid surface does not intersect
itself, while (16) ensures that (10) is well-defined throughout
$\mathbb{R}\times(-\infty,0]$. There is numerical evidence [15, 14] (see also
[13]) that solutions of (11) can be found even when (15) fails to hold,
although such solutions would be nonphysical because the fluid surface
intersects itself and the fluid flow becomes multi-valued. Such nonphysical
solutions can nevertheless be useful, and indeed we will make use of them to
construct a touching wave for nonzero gravity.

When (16) fails to hold, on the other hand, the fluid surface develops a
stagnation point, where the velocity of the fluid vanishes in the moving frame
of reference. There is numerical evidence [15, 14] that for any value of
$\varOmega$, the solutions of (11) are ultimately limited by an ‘extreme’ wave
(in an appropriate function space, for instance, the (amplitude) $\times$
(wave speed) plane), whose profile has a stagnation point at the crest,
enclosing a $120^{\circ}$ angle. Extreme waves are beyond the scope of this
work, and throughout we will require that (16) holds true.

###  3.2 Reformulation for holomorphic functions in the unit disk

It is convenient to introduce

$\zeta=e^{-i(\alpha+i\beta)},$ (17)

which maps $\mathbb{T}\times(-\infty,0)$ to
$\mathbb{D}:=\\{\zeta\in\mathbb{C}:|\zeta|<1\\}$ and $\mathbb{T}\times\\{0\\}$
to $\partial\mathbb{D}$, by (4). Abusing notation, we write (10) as
$z(\zeta)$, and let

$z(\zeta)=i\log\zeta+w(\zeta).$ (18)

Note from (10) and (4) that $w$ is a single-valued holomorphic function of
$\zeta\in\mathbb{D}$, although $z$ is not, and that

$w(\overline{\zeta})=-\overline{w(\zeta)}.$

We can then rewrite (11) as

$\frac{1}{2}\frac{(1+\varOmega(\operatorname{Im}w+\mathcal{Q}(w)))^{2}}{|1-i\zeta
w_{\zeta}|^{2}}=B-G\operatorname{Im}w\qquad\text{for $|\zeta|=1$},$ (19)

where after the change of variables, (14) becomes

$\mathcal{Q}(w(\zeta)):=-\frac{\zeta}{2\pi
i}\ointctrclockwise_{|\zeta^{\prime}|=1}\left(\frac{\operatorname{Im}(w(\zeta)-w(\zeta^{\prime}))}{\zeta-\zeta^{\prime}}\right)^{2}\leavevmode\nobreak\
d\zeta^{\prime}\qquad\text{for $|\zeta|=1$}.$ (20)

Of particular usefulness for our purpose is that when $w$ is meromorphic in
the unit disk, one can evaluate the right hand side of (20) by means of the
calculus of residues. See Section 3.3. Note that (16) becomes $|1-i\zeta
w_{\zeta}|^{2}=|z_{\alpha}|^{2}\neq 0$ for $|\zeta|=1$.

###  3.3 The exact solution revisited

For zero gravity, that is, $G=0$, we deduce from Sections 2.2 and 3.2 that,
for any $A\in[0,1/2)$,

$w(\zeta;A):=-\frac{4iA\zeta}{1+A\zeta}$ (21)

together with (6) are an exact solution of (19). Clearly, $w(A)$ is
holomorphic in $\mathbb{D}$ and
$w(\overline{\zeta};A)=-\overline{w(\zeta;A)}$.

When $\sqrt{2}-1<A<A_{\max}$, where $A_{\max}$ is in (9), the fluid surface
given by (18), where $w$ is in (21), is overhanging but does not intersect
itself. When $A=A_{\max}$, it intersects itself tangentially at one point over
the period and can make sense of a solution of (2)–(4). When $A>A_{\max}$, on
the other hand, it intersects itself transversely at two points, giving rise
to a nonphysical solution of (2)–(4).

We give some details on how (21) solves (19), for the sake of completeness and
also for future reference. We restrict attention to $A\in[0,1/2)$. Since
$\overline{\zeta}=1/\zeta$ for $|\zeta|=1$,

$\operatorname{Im}w(\zeta;A)=2\left(-1+\frac{A}{\zeta+A}+\frac{1/A}{\zeta+1/A}\right)\qquad\text{for
$|\zeta|=1$}.$ (22)

Substituting (22) into (20), we evaluate the integral by means of the calculus
of residues to arrive at

$\mathcal{Q}(w(\zeta;A))=-8(A-1/A)^{-2}\left(\frac{A}{\zeta+A}-\frac{1/A}{\zeta+1/A}\right)$

for $|\zeta|=1$, whence

$1+\varOmega(A)(\operatorname{Im}w(\zeta;A)+\mathcal{Q}(w(\zeta;A)))=(1-2\varOmega(A))\frac{(\zeta-A)(\zeta-1/A)}{(\zeta+A)(\zeta+1/A)}$
(23)

for $|\zeta|=1$. On the other hand,

$1-i\zeta w_{\zeta}(\zeta;A)=\left(\frac{\zeta-1/A}{\zeta+1/A}\right)^{2},$
(24)

whence

$|1-i\zeta
w_{\zeta}(\zeta;A)|^{2}=\left(\frac{(\zeta-A)(\zeta-1/A)}{(\zeta+A)(\zeta+1/A)}\right)^{2}\qquad\text{for
$|\zeta|=1$}.$ (25)

A straightforward calculation reveals that (19) holds true for (6) and $G=0$.
Note that $|1-i\zeta w_{\zeta}(A)|^{2}\neq 0$ for $|\zeta|=1$.

