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This article is cited in 4 scientific papers (total in 4 papers) The finite-gap method and the periodic Cauchy problem for P. G. Grinevicha, P. M. Santinibc a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b Dipartimento di Fisica, Università di Roma "La Sapienza", Roma, Italy c Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma, Roma, Italy
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    1. IntroductionAnomalous waves, also called rogue or freak waves, are extreme waves of anomalously large amplitude with respect to the surrounding waves, arising apparently from nowhere and disappearing without leaving any trace. Deep sea water was the environment where anomalous waves were studied first, and the term rogue wave was originally coined by oceanographers (a rogue wave can exceed the height of 20 meters and be very dangerous). Although the existence of anomalous waves was notorious even to the ancients, and has been a recurring theme in the literature, the first scientific observation and measurement of an anomalous wave was only made in 1995 at the Draupner oil platform in the North Sea [36]. Starting from the pioneering work on optical fibres [67], it was understood that anomalous waves are not confined to oceanography, and their presence is ubiquitous in nature: they have been observed or predicted in nonlinear optics [67], [40], [41], [4], [51], in Bose–Einstein condensates [81], [13], in plasma physics [7], [55], and in other physical contexts. In the presence of nonlinearity, the accepted explanation for the formation of anomalous waves is the modulational instability of some basic solutions, discovered originally in nonlinear optics [12] and ocean waves [10], [82], but associated with the appearance of anomalous waves only in the last 20 years. In the understanding of the analytic properties of anomalous waves, it is of great importance that nonlinear stages of modulational instability in $1+1$ dimensions are well described by exact solutions of the integrable [85], [86] focusing nonlinear Schrödinger (NLS) equation in $1+1$ dimensions
$$
\begin{equation}
i u_t +u_{xx}+2 |u|^2 u=0, \qquad u=u(x,t)\in\mathbb{C}, \quad x,t\in\mathbb{R},
\end{equation}
\tag{1}
$$
which are known as breathers [50], [65], [5] and can be considered as prototypes of anomalous waves [24], [37], [62], [83], [4], [8]. Such solutions can be reproduced in wave tank laboratories, optical fibres, and photorefractive crystals with a high degree of accuracy [15], [40], [66]. Using the finite-gap method, the NLS Cauchy problem for periodic initial perturbations of an unstable background, which we call the Cauchy problem for anomalous waves, was recently solved to the leading order [33], [35] in the case of a finite number of unstable modes, leading to a quantitative description of the recurrence properties of the multi-breather generalization [39] of the Akhmediev breather [5]. In the simplest case of a single unstable mode this theory describes quantitatively the Fermi–Pasta–Ulam–Tsingou recurrence of anomalous waves in terms of a sequence of Akhmediev breathers [33], [34]. In addition, a finite-gap perturbation theory for $(1+ 1)$- dimensional anomalous waves has also been developed [17], [19] to describe analytically the order-one effect of small physical perturbations of the NLS model on the dynamics of anomalous waves (also see [18]). We observe that in this paper the term anomalous wave is used in an extended sense, and we mean by it order-one (or higher-order) coherent structures over an unstable background which are generated by modulational instability, and the above results on anomalous waves for NLS deal with the analytic aspects of the deterministic theory of periodic anomalous waves, for a finite number of unstable modes. A coherent structure of anomalously large amplitude with respect to the average amplitude arises from an event when the nonlinear interaction of many unstable modes is constructive. Therefore, the formation of an anomalous wave is, strictly speaking, a statistical event. Statistical aspects of the theory of anomalous waves for NLS in $1+1$ dimensions can be found in [31], [21], [26], and [25]. While optical fibres offer a natural testbed for $(1+1)$-dimensional anomalous waves, ocean wind waves are intrinsically two-dimensional, and the extent to which the above analytic solutions of the $1+1$ NLS equation can be observed in the ocean and are relevant to multidimensional nonlinear optics is not yet clear [28], [61]. A good understanding of the deterministic and statistical properties of anomalous waves in $2+1$ dimensions is lacking so far; the main difficulty in generalizing the $1+1$ theory to a multidimensional context is related to the fact that the large majority of physically relevant $2+1$ generalizations of the NLS equation are not integrable, and exact solution techniques are not available. We have in mind, for instance, the elliptic NLS equation, relevant to nonlinear optics [57], [42], the hyperbolic NLS equation, relevant to deep-water gravitational waves [82], and the majority of Davey–Stewartson type equations [20], relevant to nonlinear optics, water waves, plasma physics, and Bose condensates [11], [20], [1], [58], [38]. The integrable Davey–Stewartson (DS) equations [6], [2] can be written in the following form
$$
\begin{equation}
