2.1. Boundary values of analytic function at the cut

Before proceeding, one must be able to evaluate the complex analytic function by its associated jump on the cut. Given $\varepsilon > 0$, one finds that

(2.12a,b)\begin{align} R({\rm i}\tau \pm \varepsilon) =1 - \frac{1}{\rm \pi}\int_a^b \frac{r(\tau')\,{\rm d}\tau'}{\tau' - \tau \pm {\rm i}\varepsilon}\quad\mbox{and}\quad V({\rm i}\tau \pm \varepsilon) ={-} \frac{1}{\rm \pi}\int_a^b \frac{v(\tau')\,{\rm d}\tau'}{\tau' - \tau \pm {\rm i}\varepsilon}, \end{align}

and we may apply the Sokhotskii–Plemelj theorem to obtain

(2.13)$$\begin{gather} R^{{\pm}}({\rm i}\tau) = \lim_{\varepsilon\to 0^+} R({\rm i}\tau \pm \varepsilon) = 1 - (\hat H \mp {\rm i}) r(\tau), \end{gather}$$

(2.14)$$\begin{gather}V^{{\pm}}({\rm i}\tau) = \lim_{\varepsilon\to 0^+} V({\rm i}\tau \pm \varepsilon) ={-}(\hat H \mp {\rm i}) v(\tau), \end{gather}$$

where we have defined the integral operator $\hat H$ as follows:

(2.15)\begin{equation} \hat H f(\tau) = \frac{1}{\rm \pi}v.p.\int_a^b\frac{f(\tau')\,{\rm d}\tau'}{\tau'-\tau}. \end{equation}

Similar relations hold for the functions $U$ and $B$.

2.2. Equations of motion on the cut

The equations of motion in $w$-plane have been derived previously, and are given by

(2.16)$$\begin{gather} \partial_t R = {\rm i}\left(U R_w - U_w R\right), \end{gather}$$

(2.17)$$\begin{gather}\partial_t V = {\rm i}\left(U V_w - B_w R\right), \end{gather}$$

and can be written in terms of the jumps of the associated functions as follows:

(2.18)\begin{equation} \left.\begin{gathered} r_t + u\hat H r' + r'\hat H u + u'(1-\hat Hr) - r\hat Hu' = 0 \\ v_t + u \hat H v' + v'\hat H u + b'(1-\hat Hr) - r\hat Hb' = 0 \end{gathered}\right\}. \end{equation}

Given a square-root branch point in $z(w) = \sqrt {w-\textrm {i}a}$, the function $R = 1/z_u \sim \sqrt {w-\textrm {i}a}$ vanishes at the branch points. Similarly, it is trivial to show that $R(w,t)$ has zeros at the branch points $\textrm {i}a$ and $\textrm {i}b$

(2.19a,b)\begin{equation} R({\rm i}a) = R({\rm i}b) = 0\quad\mbox{and}\quad R(w\to -{\rm i}\infty) \to 1, \end{equation}

as the solution evolves the branch points do not vanish, but move in the complex plane, see Dyachenko et al. (Reference Dyachenko, Dyachenko, Lushnikov and Zakharov2021).

One can use the shifted Chebyshev basis to efficiently represent the complex analytic functions with a pair of square-root singularities. We introduce the centre of the cut, $c(t) = {(a(t)+b(t))}/{2}$, and its half-length, $l(t)= {(b(t)-a(t))}/{2}$, and use $f_n(w,t)$ as a basis for expansion of the analytic functions $R(w,t)$ and $V(w,t)$. It is convenient to work with the variable $\xi (w,t) = {(c+\textrm {i}w)}/{l}$, then

(2.20)\begin{equation} f_n(\xi) = \left[\xi - \sqrt{\xi^2 - 1}\right]^n = ({-}1)^n \left[T_n(-\xi) + U_{n-1}\left(-\xi\right)\sqrt{\xi^2 - 1}\right], \end{equation}

where $T_n(\xi )$ and $U_{n-1}(\xi )$ for $n = 1,2,\ldots$ are the Chebyshev polynomials of the first and the second kinds, respectively. Note that $w\in [\textrm {i}a,\textrm {i}b]$ is mapped to $\xi \in [-1,1]$ for convenience.