##  4 Proof of Theorems 1 and 2

For $a\in(0,1)$ and fixed, we define the Banach spaces

$\displaystyle X$ $\displaystyle=\\{w\in
C^{3+a}(\mathbb{D},\mathbb{C}):\text{$w$ is holomorphic in $\mathbb{D}$ and
$w(\overline{\zeta})=-\overline{w(\zeta)}$}\\},$ $\displaystyle Y$
$\displaystyle=\\{f\in
C^{2+a}(\partial\mathbb{D},\mathbb{R}):f(\overline{\zeta})=f(\zeta)\\}$ and
the open set $\displaystyle U$ $\displaystyle=\\{w\in X:\text{$1-i\zeta
w_{\zeta}(\zeta)\neq 0$ for $|\zeta|=1$}\\}\subset X.$

We define $\mathcal{F}\colon U\times\mathbb{R}^{2}\to Y$ as

$\mathcal{F}(w;G,A)=\frac{1}{2}\frac{(1+\varOmega(A)(\operatorname{Im}w+\mathcal{Q}(w)))^{2}}{|1-i\zeta
w_{\zeta}|^{2}}+G\operatorname{Im}w-B(A)\qquad\text{for $|\zeta|=1$},$

where $\varOmega(A)$ and $B(A)$ are in (6). Since $\overline{\zeta}=1/\zeta$
for $|\zeta|=1$, note from (20) that
$\mathcal{Q}(w(\overline{\zeta}))=\mathcal{Q}(w(\zeta))$ for $w\in X$. Since
(14) is real analytic and (17) is holomorphic, $\mathcal{Q}$ is real analytic.
Therefore $\mathcal{F}$ is well-defined and real analytic.

Recall from Section 3.3 that for any $A\in[0,1/2)$,

$\mathcal{F}(w(A);0,A)=0,$ (26)

and our task is to construct nearby solutions of $\mathcal{F}(w;G,A)=0$ for
$G\neq 0$ small.

######  Theorem 3.

For each $A_{0}\in(0,1/2)$ there exists $\varepsilon>0$ and a real-analytic
map

$W\colon(-\varepsilon,\varepsilon)\times(A_{0}-\varepsilon,A_{0}+\varepsilon)\to
U$

such that $W(0,A)=w(A)$ and

$\mathcal{F}(W(G,A);G,A)=0.$

Moreover, there exists $\delta>0$ such that $W(G,A)$ is the unique solution of
$\mathcal{F}(w;G,A)=0$ for which $\|w-w(A)\|_{X}<\delta$.

Our proof of Theorem 3 is based on the implicit function theorem. For $w\in
U$, $G\in\mathbb{R}$ and $A\in[0,1/2)$, we calculate that the linearized
operator $\mathcal{F}_{w}(w;G,A)\colon X\to Y$ is given as

$\displaystyle\mathcal{F}_{w}(w;G,A)v=$
$\displaystyle\frac{1+\varOmega(A)(\operatorname{Im}w+\mathcal{Q}(w))}{|1-i\zeta
w_{\zeta}|^{2}}\varOmega(A)(\operatorname{Im}v+\mathcal{Q}_{w}(w)v)$
$\displaystyle-\frac{(1+\varOmega(A)(\operatorname{Im}w+\mathcal{Q}(w)))^{2}}{|1-i\zeta
w_{\zeta}|^{4}}\operatorname{Im}(\overline{(1-i\zeta w_{\zeta})}\zeta
v_{\zeta})+G\operatorname{Im}v\qquad\text{for $|\zeta|=1$}.$

Recalling (23), (24) and (25), after some algebra we arrive at

$\displaystyle\mathcal{F}_{w}(w(A);0,A)v=$
$\displaystyle(1-2\varOmega(A))\frac{(\zeta+A)(\zeta+1/A)}{(\zeta-A)(\zeta-1/A)}\varOmega(A)(\operatorname{Im}v+\mathcal{Q}_{w}(w(A))v)$
$\displaystyle-\left((1-2\varOmega(A))\frac{(\zeta+A)(\zeta+1/A)}{(\zeta-A)(\zeta-1/A)}\right)^{2}\operatorname{Im}\bigg{(}\left(\frac{\zeta-A}{\zeta+A}\right)^{2}\zeta
v_{\zeta}\bigg{)}$

for $|\zeta|=1$. On the other hand, linearizing (20) about $w(A)$,

$\displaystyle\mathcal{Q}_{w}(w(A))v(\zeta)$ $\displaystyle=-\frac{\zeta}{\pi
i}\ointctrclockwise_{|\zeta^{\prime}|=1}\frac{\operatorname{Im}(w(\zeta;A)-w(\zeta^{\prime};A))\operatorname{Im}(v(\zeta)-v(\zeta^{\prime}))}{(\zeta-\zeta^{\prime})^{2}}\leavevmode\nobreak\
d\zeta^{\prime}$ $\displaystyle=\operatorname{Im}\left(-\frac{\zeta}{\pi
i}\ointctrclockwise_{|\zeta^{\prime}|=1}\frac{\operatorname{Im}(w(\zeta;A)-w(\zeta^{\prime};A))}{(\zeta-\zeta^{\prime})^{2}}(v(\zeta)-v(\zeta^{\prime}))\leavevmode\nobreak\
d\zeta^{\prime}\right)$

for $|\zeta|=1$. Recalling (22), since

$\frac{\operatorname{Im}(w(\zeta;A)-w(\zeta^{\prime};A))}{(\zeta-\zeta^{\prime})^{2}}=\frac{2}{\zeta-\zeta^{\prime}}\left(\frac{A}{(\zeta+A)(\zeta^{\prime}+A)}+\frac{1/A}{(\zeta+1/A)(\zeta^{\prime}+1/A)}\right)$

has poles at $\zeta^{\prime}=\zeta$ and $-A$ in $\mathbb{D}$, we evaluate the
integral by means of the calculus of residues to arrive at a strikingly
simple, albeit non-local, formula