\begin{gathered} \, i u_t +u_{xx}-\nu^2 u_{yy}+2\eta q u=0, \qquad \eta=\pm 1, \quad \nu^2=\pm 1, \\ q_{xx}+\nu^2 q_{yy}=(|u|^2)_{xx}-\nu^2(|u|^2)_{yy}, \\ u=u(x,y,t)\in\mathbb{C}, \quad q(x,y,t)\in\mathbb{R}, \quad x,y,t\in\mathbb{R}, \end{gathered}
\end{equation}
\tag{2}
$$
where $u$ is the complex amplitude of a monochromatic wave, and the real field $q(x,y,t)$ is related to the mean flow. If $\nu^2=-1$, then we have the DS1 equation (surface tension prevails over gravity in the water wave derivation); in this case the sign of $\eta$ is irrelevant, since one can go from the equation with $\eta=-1$ to the equation with $\eta=1$ via the substitutions $q\to -q$ and $x\leftrightarrow y$; therefore there exists only one DS1 equation. If $\nu^2=1$, then gravity prevails over surface tension, and we have the DS2 equations; in this case the sign of $\eta$ cannot be rescaled away, and we distinguish between the focusing and defocusing DS2 equations for $\eta=1$ and $\eta=-1$, respectively. It turns out that the shallow-water limit of the Benney–Roskes/Zakharov–Rubenchik equations [11], [84] leads to the DS1 and defocusing DS2 equations [3]. Although some examples of exact anomalous-wave solutions of the DS equations (2) are present in the literature (see, for instance, [52], [53], [59], and [60]), their importance for physically relevant Cauchy problems is still to be understood. As we will see in the next section, (i) the doubly periodic Cauchy problem is well posed for the focusing and defocusing DS2 equations; (ii) a linear stability analysis shows that, as in the NLS case, the DS2 background solution is linearly stable in the (defocusing) case $\eta=-1$ and unstable for sufficiently small wave vectors in the (focusing) case $\eta=1$. It follows that, although its physical relevance is not clear at the moment, the integrable focusing DS2 equation (with $\nu^2=\eta=1$) is the best mathematical model on which to construct an analytic theory of $(2+1)$-dimensional anomalous waves. This is the goal of this paper. The focusing DS2 equation also has important applications to the differential geometry of surfaces in $\mathbb{R}^4$. The fact that surfaces in $\mathbb{R}^3$ can locally be immersed using squared eigenfunctions of the 2D Dirac operator with real potential $u(x,y)$, and that the modified Novikov–Veselov hierarchy (which is a part of the focusing DS2 hierarchy with extra reality reduction) acts on such immersions was pointed out in [46]. An extension of this construction to surfaces in $\mathbb{R}^4$ was proposed in [64] (also see [47]). For surfaces in $\mathbb{R}^4$ symmetries of immersions are generated by the DS2 hierarchy. If the surface is compact and has the topology of a torus, the corresponding Dirac operators are doubly periodic. The existence of a global representation for immersions of tori into $\mathbb{R}^3$ was proved in [68] (also see [69] and [70]). In this case, in particular, the famous Willmore functional coincides with the energy conservation law for the Novikov–Veselov hierarchy [68]; therefore, it is possible to apply soliton theory to classical geometrical problems (see the survey [73] and the references there for more details). We point out that, in contrast to the 1-dimensional case, the analytic aspects of spectral theory in two dimensions are deeply non-trivial; for example, the proof of the existence of a zero-energy spectral curve obtained in [70] (also see [73]) is based on a deep result of Keldysh. Note that a different approach to constructing the spectral curve, which provides more information about its asymptotic properties, was developed in [48] and [49]. Note also that, in contrast to immersions in $\mathbb{R}^3$, a parametrization of surfaces in $\mathbb{R}^4$ in terms of the Dirac operator is not unique, and different representations can give rise to different dynamics of surfaces. In addition, it is not a priori clear if the DS dynamics is compatible with periodicity. These questions were discussed in [72], where it was shown, in particular, that it is possible to define dynamics preserving the conformal classes of toric surfaces as well as their Willmore functional. In 2021 professor Iskander Taimanov observed his 60th birthday. To honour his very important results in the area of mathematical physics, and, in particular, his works devoted to the doubly-periodic theory for the 2D Dirac operator and DS2 equations and their applications to geometry, we would like to dedicate this article to Taimanov’s 60th birthday. 2. PreliminariesThe DS equations (2) can be written as compatibility conditions for the following pair of auxiliary linear operators (see [6] and [2]):
$$
\begin{equation}
\begin{aligned} \, \nu\vec\psi_y&=i\sigma_3\vec\psi_x+U\vec\psi, \\ \vec\psi_t&=2i\sigma_3\vec\psi_{xx}+2U\vec\psi_x+V\vec\psi, \end{aligned}
\end{equation}
\tag{3}
$$
where
$$
\begin{equation}
\begin{aligned} \, U&=\begin{pmatrix} 0 & u \\ -\eta \overline u & 0 \end{pmatrix}, &\qquad V&=\begin{pmatrix} -\eta(w-iq) & u_x-i\nu u_y \\ -\eta (\overline u_x+i\nu \overline u_y) & -\eta(w+iq) \end{pmatrix}, \\ \nu w_y&=(q -|u|^2 )_x, &\qquad w_x&=-\nu (q+|u|^2)_y. \end{aligned}
\end{equation}
\tag{4}
$$
Equations (2) are non-local, and the function $q$ is defined up to an arbitrary integration constant: We point out that this change of the constant of integration corresponds to the standard gauge transformation
$$
\begin{equation}
u(x,y,t) \mapsto u(x,y,t)\exp\biggl(-i\frac{\eta}{2} \int^t f(\tau)\,d\tau\biggr).