2.3. Expansion of $R(w,t)$ and $V(w,t)$

The complex velocity, $V$, and $R$ may be expanded in the form

(2.21a,b)\begin{equation} V(\xi,t) = \sum_{k=1}^{\infty} v_k(t) f_k(\xi)\quad\mbox{and}\quad R(\xi,t) = 1 + \sum_{k=1}^{\infty} r_k(t) f_k(\xi). \end{equation}

In order to satisfy the conditions (2.19a,b) we have two additional constraints on the coefficients of $R$ that must be satisfied for any $t>0$

(2.22)\begin{equation} \sum_{k=1}^\infty r_k(t) = \sum_{k=1}^{\infty} ({-}1)^k r_k(t) = -1. \end{equation}

It is convenient to rewrite the series defined in relations (2.21a,b) and (2.20) using the following substitution:

(2.23)\begin{equation} \xi = \cos{\chi}\quad\mbox{or, equivalently}\quad w = {\rm i}c - {\rm i}l\cos{\chi},\quad \chi = \eta +{\rm i} \zeta ,\end{equation}

where $\chi \in [-{\rm \pi},{\rm \pi} ]$ is mapped to $\xi \in [-1,1]$. Moreover, the relation (2.23) defines a conformal mapping for complex $w$, such that $\chi (w\in \mathbb {C})\to \mathbb {C^-}$ (see also the illustration in figure 2), and the free surface (blue–white boundary) is located at

(2.24)\begin{equation} \chi(u) = {\rm i}\ln\left[ \xi - \sqrt{\xi^2 - 1}\right],\quad \mbox{where}\ \xi = \frac{c+{\rm i}u}{l}. \end{equation}

Note that the square root in (2.24) is so that $\sqrt {a^2 \,\textrm {e}^{\textrm {i}\phi }}=|a|\exp ({\textrm {i}({\phi }/{2})})$.

Figure 2.

Schematics of $\chi$-map. A periodic strip $-{\rm \pi} \leq \operatorname {Re} \chi < {\rm \pi}$ and $\operatorname {Im} \chi < 0$ (right) is mapped to $w\in \mathbb {C}\cap [\textrm {i}a,\textrm {i}b]$ (left), and the $\operatorname {Im} \chi > 0$ (right) is mapped to the second sheet of the $w$-plane. (a) The blue region is mapped to the fluid domain (the real fluid), and the upper half-plane marked white excluding the cut is the virtual fluid in the first sheet outside the cut. The red circles mark the locations of the branch points, and arrows indicate the positive orientation for a Cauchy-type integral (2.5). An observer marked with an eye that is located far away from the cut (large distance from real line is illustrated by the lightning symbols) sees the dominant term of a far field $1/w$ whose coefficient is a motion integral defined in (2.34a,b). The green domain observed on the left is the virtual fluid of the second Riemann sheet which is seen through the cut. (b) After the conformal map (2.23) the image of a periodic contour around the cut is mapped to the interval $[-{\rm \pi},{\rm \pi} ]$, and the Chebyshev function basis (2.20) becomes the standard Fourier basis (2.25). The image of the real fluid is located in the lower complex plane enclosed within vertical asymptotes $\chi = \pm {{\rm \pi} }/{2}$ (dashed lines), and the virtual fluid from the first sheet is mapped to the white region in the lower half-plane. The second Riemann sheet is unfolded into the upper half-plane of the $\chi$-plane. The singularities in the upper half-plane of $\chi$ are located in the second Riemann sheet of $w$-plane, and can be studied numerically and theoretically.

The components of the Chebyshev series

(2.25)\begin{equation} f_n(\xi) = \left[\xi - \sqrt{\xi^2 - 1}\right]^n = \exp({-{\rm i}n\chi}), \end{equation}

become the standard Fourier series in $\chi$ variable in negative Fourier harmonics. Note that all functions $R$, $V$, $U$, $B$ and their complex conjugates in the $w$-plane can be expanded in the series (2.25) with $n \geq 0$

(2.26a,b)\begin{equation} V(\chi,t) = \sum_{k=1}^{\infty} v_k(t) \exp({-{\rm i}k\chi})\quad\mbox{and}\quad R(\chi,t) = 1 + \sum_{k=1}^{\infty} r_k(t) \exp({-{\rm i}k\chi}). \end{equation}