$\mathcal{Q}_{w}(w(A))v(\zeta)=\operatorname{Im}\left(-4A\zeta\frac{v(\zeta)-v(-A)}{(\zeta+A)^{2}}\right)\qquad\text{for
$|\zeta|=1$}.$ (27)

Therefore

$\mathcal{F}_{w}(w(A);0,A)v=-(1-2\varOmega(A))\frac{(\zeta+A)(\zeta+1/A)}{(\zeta-A)(\zeta-1/A)}\mathcal{L}(A)v\qquad\text{for
$|\zeta|=1$},$ (28)

where

$\mathcal{L}(A)v:=\operatorname{Im}\left((1-2\varOmega(A))\zeta\frac{(\zeta-A)(\zeta+1/A)}{(\zeta+A)(\zeta-1/A)}v_{\zeta}-\varOmega(A)v+4A\varOmega(A)\zeta\frac{v-v(-A)}{(\zeta+A)^{2}}\right).$
(29)

We pause to remark that $\mathcal{L}(A)$ is a ‘generalized’ Riemann–Hilbert
operator, with the difference that in addition to $v$ itself, $v_{\zeta}$
appears as well as the non-local term $v(-A)$. General classes of
Riemann–Hilbert operators including those of the form (29) with Hölder
continuous coefficients are discussed in [16, Section 34] and [19, Section
71], among many others, and $\mathcal{L}(A)v=f$ can be transformed into
various integral equations. We emphasize that the particularly simple form of
(29) comes from the use of the commutator formula (20), introduced by [4, 5]
for zero vorticity.

######  Lemma 4.

For any $A\in[0,1/2)$, $\mathcal{L}(A)\colon X\to Y$ is Fredholm with index
zero.

###### Proof.

The principal part of $\mathcal{L}(A)$ is

$\mathcal{L}_{0}(A)v:=\operatorname{Im}\left((1-2\varOmega(A))\zeta\frac{(\zeta-A)(\zeta+1/A)}{(\zeta+A)(\zeta-1/A)}v_{\zeta}\right),$
(30)

which is a classical Riemann–Hilbert operator acting on $v_{\zeta}$. Since

$\inf_{\zeta\in\partial\mathbb{D}}\left|(1-2\varOmega(A))\zeta\frac{(\zeta-A)(\zeta+1/A)}{(\zeta+A)(\zeta-1/A)}\right|>0$

whenever $A\in[0,1/2)$, classical elliptic theory (see, for instance, [25,
Chapter 1, Section 2]) implies that

$\|v_{\zeta}\|_{C^{2+a}(\mathbb{D})}\leqslant
C(\|\mathcal{L}_{0}(A)v\|_{C^{2+a}(\partial\mathbb{D})}+\|v\|_{C^{0}(\mathbb{D})})$

for some constant $C$ independent of $v$ and, hence, by interpolation,

$\|v\|_{X}\leqslant C(A)(\|\mathcal{L}(A)v\|_{Y}+\|v\|_{C^{0}(\mathbb{D})}).$
(31)

(Alternatively, one can obtain (31) by reformulating (29) as an elliptic
system for the real and imaginary parts of $v$ and verifying that it satisfies
the hypotheses of [1].) Since $X$ is compact in
$C^{0}(\overline{\mathbb{D}})$, we conclude that $\mathcal{L}(A)\colon X\to Y$
is semi-Fredholm, with closed range and a finite-dimensional kernel. It
remains to show that the index of $\mathcal{L}(A)$ is zero.

Since the index of $\mathcal{L}(A)$ is continuous and, hence, independent of
$A\in[0,1/2)$, it suffices to show that the index of $\mathcal{L}(0)$ is zero.
We write $v\in X$ as

$v(\zeta)=\sum_{n=0}^{\infty}v_{n}\zeta^{n},$

where the coefficients $v_{n}$ are purely imaginary by symmetry, so that

$\mathcal{L}(0)v=\operatorname{Im}(\zeta
v_{\zeta}-v)=\operatorname{Im}\left(\sum_{n=0}^{\infty}(n-1)v_{n}\zeta^{n}\right)$

by (29) and (6). A straightforward calculation reveals that the kernel of
$\mathcal{L}(0)$ is one dimensional, spanned by $i\zeta$, and the co-kernel is
one dimensional, spanned by $\operatorname{Re}\zeta$. Therefore
$\mathcal{L}(0):X\to Y$ is Fredholm with index zero, and the proof is
complete. ∎

Therefore $\mathcal{L}(A)\colon X\to Y$ is invertible provided that the kernel
is trivial. We will study the kernel of $\mathcal{L}(A)$ by relating it to a
complex ODE, for which the following is useful.

######  Lemma 5.

Suppose that $v$ is holomorphic in a neighborhood of $\zeta_{0}\in\mathbb{C}$
and that

$v_{\zeta}+pv=q,$ (32)

where $p$ and $q$ are meromorphic with at most simple poles at $\zeta_{0}$.