\end{equation}
\tag{6}
$$
The DS equation has the following real forms:
In the Fourier representation we have
$$
\begin{equation}
\widehat q (\vec k)=Q\cdot \widehat{|u|^2}(\vec k)\quad\text{and} \quad Q=\frac{k_x^2-\nu^2k_y^2}{k_x^2+\nu^2k_y^2}\,.
\end{equation}
\tag{7}
$$
In the DS1 case the linear operator $Q$ is unbounded, and this results in very non-trivial analytic effects. For the problem in the infinite $(x,y)$-plane one has to impose additional boundary conditions at infinity; in particular, making a proper choice of the boundary conditions at infinity one can produce exponentially localized solutions (dromions) [29], [30], discovered originally via Bäcklund transformations [14]. In the doubly-periodic case it is not clear if the DS1 system is well defined and, as far as we know, this problem has never been studied properly in the literature. In contrast to the DS1 case, for the DS2 with spatially doubly-periodic boundary conditions the linear operator $Q$ is well defined and has the unit norm on the space of functions with zero mean $L^2_0(T^2)$. It can naturally be extended to a linear map by assuming that
$$
\begin{equation}
\int\!\!\!\int_{T^2} q(x,y,t)\,dx\,dy=0 \quad \text{for all } t.
\end{equation}
\tag{9}
$$
Due to the gauge freedom (6), constraint (9) is not restrictive, and after imposing it, the DS2 flow becomes well defined. Recall that $2+1$ integrable systems are usually non-local. The most famous example is the Kadomtsev–Petviashvily hierarchy. Consider a small perturbation of a constant DS2 solution: The linearized DS2 equation has the following form:
$$
\begin{equation}
i v_t+v_{xx}-v_{yy}+2\eta Q\cdot [|a|^2 v+a^2 \overline v]=0.
\end{equation}
\tag{11}
$$
For a monochromatic perturbation
$$
\begin{equation}
v(x,y,t)=u_1\exp(i[k_x x+k_y y]+\sigma t)+u_{-1} \exp(-i[k_x x+k_y y]+ \sigma t)
\end{equation}
\tag{12}
$$
we obtain
$$
\begin{equation}
[i\sigma-k_x^2+k_y^2]u_1+2 \eta\,\frac{k_x^2-k_y^2}{k_x^2+k_y^2} [|a|^2 u_1+a^2 \overline u_{-1}]=0
\end{equation}
\tag{13}
$$
and
$$
\begin{equation}
[i\sigma+k_x^2-k_y^2] \overline u_{-1}- 2\eta\,\frac{k_x^2-k_y^2}{k_x^2+k_y^2}[|a|^2\overline u_{-1}+ \overline a^2 u_{1}]=0.
\end{equation}
\tag{14}
$$
Therefore,
$$
\begin{equation}
\begin{aligned} \, \sigma=\pm \frac{(k_x^2-k_y^2)\sqrt{4\eta|a|^2-(k_x^2+k_y^2)}} {\sqrt{k_x^2+k_y^2}}\,. \end{aligned}
\end{equation}
\tag{15}
$$
We see that in the defocusing case, when $\eta=-1$, the increment $\sigma$ is imaginary for all wave vectors $(k_x,k_y)$ and the constant solution is linearly stable. In the self-focusing case, when $\eta=1$, harmonic perturbations in the disc $k^2 \leqslant 4 |a|^2$ are unstable, and harmonic perturbations outside this disc are stable. In the doubly- periodic problem we have a finite number of unstable modes. Therefore, from the point of view of the theory of anomalous waves the most important real form of the DS equation is the focusing DS2 equation, where $\nu=1$ and $\eta=1$. This real form also appears in the theory of surfaces in $\mathbb{R}^4$ defined using the generalized Weierstrass representation (see [73] and the references there). In contrast to the focusing NLS equation, DS2 solutions with smooth Cauchy data can blow up in finite time [63], [60], and this fact can be important for physical applications. This problem was discussed, in particular, in [44]. It is possible to construct blow-up solutions using Moutard transformations; for the modified Novikov–Veselov equation this was done in [78]. A beautiful geometric model for this generation of singularities was proposed in [54] and [75]–[77]. Consider an immersion of a surface in $\mathbb{R}^4$ defined in terms of the generalized Weierstrass representation. It is possible to construct explicitly the Moutard transformation corresponding to the inversion of $\mathbb{R}^4$. If the original family of surfaces passes through the origin, then after the inversion this family, as well as the corresponding DS2 solution, blows up for some $t=t_0$. In our text we do not discuss the blow-up of DS2 solutions corresponding to the Cauchy problem for anomalous waves. We defer this interesting question to forthcoming papers. 3. Doubly periodic Cauchy problem for anomalous wavesWe study the spatially doubly-periodic Cauchy problem for the focusing DS2 equation
$$
\begin{equation}
\begin{gathered} \, iu_t+u_{xx}-u_{yy}+2q u=0, \\ q_{xx}+q_{yy}=(|u|^2)_{xx}-(|u|^2)_{yy}, \\ u=u(x,y,t) \in \mathbb{C}, \quad q=q(x,y,t) \in \mathbb{R}, \\ u(x+L_x,y,t)=u(x,y+L_y,t)=u(x,y,t), \\ q(x+L_x,y,t)=q(x,y+L_y,t)=q(x,y,t), \end{gathered}
\end{equation}
\tag{16}
$$
under the assumption that the Cauchy data are a small perturbation of a constant solution:
$$
\begin{equation}
\begin{gathered} \, u(x,y,0)=a+\varepsilon v_0(x,y), \qquad \varepsilon\in\mathbb{R}, \quad \varepsilon \ll 1, \\ v_0(x+L_x,y)=v_0(x,y+L_y)=v_0(x,y). \notag \end{gathered}
\end{equation}
\tag{17}
$$
We call equation (16) with the initial data (17) the doubly-periodic Cauchy problem for anomalous waves. The first auxiliary linear problem can be rewritten as
$$
\begin{equation}
\begin{bmatrix} \partial_x+ i\partial_y & u \\ -\overline u & \partial_x- i\partial_y \end{bmatrix} \begin{bmatrix} \psi_1 \\ \psi_2 \end{bmatrix}=0,
\end{equation}
\tag{18}
$$
where
$$
\begin{equation}
\vec\psi=\begin{bmatrix} \psi_1 \\ i \psi_2 \end{bmatrix}.
\end{equation}
\tag{19}
$$
By analogy with [68], in some situations it is convenient to introduce the complex notation
$$
\begin{equation}
z=x+i y, \quad \overline z=x-i y, \quad \partial_z=\frac{1}{2}(\partial_x-i\partial_y), \quad \partial_{\overline z}=\frac{1}{2}(\partial_x+i\partial_y).
\end{equation}
\tag{20}
$$
To simplify our formulae we write $u(z)$ instead of $u(z,\overline z)$, without assuming that $u(z)$ is holomorphic. In the complex notation equation (18) reads
$$
\begin{equation}
\begin{bmatrix} 2\partial_{\overline z} & u(z) \\ -\overline u(z) & 2\partial_{z} \end{bmatrix} \begin{bmatrix} \psi_1 \\ \psi_2 \end{bmatrix}=0.
\end{equation}
\tag{21}
$$
4. Finite-gap DS2 solutionsIn this section we recall the construction of 2D Dirac operators which are finite- gap at zero energy and of the corresponding DS2 solutions. For 2D Dirac operators with the additional constraint $u=\overline u$ finite-gap formulae were obtained in [70] (also see [73]). We point out that in the periodic theory of the 2D Dirac operator we can use various normalizations of the wave function. In [70], [72], and [73] the following normalization was used: In this text (similarly to [33] and [35]) we work with a non-symmetric normalization of the wave function: The main differences between these two normalizations are as follows.I. If the symmetric normalization (22) is used, then the degree of the divisor is $g+1$, where $g$ is the genus of the spectral curve, and for small perturbations of the constant potential at least one point in the divisor is far away from the resonant point. If one uses (23) instead, then the degree of the divisor is equal to the genus of the spectral curve, and for a small perturbation of the constant potential all divisor points are positioned near the resonant points. II. If the normalization (22) is used, then the spectral data determine the potential $u(z)$ completely. If one uses (23), then the potential is determined by the spectral data only up to a constant phase factor, which should be added as an additional parameter. As we pointed out above, DS2 solutions are defined up to a phase factor, which is an arbitrary function of time, and the non-symmetric normalization (23) ‘hides’ this gauge freedom. The construction of finite-gap solutions for the DS2 equation consists of two steps: 1. Starting from a spectral curve with marked points and divisor, the ‘complex’ Dirac operator
$$
\begin{equation}
\begin{bmatrix} 2\partial_{\overline z} & u(z) \\ v(z) & 2\partial_{z} \end{bmatrix} \begin{bmatrix} \psi_1 \\ \psi_2 \end{bmatrix}=0
\end{equation}
\tag{24}
$$
is constructed. Here ‘complex’ means that the functions $u(z)$ and $v(z)$ are independent. (See § 4.1 below.) 2. Additional conditions are imposed on the spectral curve and the divisor, which imply that $v(z) = - \overline u(z)$ (or $v(z) = \overline u(z)$ in the defocusing case). It is easy to check that these conditions are invariant with respect to time evolution. (See § 4.2.) 4.1. Complex Dirac operatorsConsider the following set of spectral data:
(i) they are holomorphic on $\Gamma$ away from the points $\infty_1$, $\infty_2$, and $\gamma_1, \dots, \gamma_g$; (ii) they have first-order poles at the points $\gamma_1, \dots, \gamma_g$ in the divisor; (iii) they have the following essential singularities at $\infty_1$ and $\infty_2$:
$$
\begin{equation}
\begin{bmatrix} \psi_1(\gamma,z) \\ \psi_2(\gamma,z) \end{bmatrix} = \begin{bmatrix} 1+\dfrac{\xi_1^+(z)}{\lambda_1}+\dfrac{\xi_2^+(z)}{\lambda_1^2}+ \cdots \\ \dfrac {\xi_1^-(z)}{\lambda_1}+\dfrac {\xi_2^-(z)}{\lambda_1^2}+\cdots \end{bmatrix} e^{\lambda_1 z} \quad\text{as } \gamma \to\infty_1,
\end{equation}
\tag{25}
$$
$$
\begin{equation}
\begin{bmatrix} \psi_1(\gamma,z) \\ \psi_2(\gamma,z) \end{bmatrix} = \begin{bmatrix} \chi_0^+(z)+\dfrac{\chi_1^+(z)}{\lambda_2}+\cdots \\ X_{-1}\lambda_2+ \chi_0^-(z)+\dfrac{\chi_1^-(z)}{\lambda_2}+ \cdots\end{bmatrix} e^{\lambda_2 \overline z} \quad \text{as } \gamma \to\infty_2.