In order to determine the functions $U$ and $B$ we will first evaluate the auxiliary functions $u(\tau )$ and $b(\tau )$ at the cut. Then, we will use the Sokhotskii–Plemelj theorem to find the values of $U$ and $B$ at the cut, $\chi = \eta + \textrm {i}0 \in [-{\rm \pi},{\rm \pi} ]$. The values of $R$ and $V$ at the reflection of the contour around the cut in the lower half-plane of $w$ can be determined in the $\chi$-plane. The reflection of the cut is located on the negative imaginary axis at $\chi = 0 + \textrm {i}\zeta _{cc}(\tau )$ given by the relation

(2.27)\begin{equation} \left.\begin{gathered} \cos {\rm i}\zeta_{cc} = \frac{c + \tau}{l}, \\ \cos {\rm i}\zeta_{cc} = 2 + 2\frac{a}{l} - \xi = 2 + 2\frac{a}{l} - \cos{\eta}, \end{gathered}\right\} \end{equation}

where $\tau \in [a,b]$ (see also the formula (2.11a,b)). Note that the right-hand side of the formula is strictly greater than $1$, and the values of $\zeta _{cc}$ are purely real and negative, given our choice of the mapping $\chi (w)$. Solving for $\zeta _{cc}$, we find

(2.28)\begin{equation} \zeta_{cc}(\xi) ={-}\ln \left[ y + \sqrt{y^2 - 1} \right] ={-}\operatorname{acosh}{y}, \end{equation}

where $y = 2 + 2({a}/{l}) - \xi$. The functions $\bar R$ and $\bar V$ can then be found on the reflection of the cut $w\in [-\textrm {i}a,-\textrm {i}b]$ (or equivalently $\chi \in \textrm {i}\zeta _{cc}$) by evaluating the formulas

(2.29)\begin{equation} \left.\begin{gathered} \bar R = 1 - {\rm \pi}\sum_{k=1} k_n \left(\sqrt{y^2 -1} -y\right)^k = 1 - {\rm \pi}\sum_{n=1} r_k ({-}1)^k \exp({k\zeta_{cc}}), \\ \bar V ={-} {\rm \pi}\sum_{k=1} v_k \left(\sqrt{y^2 -1} -y\right)^k ={-} {\rm \pi}\sum_{k=1} v_k ({-}1)^k \exp({k\zeta_{cc}}). \end{gathered}\right\} \end{equation}

We may now form the functions $u = V\bar R + \bar V R$ and $b = V \bar V$ and find the boundary values of analytic functions $U$ and $B$ as follows:

(2.30)\begin{equation} \left.\begin{gathered} U(\eta) = \frac{1}{2{\rm \pi} {\rm i}} \int_{-{\rm \pi}}^{\rm \pi} \frac{u(-\eta') - u(\eta')}{2} \cot \frac{\eta'-\eta}{2} {\rm d}\eta', \\ B(\eta) = \frac{1}{2{\rm \pi} {\rm i}} \int_{-{\rm \pi}}^{\rm \pi} \frac{b(-\eta') - b(\eta')}{2} \cot \frac{\eta'-\eta}{2} {\rm d}\eta', \end{gathered}\right\} \end{equation}

which can be efficiently computed by means of the fast Fourier transform.

2.4. Equations in $\chi$-plane and integrals of motion

By making additional transformation (2.23) the equations of motion become particularly simple, and are given by the following relations:

(2.31)$$\begin{gather} -l\sin\eta R_t - \left(\dot c + \dot l \cos\eta\right) R_{\eta} = {\rm i}\left(U R_{\eta} - U_{\eta} R\right), \end{gather}$$

(2.32)$$\begin{gather}-l\sin\eta V_t - \left(\dot c + \dot l \cos\eta\right) V_{\eta} = {\rm i}\left(U V_{\eta} - B_{\eta} R\right). \end{gather}$$

These equations have two integrals of motion which are equivalent to the ones found in Tanveer (Reference Tanveer1993). Indeed, one can integrate equations (2.31) using expansion (2.26a,b). As a result, the following expressions are valid:

(2.33)\begin{equation} \begin{cases} r_1 l(t) ={-}2Q = const \\ v_1 l(t) ={-}2\varGamma =const \end{cases}\Rightarrow \quad {\rm e}^{-{\rm i}\chi} ={-}\frac{l(t)}{2w} \quad\mbox{as}\ w\to\infty, \end{equation}

or

(2.34a,b)\begin{equation} R(w) = 1 + \frac{Q}{w} +{O}\left(\frac{1}{w^2}\right),\quad V(w) = \frac{\varGamma}{w} +{O}\left(\frac{1}{w^2}\right)\quad |w|\to\infty ,\end{equation}

where $Q$ and $\varGamma$ are called ‘winding’ and virtual ‘circulation’. Conservation of circulation can be viewed as Kelvin's theorem for a virtual fluid, and winding is related to the topology of the conformal plane.