  1. (i)

$\operatorname{Res}(q,\zeta_{0})=v(\zeta_{0})\operatorname{Res}(p,\zeta_{0})$.

  2. (ii)

If $q\equiv 0$ and $v\not\equiv 0$ then
$\operatorname{Res}(p,\zeta_{0})\leqslant 0$ is an integer.

  3. (iii)

If $\operatorname{Res}(p,\zeta_{0})=-2$, so that

$\displaystyle
p(\zeta)=\frac{-2}{\zeta-\zeta_{0}}+p_{0}+p_{1}(\zeta-\zeta_{0})+\cdots\quad\text{and}\quad
q(\zeta)=\frac{-2v(\zeta_{0})}{\zeta-\zeta_{0}}+q_{0}+q_{1}(\zeta-\zeta_{0})+\cdots$
(33)

as $\zeta\to\zeta_{0}$, by (i), for some
$p_{0},p_{1},q_{0},q_{1}\in\mathbb{C}$, then

$\displaystyle(p_{0}^{2}+p_{1})v(\zeta_{0})-p_{0}q_{0}-q_{1}=0.$ (34)

###### Proof.

The assertion (i) follows from the well-known fact that
$\operatorname{Res}(fg,\zeta_{0})=f(\zeta_{0})\operatorname{Res}(g,\zeta_{0})$
whenever $f$ is analytic and $g$ has at most a simple pole at $\zeta_{0}$. If
$q\equiv 0$ and $v\not\equiv 0$ then (32) rearranges to

$\displaystyle p=-\frac{v_{\zeta}}{v}=-\frac{d}{d\zeta}\log v.$

A simple calculation shows that $\operatorname{Res}(p,\zeta_{0})=-m$, where
$m\geqslant 0$ is the order of the zero of $v$ at $\zeta_{0}$. Finally, to see
(iii), we insert (33) into (32) to obtain

$\displaystyle\frac{-2v(\zeta_{0})}{\zeta-\zeta_{0}}+(p_{0}v(\zeta_{0})-v_{\zeta}(\zeta_{0}))+(p_{1}v(\zeta_{0})+p_{0}v_{\zeta}(\zeta_{0}))(\zeta-\zeta_{0})+\cdots=\frac{-2v(\zeta_{0})}{\zeta-\zeta_{0}}+q_{0}+q_{1}(\zeta-\zeta_{0})+\cdots$

as $\zeta\to\zeta_{0}$, where the terms involving $v_{\zeta\zeta}(\zeta_{0})$
cancel because $\operatorname{Res}(p,\zeta_{0})=-2$. Grouping like powers and
eliminating $v_{\zeta}(\zeta_{0})$, we obtain (34) as desired. ∎

######  Lemma 6.

For any $A\in(0,1/2)$, $\mathcal{L}(A)\colon X\to Y$ is invertible.

###### Proof.

By Lemma 4, it suffices to show that the kernel of $\mathcal{L}(A)$ is
trivial. Suppose for the sake of contradiction that $v\not\equiv 0,\in X$ lies
in the kernel of $\mathcal{L}(A)$. That is,

$\operatorname{Im}\left((1-2\varOmega(A))\zeta\frac{(\zeta-A)(\zeta+1/A)}{(\zeta+A)(\zeta-1/A)}v_{\zeta}-\varOmega(A)v+4A\varOmega(A)\zeta\frac{v-v(-A)}{(\zeta+A)^{2}}\right)=0\qquad\text{for
$|\zeta|=1$}.$

Clearly,

$f:=(1-2\varOmega(A))\zeta\frac{(\zeta-A)(\zeta+1/A)}{(\zeta+A)(\zeta-1/A)}v_{\zeta}-\varOmega(A)v+4A\varOmega(A)\zeta\frac{v-v(-A)}{(\zeta+A)^{2}}$
(35)

is meromorphic in $\mathbb{D}$, possibly with a simple pole at $\zeta=-A$, and
$\operatorname{Im}f=0$ on $\partial\mathbb{D}$.

Since $v(\overline{\zeta})=-\overline{v(\zeta)}$ by symmetry, $v$ is purely
imaginary on the real axis and, hence, the residue of $f$ at $\zeta=-A$ must
be purely imaginary. Therefore

$f-\frac{if_{-1}}{\zeta+A}$

is holomorphic in $\mathbb{D}$ for some $f_{-1}\in\mathbb{R}$. Note that

$\frac{if_{-1}}{\zeta+A}+\overline{\left(\frac{if_{-1}}{\zeta+A}\right)}=\frac{if_{-1}}{\zeta+A}+\frac{if_{-1}/A^{2}}{\zeta+1/A}-\frac{if_{-1}}{A}\qquad\text{for
$|\zeta|=1$}$

is real, and we consider the function

$\tilde{f}:=f-\frac{if_{-1}}{\zeta+A}-\frac{if_{-1}/A^{2}}{\zeta+1/A}+\frac{if_{-1}}{A},$
(36)

which is not only holomorphic in $\mathbb{D}$ but also real on
$\partial\mathbb{D}$. Since $v$ is purely imaginary on
$\mathbb{R}\cap\mathbb{D}$, the same is true of $\tilde{f}$. Together, these
lead to

$\tilde{f}=0\qquad\text{in }\mathbb{D}.$ (37)

Indeed, since $\operatorname{Im}\tilde{f}=0$ on $\partial\mathbb{D}$, the
maximum principle implies that $\operatorname{Im}\tilde{f}=0$ in $\mathbb{D}$.
Thus $\tilde{f}$ is a real and holomorphic function and, hence, a real
constant. Finally, since $\tilde{f}$ is purely imaginary along $\mathbb{R}$,
the only possibility is that this constant is $0$.