\end{equation}
\tag{26}
$$
In (25) and (26) we assume that the pre-exponential terms are locally meromorphic (holomorphic) in neighbourhoods of $\infty_1$ and $\infty_2$, respectively, $X_{-1}$ is a fixed constant, and the coefficients $\xi_j^+(z)$, $\xi_j^-(z)$, $\chi_j^+(z)$, and $\chi_j^-(z)$ of the expansions are a priori unknown. Then using standard arguments we prove that for generic $\gamma$ the functions $\psi_1(\gamma,z)$ and $\psi_2(\gamma,z)$ satisfy (24) for
$$
\begin{equation}
u(z)=-\frac{2\chi_0^+(z)}{X_{-1}}\quad\text{and}\quad v(z)=-2\xi_1^-(z).
\end{equation}
\tag{27}
$$
4.2. Real reductions of DS2Now assume that the finite-gap spectral data satisfy additional reality conditions.
Lemma 1. Let $f(\gamma)$ be a function with poles in ${\mathcal D}$ and at $\infty_2$ and zeros in $\sigma{\mathcal D}$ and at $\infty_1$, with the normalization Denote by $\mathcal C$ the coefficient at the leading term of the expansion for $f(\gamma)$ near $\infty_1$: Then for generic data the constant $\mathcal C$ is real. Proof. Consider the function It is holomorphic, has the same poles and zeros as $f(\gamma)$, and For generic data the function $f(\gamma)$ is unique, and therefore $f_1(\gamma)=f(\gamma)$ and $\overline{\mathcal C}= {\mathcal C}$. Lemma 2. Let the spectral data satisfy reality conditions (1) and (2) formulated in this section, and let the normalization constant $X_{-1}$ satisfy Then the potentials $u(z)$ and $v(z)$ in (24) satisfy the DS2 reduction Proof. If the spectral data satisfies reality conditions (1) and (2), then set
$$
\begin{equation}
\psi_2(\gamma,z)=X_{-1} f(\gamma)\overline{\psi_1(\sigma\gamma,z)}.
\end{equation}
\tag{35}
$$
The function $\psi_2(\gamma,z)$ defined by (35) has the required analytic properties, and Taking the relations into account we complete the proof. We observe that for $z=0$ we have:
$$
\begin{equation*}
\psi_1(\gamma,0)\equiv 1\quad\text{and} \quad \chi_0^+(0)=1,
\end{equation*}
\notag
$$
and therefore Remark 1. The antiholomorphic involution $\sigma$ in our paper coincides with the product of the holomorphic involution $\sigma$ and the antiholomorphic involution $\tau$ from [70]. Therefore, the reality condition (28) on the divisor coincides with the consequence of the corresponding pair of constraints on the divisor associated in [70] with these involutions. Remark 2. In the finite-gap theory of soliton equations we often meet with the following situation. It is sufficiently easy to construct solutions of a generalization of the original system, but the selection of the spectral data corresponding to solutions of the original system requires additional efforts. One problem of this type is the selection of real (or real regular) solutions. It is well known that for the Korteweg–de Vries equation, defocusing nonlinear Schrödinger equation, $\sinh$-Gordon equation, and Kadomtsev–Petviashvili 2 equation real solutions correspond to complex curves with antiholomorphic involution (complex conjugation), with divisors invariant with respect to this complex conjugation. However, for equations like the self-focusing nonlinear Schrödinger equation, $\sin$-Gordon equation, or Kadomtsev–Petviashvili 1 equation the characterization of divisors corresponding to real soltuions is much less trivial. For the self-focusing nonlinear Schrödinger equation and $\sin$-Gordon equation the corresponding conditions were found for the first time in [16], and the answer was formulated in terms of the a certain meromorphic differential, now known as the Cherednik differential. Note that (28) is neither a ‘naive’ reality condition as for the defocusing NLS, nor a Cherednik-type condition, because (28) does not involve the canonical class of the surface. Another problem of this type is the selection of solutions such that some fields are identically zero. An important problem of this type is the fixed-energy periodic problem for the 2-dimensional stationary Schrödinger operator. Operators with non-zero magnetic field which are finite gap at one energy were constructed in 1976 [23], but the selection of the spectral data corresponding to operators with identically zero magnetic field took almost seven years; and the solution, obtained in [79] and [80], was formulated in terms of the Cherednik differential again. The selection of spectral data generating pure magnetic operators was performed much later in [32], and it used a completely different approach. 5. Theta-functional formulaeDenote by $\omega_j$ the basis of holomorphic differentials