2.5. Formation of singularity in infinite time

We shall consider the lowest-order expansion so that the initial datum satisfies the constraints (2.19a,b) and still results in a non-trivial fluid flow. The functions $R$ and $V$ at time $t=0$ are given by

(2.35)$$\begin{gather} V(\xi) = l f_1(\xi) = {\rm i}[w - {\rm i}c - \sqrt{(w-{\rm i}c)^2 + l^2}] \end{gather}$$

(2.36)$$\begin{gather}R(\xi) = 1 - f_2(\xi) = [ 1 + \frac{1}{l^2}( w-{\rm i}c - \sqrt{(w-{\rm i}c)^2 + l^2})^2 ]. \end{gather}$$

It is assumed that, at least for some time, these functions are given by convergent series of the form (2.21a,b), or equivalently the Fourier series (2.26a,b). This is a crucial assumption that is only based on the results of numerical simulations of the short branch cut (Dyachenko et al. Reference Dyachenko, Dyachenko, Lushnikov and Zakharov2021), and preceding theoretical work (see Tanveer Reference Tanveer1993; Dyachenko et al. Reference Dyachenko, Dyachenko, Lushnikov and Zakharov2019). Approximately speaking, this assumption will hold when the only zeros of $R$ in the first Riemann sheet are located at the branch points. Naturally, one must ensure that the initial data (2.35)–(2.36) have no additional zeros in the first sheet.

In order to establish the motion of the branch points we seek the complex conjugated functions given by

(2.37)$$\begin{gather} \tilde V ={-}{\rm i}[w+{\rm i}c - \sqrt{(w+{\rm i}c)^2 + l^2}], \end{gather}$$

(2.38)$$\begin{gather}\tilde R = \left[1 + \frac{1}{l^2}(w+{\rm i}c - \sqrt{(w+{\rm i}c)^2 + l^2})^2 \right]. \end{gather}$$

Then, we must calculate all four functions $V$, $\tilde V$, $R$ and $\tilde R$ on the imaginary axis by replacing $w\to \textrm {i}v$. We introduce

(2.39)$$\begin{gather} S = \sqrt{(v-a)(v-b)} = \sqrt{l^2 - (v-c)^2}, \end{gather}$$

(2.40)$$\begin{gather}F = \sqrt{(v+a)(v+b)} = \sqrt{(v+c)^2 - l^2}. \end{gather}$$

Now

(2.41)$$\begin{gather} V ={-}v + c - {\rm i}S, \end{gather}$$

(2.42)$$\begin{gather}\tilde V = v + c - F, \end{gather}$$

and we may write

(2.43)$$\begin{gather} R = 1 - \frac{\left(v-c + {\rm i}S\right)^2}{l^2} =\frac{2}{l^2} [S^2 - {\rm i}(v-c)S ], \end{gather}$$

(2.44)$$\begin{gather}\tilde R = 1 - \frac{\left(v+c - F\right)^2 }{l^2} = \frac{2}{l^2}F\left(v+c-F\right). \end{gather}$$

Then we introduce

(2.45)\begin{equation} Q = R\tilde V + \tilde R V, \end{equation}

to end up with transport velocity $U$

(2.46)\begin{equation} U = P^- Q. \end{equation}

Notice that the singularities of $Q$ in the upper half-plane are the ones coming from the function $S$. We can replace $Q\to Q_{ess}$ where $Q_{ess}$ only includes the terms proportional to $S$. A simple calculation reveals the following expression:

(2.47)\begin{equation} Q_{ess} = \frac{2{\rm i}}{l^2} \left[2lFS - \left(c^2 + ab + 2cv\right)S\right]. \end{equation}

Then

(2.48)\begin{equation} U = \left(1 - {\rm i}\hat H\right)Q_{ess}, \end{equation}

where $\hat H$ is the Hilbert transform. To calculate the ‘Hilbert’ transform one should remember that, after projecting to the lower half-plane, $Q_{ess}$ must be replaced by its analytic continuation to the upper half-plane. This is done by restoring the regular part by the corresponding singular part as in formula (2.20). It amounts to writing