Recalling (35) and (36) we write (37) as the complex ODE in (32), where

$\displaystyle p(\zeta)$
$\displaystyle=-\frac{\varOmega(A)}{1-2\varOmega(A)}\frac{(\zeta-A)(\zeta-1/A)}{\zeta(\zeta+A)(\zeta+1/A)}=\frac{1-A^{2}}{1+A^{2}}\frac{1}{\zeta}-\frac{2}{\zeta+A}+\frac{2}{\zeta+1/A}$
and $\displaystyle q(\zeta)$
$\displaystyle=\frac{1}{1-2\varOmega(A)}\frac{(\zeta+A)(\zeta-1/A)}{\zeta(\zeta-A)(\zeta+1/A)}\left(\frac{4A\varOmega(A)v(-A)\zeta}{(\zeta+A)^{2}}+\frac{if_{-1}}{\zeta+A}+\frac{if_{-1}/A^{2}}{\zeta+1/A}-\frac{if_{-1}}{A}\right)$
$\displaystyle=\frac{i(1-3A^{2})}{A(1+A^{2})}f_{-1}\left(\frac{1}{\zeta}-\frac{(1-A^{2})^{2}}{(1+A^{2})^{2}}\frac{1}{\zeta-A}-\frac{1+6A^{2}+A^{4}}{(1+A^{2})^{2}}\frac{1}{\zeta+1/A}+\frac{2(1-A^{2})}{A(1+A^{2})}\frac{1}{(\zeta+1/A)^{2}}\right)$
$\displaystyle\qquad+v(-A)\left(\frac{(1-A^{2})^{2}}{(1+A^{2})^{2}}\frac{2}{\zeta-A}-\frac{2}{\zeta+A}+\frac{8A^{2}}{(1+A^{2})^{2}}\frac{1}{\zeta+1/A}\right),$

where we use (6) and partial fractions. Note that potential singularities in
$\mathbb{D}$ of $p$ and $q$ are simple poles at $\zeta=\pm A$ and $\zeta=0$.

At $\zeta=A$ we calculate that

$\operatorname{Res}(p,A)=0\quad\text{and}\quad\operatorname{Res}(q,A)=2\frac{(1-A^{2})^{2}}{(1+A^{2})^{2}}v(-A)-\frac{i(1-A^{2})^{2}(1-3A^{2})}{A(1+A^{2})^{3}}f_{-1}.$

Applying Lemma 5(i), we deduce that

$f_{-1}=-\frac{2iA(1+A^{2})}{1-3A^{3}}v(-A).$ (38)

Using this to eliminate $f_{-1}$ in the above formula for $q$, things simplify
considerably and we are left with

$\displaystyle
q=2v(-A)\left(\frac{1}{\zeta}-\frac{1}{\zeta+A}-\frac{1}{\zeta+1/A}+\frac{2A(1-A^{2})}{1+A^{2}}\frac{1}{(\zeta+1/A)^{2}}\right).$

In particular, at $\zeta=-A$, we calculate

$\operatorname{Res}(p,-A)=-2\quad\text{and}\quad\operatorname{Res}(q,-A)=-2v(-A).$

Writing

$\displaystyle
p(\zeta)=\frac{-2}{\zeta+A}+p_{0}+p_{1}(\zeta+A)+\cdots\quad\text{and}\quad
q(\zeta)=\frac{-2v(-A)}{\zeta+A}+q_{0}+q_{1}(\zeta+A)+\cdots$

as $\zeta\to-A$, where

$\displaystyle p_{0}$
$\displaystyle=-\frac{1-4A^{2}-A^{4}}{A(1-A^{2})(1+A^{2})},$ $\displaystyle
p_{1}$
$\displaystyle=-\frac{1-3A^{2}+5A^{4}+A^{6}}{A^{2}(1-A^{2})^{2}(1+A^{2})},$
$\displaystyle q_{0}$ $\displaystyle=-\frac{2v(-A)}{A(1+A^{2})},$
$\displaystyle q_{1}$
$\displaystyle=-\frac{2(1-A^{2}+2A^{4})v(-A)}{A^{2}(1-A^{2})^{2}(1+A^{2})},$

we conclude from Lemma 5(iii) that

$\displaystyle(p_{0}^{2}+p_{1})v(-A)-p_{0}q_{0}-q_{1}=\frac{4v(-A)}{(1-A^{2})^{2}}=0.$

This forces $v(-A)=0$ and, hence, $f_{-1}=0$ as well by (38). Thus $q\equiv 0$
and, therefore, Lemma 5(ii) implies that $\operatorname{Res}(p,0)\leqslant 0$
is an integer. But we calculate

$\operatorname{Res}(p,0)=\frac{1-A^{2}}{1+A^{2}}\in(0,1),$

which is the desired contradiction. ∎

######  Remark.

Recall $v(-A)=0$ and $f_{-1}=0$ in the course of the proof of Lemma 6, whence
(37) becomes

$v_{\zeta}+\frac{1-A^{2}}{1+A^{2}}\frac{(\zeta-A)(\zeta-1/A)}{\zeta(\zeta+A)(\zeta+1/A)}v=0,$

whose general solution is

$v(\zeta)=C\left(\frac{\zeta+A}{\zeta+1/A}\right)^{2}\zeta^{\frac{A^{2}-1}{A^{2}+1}},$

where $C\in\mathbb{C}$ is an arbitrary constant. This is multi-valued in
$\mathbb{D}$ unless $A=0$ or $C=0$.