$$
\begin{equation}
\oint_{a_j}\omega_k=2\pi i \delta_{j,k}\quad\text{and} \quad \oint_{b_j} \omega_k=b_{jk},
\end{equation}
\tag{39}
$$
where $b_{jk}$ denotes the Riemann matrix of periods. Recall that this matrix is symmetric: $b_{jk}=b_{kj}$, and its real part is negative definite. Remark 3. Two standard normalizations of basic holomorphic differentials are known in the literature, namely, (39) (see [9], [22], and [27]) and the following one (see [56]): Before using theta-functional formulae it is important to check which of these normalizations is used. We also need the following meromorphic differentials $\Omega_0$, $\Omega_z$, $\Omega_{\overline z}$, and $\Omega_t$ such that: 1. They are holomorphic outside the marked points $\infty_1$ and $\infty_2$. 2. They have the following asymptotic behaviour near the marked points:
$$
\begin{equation}
\Omega_0 =\begin{cases} [-\lambda_1^{-1}+O(\lambda_1^{-2})]\,d\lambda_1 & \text{ at } \infty_1, \\ [\lambda_2^{-1}+O(\lambda_2^{-2})]\,d\lambda_2 & \text{ at } \infty_2; \end{cases}
\end{equation}
\tag{41}
$$
$$
\begin{equation}
\Omega_z =\begin{cases} [1+O(\lambda_1^{-2})]\,d\lambda_1 & \text{ at } \infty_1, \\ O(\lambda_2^{-2})\,d\lambda_2 & \text{ at } \infty_2; \end{cases}
\end{equation}
\tag{42}
$$
$$
\begin{equation}
\Omega_{\overline z} =\begin{cases} O(\lambda_1^{-2})\,d\lambda_1 & \text{ at } \infty_1, \\ [1+O(\lambda_2^{-2})]\,d\lambda_2 & \text{ at } \infty_2\,; \end{cases}
\end{equation}
\tag{43}
$$
$$
\begin{equation}
\Omega_t =\begin{cases} [4i\lambda_1+O(\lambda_1^{-2})]\,d\lambda_1 & \text{ at } \infty_1, \\ [-4i\lambda_2+O(\lambda_2^{-2})]\,d\lambda_2 & \text{ at } \infty_2. \end{cases}
\end{equation}
\tag{44}
$$
3. All the $a$-periods of $\Omega_0$, $\Omega_z$, $\Omega_{\overline z}$, and $\Omega_t$ are equal to zero:
$$
\begin{equation}
\oint_{a_j} \Omega_0=\oint_{a_j} \Omega_z=\oint_{a_j} \Omega_{\overline z}= \oint_{a_j} \Omega_{t}= 0, \qquad j=1,\dots,g.
\end{equation}
\tag{45}
$$
Without loss of generality we may assume that the base point of the Abel–Jacobi map coincides with $\infty_1$, and we fix a path ${\mathcal P}_0$ connecting $\infty_1$ with $\infty_2$. Consider the antiderivatives of these differentials in a neighbourhood of this path,
$$
\begin{equation}
\Omega_0=dF_0, \quad \Omega_z=dF_z, \quad \Omega_{\overline z}=dF_{\overline z}, \quad\text{and}\quad \Omega_t=dF_t,
\end{equation}
\tag{46}
$$
and assume that the constants of integration are chosen by assuming that near $\infty_1$ we have
$$
\begin{equation}
\begin{alignedat}{2} F_0&=-\log{\lambda_1}+o(1), &\qquad F_z&=\lambda_1 +o(1), \\ F_{\overline z}&=o(1), &\qquad F_t&=2i \lambda_1^2+o(1). \end{alignedat}
\end{equation}
\tag{47}
$$
Then the constants ${\mathcal C}_0$, ${\mathcal C}_{z}$, ${\mathcal C}_{\overline z}$, and ${\mathcal C}_{t}$ are defined in terms of the expansions of these aniderivatives near $\infty_2$:
$$
\begin{equation}
\begin{alignedat}{2} F_0&=\log{\lambda_2}+{\mathcal C}_0+ o(1),&\qquad F_z&={\mathcal C}_{z} +o(1), \\ F_{\overline z}&=\lambda_2+{\mathcal C}_{\overline z}+o(1), &\qquad F_t&=-2i \lambda_2^2+{\mathcal C}_{t}+o(1). \end{alignedat}
\end{equation}
\tag{48}
$$
The eigenfunctions of the zero-curvature representation are given by Its’s formula, provided in [70] for the 2D Dirac operator:
$$
\begin{equation}
\psi_1(\gamma,z,t) =\exp\biggl[z\int^{\gamma}\Omega_z+ \overline z\int^{\gamma}\Omega_{\overline z}+t\int^{\gamma}\Omega_t\biggr] \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad \times \frac{\theta(\vec A(\gamma)+\vec W_z z+ \vec W_{\overline z}\overline z+\vec W_{t}t-\vec A({\mathcal D})- \vec K)\theta(-\vec A({\mathcal D})-\vec K)}{\theta(\vec A(\gamma)- \vec A({\mathcal D})-\vec K)\theta(\vec W_z z+\vec W_{\overline z}\overline z+ \vec W_{t}t-\vec A({\mathcal D})-\vec K)}\,,