(2.49)$$\begin{gather} {\rm i}S \to {\rm i}S - (v-c), \end{gather}$$

(2.50)$$\begin{gather}{\rm i}vS \to {\rm i}vS - v(v-c) + \tfrac{1}{2}l^2. \end{gather}$$

Collecting all terms together we find following expression for transport velocity $U$:

(2.51)\begin{equation} U = Q_{ess} + \frac{2}{l^2}\left[ 2cv^2 + abv - c \left(c^2 + \frac{l^2}{2} + ab\right) + 2cI\right], \end{equation}

where

(2.52)\begin{equation} I(v) = \frac{1}{\rm \pi} v. p. \int_{a}^{b} \frac{\sqrt{(s^2-a^2)(b^2-s^2)}\,{\rm d} s}{s-v}. \end{equation}

Now we calculate $U(a)$. As long as $S(a) = 0$ this expression is purely real. After tedious calculations we find that

(2.53)\begin{equation} U(\varepsilon) = \frac{8b}{(1-\varepsilon)^2} \left[-\frac{1}{4} - \varepsilon + \frac{1}{2} \varepsilon^2 + \varepsilon^3 + I(\varepsilon)\right], \end{equation}

here, $\varepsilon = {a}/{b}$ and

(2.54)\begin{equation} I(\varepsilon) = \frac{1}{\rm \pi} \int_{\varepsilon}^1 \sqrt{\frac{(q+\varepsilon)(1-q^2)}{q-\varepsilon}}{\rm d} q. \end{equation}

This integral is expressed in terms of elliptic functions, but we will not provide explicit formulas. We study the asymptotic behaviour of $I(\varepsilon )$ at $\varepsilon \to 0$. To do this we use the expansion

(2.55)\begin{equation} \sqrt{\frac{q+\varepsilon}{q - \varepsilon}} = 1 + \frac{\varepsilon}{q} + \sum_{k=0}^{\infty}c_k \left(\frac{\varepsilon}{q}\right)^{k+2}. \end{equation}

Thus we must estimate the integral

(2.56)\begin{equation} I(\varepsilon) = \frac{1}{\rm \pi} \int_{\varepsilon}^1 \left(1 + \frac{\varepsilon}{q} + \sum_{k=0}^{\infty}c_k \left(\frac{\varepsilon}{q}\right)^{k+2}\right)\sqrt{1-q^2}\,{\rm d} q. \end{equation}

Now we mention that

(2.57)\begin{equation} \frac{1}{\rm \pi} \int_0^1\sqrt{1-q^2}\,{\rm d} q = \frac{1}{4}. \end{equation}

One can show that

(2.58)\begin{equation} U(\varepsilon) = 8b\left(-\frac{1}{4} + \frac{1}{4} + \frac{1}{\rm \pi}\varepsilon\ln\varepsilon + O(\varepsilon)\right), \end{equation}

and therefore

(2.59)\begin{equation} U(\varepsilon) = \frac{8b}{\rm \pi}\varepsilon\ln\varepsilon + O(\varepsilon), \end{equation}

where $O(\varepsilon )$ denotes terms of order $\varepsilon$ and all higher orders. It is not necessary to determine them precisely.

We end up with the following result:

(2.60)\begin{equation} U(a) \to \frac{8}{\rm \pi}a\ln\frac{a}{b} + O(a)\quad \mbox{at}\ a\to 0. \end{equation}

The differential equation

(2.61)\begin{equation} \dot{a} = \frac{8}{\rm \pi} a\ln\frac{a}{b} \end{equation}

with initial data

(2.62)\begin{equation} \ln\frac{a}{b} ={-}\ln\frac{b}{a_0} ={-}c, \end{equation}

gives solution

(2.63)\begin{equation} \ln\frac{a}{b} ={-} c \exp\left({\frac{8t}{\rm \pi}}\right), \end{equation}

and

(2.64a,b)\begin{equation} a = b\exp\left({-c\exp\left({\frac{8t}{\rm \pi}}\right)}\right)\quad c ={-}\ln\frac{b}{a_0}. \end{equation}

This means that singularity $a=0$ is never achieved in a finite time, while it approaches the real line faster than exponential estimates predict.

It is important to emphasize that this conclusion will hold only provided that no singularity from the second sheet crosses the branch cut and invalidates the assumption of a single pair of branch points in the first Riemann sheet. Nevertheless, numerical evidence suggests that at least one such solution does exist, see the figure 2(a) in Dyachenko et al. (Reference Dyachenko, Dyachenko, Lushnikov and Zakharov2021)