######  Remark.

An earlier version of this paper contained an error in Lemma 5, which led to
an overly simplistic analysis of the complex ODE in the proof of Lemma 6.
Thankfully, this error was identified by an anonymous referee, who also
suggested the current statement and proof of Lemma 5(i)–(ii).

For each $A\in(0,1/2)$ and fixed, recall (26). We deduce from (28) and Lemma 6
that

$\mathcal{F}_{w}(w(A);0,A)\colon X\to Y$ is invertible.

Theorem 3 then follows at once from the implicit function theorem for real-
analytic operators.

###### Proof of Theorem 1.

For $A\in(\sqrt{2}-1,A_{\max})$ and fixed, Theorem 3 gives a one-parameter
family of solutions $w=W(G,A)$ to (19), where $\varOmega=\varOmega(A)$ and
$B=B(A)$, for $G$ sufficiently small, depending real analytically on $G$.
Correspondingly, (18) gives a one-parameter family of holomorphic functions
$z=Z(G,A)$ whose imaginary part solves (11). Since $W(G,A)\in U$ by
construction, (16) holds true, whereby they give rise to physical solutions of
(2) provided (15) holds true.

When $G=0$, recall from Section 2.2 that (15) holds true. For $G$ sufficiently
small, we consider

$f(\alpha,\alpha^{\prime})=\frac{z(\alpha)-z(\alpha^{\prime})}{\alpha-\alpha^{\prime}},\qquad
f\colon\mathbb{T}^{2}\to\mathbb{C}.$

Since $w\in U$, this is well-defined and continuous and, moreover, (15) holds
true if and only if $f\neq 0$. Since $\mathbb{T}^{2}$ is compact, this
condition is preserved under small $C^{1}$ perturbations of $z$ and, hence,
small $C^{1}$ perturbations of $w$.

It remains to show that the solutions indeed give overhanging waves. When
$G=0$, we can find $x^{*}\in\mathbb{T}$ and
$\alpha,\alpha^{\prime}\in\mathbb{T}$ such that
$x(\alpha)=x(\alpha^{\prime})=x^{*}$,
$y_{\alpha}(\alpha),y_{\alpha}(\alpha^{\prime})>0$, and
$x_{\alpha}(\alpha)<0<x_{\alpha}(\alpha^{\prime})$. We then deduce from the
implicit function theorem that these conditions continue to hold at some
$\alpha(G),\alpha^{\prime}(G)\in\mathbb{T}$ for $G$ sufficiently small. This
completes the proof. ∎

###### Proof of Theorem 2.

When $G=0$, the profile of (5) does not intersect itself for $A<A_{\max}$,
intersects itself tangentially at one point for $A=A_{\max}$, and intersects
itself transversally at two points for $A>A_{\max}$. By continuity, the
profile corresponding to the solution $W(G,A_{\max}-\varepsilon/2)$ to (19)
does not intersect itself for $G$ sufficiently small, while the profile
corresponding to $W(G,A_{\max}+\varepsilon/2)$ intersects itself transversally
at two points. We can then find some $A\approx A_{\max}$ such that the profile
corresponding to $w(G,A)$ intersects itself tangentially at one point for $G$
sufficiently small. Details are found, for instance, in the proof of [8,
Proposition 10]. We remark that this is the only place in this work where the
$w\in C^{3+a}(\overline{\mathbb{D}})$ is needed. Elsewhere it is sufficient to
work with $w\in C^{1+a}(\overline{\mathbb{D}})$.

∎

##  5 Extensions

There are several directions in which one might take matters further.

#### Finite depth.

In the finite depth setting, we replace (1d) by

$\psi=\text{const.}\qquad\text{on $y=-h$}$

for some $h>0$. We can follow along the same line of argument as in Section
3.1 to arrive at

$(1+\varOmega(y+y\mathcal{T}_{H}y_{\alpha}-\mathcal{T}_{H}(yy_{\alpha})))^{2}=(B-2Gy)((1+\mathcal{T}_{H}y_{\alpha})^{2}+y_{\alpha}^{2})\qquad\text{for
$\beta=0$},$

instead of (11), where

$\mathcal{T}_{H}e^{in\alpha}=\begin{cases}-i\coth(nH)e^{in\alpha}\quad&\text{if
$n\neq 0$}\\\ 0&\text{if $n=0$},\end{cases}$

instead of (13). We may assume without loss of generality $H=kh$. See, for
instance, [15] for details. Since $\mathcal{T}_{H}\to\mathcal{H}$ as
$H\to\infty$ (see, for instance, [6] for details), one can argue as in Section
4 to deduce that for $G$ sufficiently small and $H$ sufficiently large, there
exists a solution to (2)–(4), replacing the last equation of (2) by
$\psi=\text{const.}$ on $y=-H$, for which the fluid surface does not intersect
itself but is overhanging. Also there exists a solution whose fluid surface
intersects itself tangentially along the trough line, enclosing a bubble of
air, namely a touching wave. We omit the details.

#### Point vortices.

When vorticity is concentrated at a point in the fluid region of one period in
the moving frame of reference, rather than constant throughout the fluid
region, (21) also gives an exact solution for zero gravity (see [12, Section
4] for details), whereby one may be able to follow along the same line of
argument as in Section 4 for the existence of overhanging and touching waves
for sufficiently weak gravity, either in the infinite or finite but
sufficiently large depth.