\end{equation}
\tag{49}
$$
$$
\begin{equation}
\psi_2(\gamma,z,t) =X_{-1}f(\gamma)\overline{\psi_1(\gamma,z,t)},
\end{equation}
\tag{50}
$$
where
$$
\begin{equation}
f(\gamma)={\mathcal C}_0^{-1} \exp\biggl[\int^{\gamma}\Omega_0\biggr] \frac{\theta(\vec A(\gamma)+\vec W_0-\vec A({\mathcal D})-\vec K) \theta(\vec A(\infty_2)-\vec A({\mathcal D})-\vec K)} {\theta(\vec A(\gamma)- \vec A({\mathcal D})-\vec K)\theta(\vec A(\infty_2)+\vec W_0- \vec A({\mathcal D})-\vec K)}\,.
\end{equation}
\tag{51}
$$
Here we assume that the constants of integration for the integrals in (49) and (51) are chosen as in (47). We also assume that in (51) the path connecting $\infty_1$ with $\gamma$ is ${\mathcal P}_0$ plus a path connecting $\infty_2$ with $\gamma$. Expanding (49) about $\infty_2$ we obtain
$$
\begin{equation}
\begin{aligned} \, \chi_0^+(z,t)&=\exp[z{\mathcal C}_z+\overline z{\mathcal C}_{\overline z}+ t{\mathcal C}_t] \notag \\ &\qquad \times \frac{\theta(\vec A(\infty_2)+\vec W_z z+ \vec W_{\overline z}\overline z+\vec W_{t}t-\vec A({\mathcal D})-\vec K) \theta(-\vec A({\mathcal D})-\vec K)}{\theta(\vec A(\infty_2)- \vec A({\mathcal D})-\vec K)\theta(\vec W_zz+\vec W_{\overline z}\overline z+ \vec W_{t}t-\vec A({\mathcal D})-\vec K)} \end{aligned}
\end{equation}
\tag{52}
$$
and then, taking (38) into account,
$$
\begin{equation}
\begin{aligned} \, u(z,t)&=\exp[z{\mathcal C}_z+\overline z{\mathcal C}_{\overline z}+ t{\mathcal C}_t] \notag \\ &\qquad \times \frac{\theta(\vec A(\infty_2)+\vec W_z z+ \vec W_{\overline z}\overline z+\vec W_{t}t-\vec A({\mathcal D})-\vec K) \theta(-\vec A({\mathcal D})-\vec K)}{\theta(\vec A(\infty_2)- \vec A({\mathcal D})-\vec K)\theta(\vec W_zz+\vec W_{\overline z}\overline z+ \vec W_{t} t-\vec A({\mathcal D})-\vec K)} u(0,0). \end{aligned}
\end{equation}
\tag{53}
$$
Recall that the Riemann theta function is defined as the following Fourier series [9], [22], [27]:
$$
\begin{equation}
\theta(\vec z)=\theta(\vec z|B)=\sum_{n_l\in\mathbb{Z},\,l=1,\dots,g} \exp\biggl[\frac{1}{2}\sum_{j,k=1}^g b_{jk}n_j n_k+\sum_{j=1}^g n_j z_j\biggr],
\end{equation}
\tag{54}
$$
where $\vec z=(z_1,\dots,z_g) \in \mathbb{C}^g$.
6. Direct spectral transformFor spatially one-dimensional problems the existence of a spectral curve for a generic periodic potential can be proved rather easily using the classical theory of ordinary differential equations. In contrast to the one-dimensional case, in the two-dimensional case one requires much more serious analytic tools. Two main approaches are currently used. In [69] and [73] the spectral curve for the doubly- periodic problem (21) was constructed using Keldysh’s theorem on analytic pencils of compact operators. In [48] and [49] the spectral curves for the non-stationary heat conductivity operator and two-dimensional doubly-periodic Schrödinger operator at a fixed energy level were constructed using perturbation theory, and we follow this approach in our paper. 6.1. The unperturbed spectral curveWe observe that, in contrast to the one-dimensional case, the existence of a ‘good’ spectral curve depends essentially on the analytic properties of operators. For example, for the Lax operators associated with the DS1 equation or Kadomtsev–Petviashvili 1 equation, the curve constructed using the methods mentioned above does not admit a good local compactification [73]. Using the scaling properties of the DS equation we may assume without loss of generality that $a=1$. Let $\mathcal L$ denote the Dirac operator multiplied by $\sigma_3$:
$$
\begin{equation}
{\mathcal L}={\mathcal L}_0+\epsilon {\mathcal L}_1,\ \ \text{where} \ \ {\mathcal L}_0 =\begin{bmatrix} \partial_x+ i\partial_y & 1 \\ 1 &-\partial_x+ i\partial_y \end{bmatrix} \ \ \text{and}\ \ {\mathcal L}_1 =\begin{bmatrix} 0 & v(x,y) \\ \overline v(x,y) & 0 \end{bmatrix}.