#### Hollow vortices.

Last but not least we consider a hollow vortex, a bounded region of constant
pressure with some nonzero circulation around it. Exact solutions have
recently been found for rotating hollow vortices with $N$-fold symmetry for
any integer $N\geqslant 2$ [11]. Interestingly, the same conformal mapping
also gives exact solutions for non-rotating hollow vortices with the effects
of surface tension [10, 26]. (To compare, Crapper’s capillary waves in an
irrotational flow give rise to exact solutions in constant vorticity flows
without the effects of surface tension.) One then may be able to argue
similarly as above for the existence of hollow vortices in fluid regions with
sufficiently large but bounded area, either rotating with $N$-fold symmetry or
non-rotating with the effects of surface tension.

## Acknowledgments

The work of VMH was supported by the NSF through the award DMS-2009981. The
authors are also indebted to the anonymous referee for their helpful comments,
and especially for identifying and resolving an error in an earlier version of
Lemma 5.

##  Appendix A Exact solution of (2) with zero gravity

Here we detail how to correct errors in [18] and show that when $G=0$, (5) and
(6) solve (2) for $A\in[-A_{\max},A_{\max}]$ (see (9)). Recall that we
identify $\mathbb{R}^{2}$ with $\mathbb{C}$ and employ the notation $z=x+iy$.
Note that (5) extends holomorphically to the lower-half plane as

$z(\alpha+i\beta;A)=\alpha+i\beta-\frac{4iAe^{-i(\alpha+i\beta)}}{1+Ae^{-i(\alpha+i\beta)}}.$
(39)

We begin by writing

$\psi=-\tfrac{1}{2}\varOmega y^{2}-y-f,$ (40)

so that the first, second and last equations of (2) become

$\displaystyle\nabla^{2}f=0$ $\displaystyle\text{in $D$},$ (41) $\displaystyle
f=-\tfrac{1}{2}\varOmega y^{2}-y\quad$ $\displaystyle\text{on $S$},$
$\displaystyle\nabla f\to(0,0)$ $\displaystyle\text{as $y\to-\infty$}.$

Similarly as in Section 3.2, we introduce

$\zeta=e^{-i(\alpha+i\beta)},$

which takes the values inside the unit disk, and we recast (41) in
$\mathbb{D}$. (In [18], the incorrect formula $\zeta=e^{i(\alpha+i\beta)}$ was
used instead, which takes values outside $\mathbb{D}$. This is the source of
most of the errors in [18].)

The Poisson integral formula gives

$f(\zeta)=\text{Re}\left(\frac{1}{2\pi
i}\ointctrclockwise_{|\zeta^{\prime}|=1}f(\zeta^{\prime})\frac{\zeta^{\prime}+\zeta}{\zeta^{\prime}-\zeta}\leavevmode\nobreak\
\frac{d\zeta^{\prime}}{\zeta^{\prime}}\right)\qquad\text{for $|\zeta|<1$},$

where

$f(\zeta^{\prime})=-\frac{1}{2}\varOmega
y^{2}-y=-\frac{2\varOmega(\zeta^{\prime
2}+2A\zeta^{\prime}+1)^{2}}{(\zeta^{\prime}+A)^{2}(\zeta^{\prime}+1/A)^{2}}+\frac{2(\zeta^{\prime
2}+2A\zeta^{\prime}+1)}{(\zeta^{\prime}+A)(\zeta^{\prime}+1/A)}\qquad\text{for
$|\zeta^{\prime}|=1$}$

by (5) (see also (22)). This agrees with the corresponding formula in [18]
except that the latter has a typographical error where the exponent $2$ is
missing in the numerators. When $A\in[-A_{\max},A_{\max}]$, since
$A_{\max}<1/2$,
$f(\zeta^{\prime})\frac{\zeta^{\prime}+\zeta}{\zeta^{\prime}-\zeta}\frac{1}{\zeta^{\prime}}$
has poles at $\zeta^{\prime}=-A$, $\zeta$ and $0$ in $\mathbb{D}$, and we
evaluate the integral by means of the calculus of residues to arrive at

$f(\zeta)=\text{Re}\left(\frac{4\varOmega}{A^{2}-1}\frac{(1-2A^{2})\zeta^{2}+1}{(\zeta+1/A)^{2}}+\frac{4\zeta}{\zeta+1/A}\right)\qquad\text{for
$|\zeta|<1$},$

so that (40) becomes

$\psi=-\beta-\frac{1}{2}\varOmega
y^{2}-\frac{4\varOmega}{A^{2}-1}\text{Re}\left(\frac{(1-2A^{2})\zeta^{2}+1}{(\zeta+1/A)^{2}}\right).$
(42)

Differentiating (42),

$\psi_{\beta}=-1-\varOmega
yx_{\alpha}-\frac{8\varOmega}{A(A^{2}-1)}\text{Re}\left(\frac{\zeta((1-2A^{2})\zeta-A)}{(\zeta+1/A)^{3}}\right).$
(43)

All of these formulas differ from those in [18] by $\zeta\mapsto 1/\zeta$. We
then use the second equation of (2) and make a chain rule calculation to see
that

$|\nabla_{(x,y)}\psi|^{2}=\frac{\psi_{\beta}^{2}}{|z_{\alpha}|^{2}}\qquad\text{on
$S$}.$

Inserting (43) into the third equation of (2), after some algebra we arrive at

$\left(1-2\varOmega\frac{\zeta^{2}-2A\frac{1+A^{2}}{1-A^{2}}\zeta+1}{(\zeta+A)(\zeta+1/A)}\right)^{2}=2B\left(\frac{(\zeta-A)(\zeta-1/A)}{(\zeta+A)(\zeta+1/A)}\right)^{2}.$
(44)

This agrees with [18, (4.7)], which remains invariant under $\zeta\mapsto
1/\zeta$. A straightforward calculation reveals that (44) holds true for (6).

## References

  * [1] S. Agmon, A. Douglis, and L. Nirenberg, _Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II_ , Comm. Pure Appl. Math. 17 (1964), 35–92.
  * [2] Benjamin F. Akers, David M. Ambrose, and J. Douglas Wright, _Gravity perturbed Crapper waves_ , Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2161, 20130526, 14.
  * [3] M. G. Blyth and E. I. Părău, _Stability of waves on fluid of infinite depth with constant vorticity_ , J. Fluid Mech. 936 (2022), Paper No. A46, 14.
  * [4] B. Buffoni, E. N. Dancer, and J. F. Toland, _The regularity and local bifurcation of steady periodic water waves_ , Arch. Ration. Mech. Anal. 152 (2000), no. 3, 207–240.
  * [5] , _The sub-harmonic bifurcation of Stokes waves_ , Arch. Ration. Mech. Anal. 152 (2000), no. 3, 241–271.
  * [6] Adrian Constantin, Walter Strauss, and Eugen Vărvărucă, _Global bifurcation of steady gravity water waves with critical layers_ , Acta Math. 217 (2016), no. 2, 195–262.
  * [7] Diego Córdoba and Elena Di Iorio, _Existence of gravity-capillary crapper waves with concentrated vorticity_ , 2021.
  * [8] Diego Córdoba, Alberto Enciso, and Nastasia Grubic, _On the existence of stationary splash singularities for the Euler equations_ , Adv. Math. 288 (2016), 922–941.
  * [9] G. D. Crapper, _An exact solution for progressive capillary waves of arbitrary amplitude_ , J. Fluid Mech. 2 (1957), 532–540.
  * [10] D. G. Crowdy, _Exact solutions for steady capillary waves on a fluid annulus_ , J. Nonlinear Sci. 9 (1999), no. 6, 615–640.
  * [11] D. G. Crowdy, R. B. Nelson, and V. S. Krishnamurthy, _‘H-states’: exact solutions for a rotating hollow vortex_ , J. Fluid Mech. 913 (2021), R5, 11.
  * [12] Darren G. Crowdy and Johan Roenby, _Hollow vortices, capillary water waves and double quadrature domains_ , Fluid Dyn. Res. 46 (2014), no. 3, 031424, 12.
  * [13] Sergey A. Dyachenko and Vera Mikyoung Hur, _Stokes waves in a constant vorticity flow_ , Nonlinear Water Waves, Tutorials, Schools, and Workshops in the Mathematical Sciences, Birkhäuser, Cham, 2019, pp. 71–86.
  * [14] , _Stokes waves with constant vorticity: folds, gaps and fluid bubbles_ , J. Fluid Mech. 878 (2019), 502–521.
  * [15] , _Stokes waves with constant vorticity: I. Numerical computation_ , Stud. Appl. Math. 142 (2019), no. 2, 162–189.
  * [16] F. D. Gakhov, _Boundary value problems_ , Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966, Translation edited by I. N. Sneddon.
  * [17] Vera Mikyoung Hur and Jean-Marc Vanden-Broeck, _A new application of Crapper’s exact solution to waves in constant vorticity flows_ , Eur. J. Mech. B Fluids 83 (2020), 0–4.
  * [18] Vera Mikyoung Hur and Miles H. Wheeler, _Exact free surfaces in constant vorticity flows_ , J. Fluid Mech. 896 (2020), R1, 10.
  * [19] N. I. Muskhelishvili, _Singular integral equations_ , Wolters-Noordhoff Publishing, Groningen, 1972, Boundary problems of functions theory and their applications to mathematical physics, Revised translation from the Russian, edited by J. R. M. Radok, Reprinted.
  * [20] Hisashi Okamoto and Mayumi Shoji, _Nonexistence of bifurcation from Crapper’s pure capillary waves (Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics)_ , Res. Inst. Math. Sci. Kokyuroku Kyoto Univ. (1991), no. 745, 21–38.
  * [21] , _The mathematical theory of permanent progressive water-waves_ , Advanced Series in Nonlinear Dynamics, vol. 20, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
  * [22] J. A. Simmen and P. G. Saffman, _Steady deep-water waves on a linear shear current_ , Stud. Appl. Math. 73 (1985), no. 1, 35–57.
  * [23] E. R. Spielvogel, _A variational principle for waves of infinite depth_ , Arch. Rational Mech. Anal. 39 (1970), 189–205.
  * [24] A. F. Teles da Silva and D. H. Peregrine, _Steep, steady surface waves on water of finite depth with constant vorticity_ , J. Fluid Mech. 195 (1988), 281–302.
  * [25] V. Volpert, _Elliptic partial differential equations. Volume 1: Fredholm theory of elliptic problems in unbounded domains_ , Monographs in Mathematics, vol. 101, Birkhäuser/Springer Basel AG, Basel, 2011.
  * [26] Rudolf Wegmann and Darren Crowdy, _Shapes of two-dimensional bubbles deformed by circulation_ , Nonlinearity 13 (2000), no. 6, 2131–2141.