\end{equation}
\tag{55}
$$
The zero Bloch eigenfunctions for ${\mathcal L}_0$ are
$$
\begin{equation}
\Psi_0(p,x,y,t)=\begin{bmatrix} 1 \\-ip +q \end{bmatrix} \exp[i(px+q y)-2pqt], \quad p^2+q^2=1, \quad p,q\in\mathbb{C}.
\end{equation}
\tag{56}
$$
Therefore, the unperturbed spectral curve $\Gamma_0$ is defined by and it can conveniently be parametrized as
$$
\begin{equation}
p=\frac{1}{2}\biggl[\tau+\frac{1}{\tau}\biggr]\quad\text{and} \quad q=-\frac{i}{2}\biggl[\tau-\frac{1}{\tau}\biggr],
\end{equation}
\tag{58}
$$
that is, The unperturbed wave function can also be written as
$$
\begin{equation}
\Psi_0(\tau,z,t)=\begin{bmatrix} 1 \\ -i\tau \end{bmatrix} \exp\biggl[\frac{i}{2}\biggl(\tau\overline z+\frac{z}{\tau}+ \biggl(\tau^2-\frac{1}{\tau^2}\biggr)t \biggr)\biggr].
\end{equation}
\tag{60}
$$
The marked points and the local parameters are, respectively,
$$
\begin{equation}
\infty_1\colon\tau=0, \quad \infty_2\colon \tau=\infty, \quad \lambda_1=\frac{i}{2\tau},\quad\text{and} \quad \lambda_2=\frac{i\tau}{2}\,.
\end{equation}
\tag{61}
$$
The eigenfunctions (56) are Bloch periodic:
$$
\begin{equation}
\begin{aligned} \, \Psi_0(p,x+L_x,y,t)&=\varkappa_x \Psi_0(p,x,y,t), \\ \Psi_0(p,x,y+L_y,t)&=\varkappa_y \Psi_0(p,x,y,t), \end{aligned}
\end{equation}
\tag{62}
$$
with Bloch multipliers
$$
\begin{equation}
\varkappa_x=\exp[ip L_x]\quad\text{and} \quad \varkappa_y=\exp[iq L_y].
\end{equation}
\tag{63}
$$
Following [48] and [49], consider the image of $\Gamma_0$ under the map (63). A pair of points $\tau_1=p_1+iq_1$, $\tau_2=p_2+iq_2$ is called resonant if their images coincide, namely,
$$
\begin{equation}
\varkappa_x(\tau_1)=\varkappa_x(\tau_2)\quad\text{and} \quad \varkappa_y(\tau_1)=\varkappa_y(\tau_2).
\end{equation}
\tag{64}
$$
Doubly-periodic small perturbations of operators result in transformations of double points into thin handles [48], [49], [71], [73], so it is natural to develop perturbation theory near such pairs. By analogy with [35], using the scaling symmetry of DS2 we assume that This assumption simplifies the calculations significantly.The Fourier harmonics of a perturbation are enumerated by pairs of integers:
$$
\begin{equation}
k_x=n_x\frac{2\pi}{L_x}\,, \quad k_y=n_y\frac{2\pi}{L_y}\,, \qquad n_x,n_y\in\mathbb{Z}.
\end{equation}
\tag{66}
$$
In our paper we assume that the periods $L_x$ and $L_y$ are generic, and therefore
Equations (64) are equivalent to the following pair of equations:
$$
\begin{equation}
\begin{cases} \tau_2-\tau_1=k_x+ i k_y, \\ \dfrac{1}{\tau_2}-\dfrac{1}{\tau_1}=k_x- i k_y, \end{cases}
\end{equation}
\tag{67}
$$
where $k_x$ and $k_y$ are defined by (66) for some integers $n_x$ and $n_y$. Remark 4. Equation (67) has the following interpretation: for a monochromatic perturbation with fixed $n_x$ and $n_y$ the matrix elements are non-zero only if (67) is fulfilled. Therefore, only the wave functions of the corresponding resonant pairs appear in the leading order of perturbation theory. If $k_x^2+k_y^2<4$, then the mode is unstable, otherwise it is stable. We have two types of resonant pairs $(\tau_1,\tau_2)$. 1. Resonant pairs corresponding to unstable modes ($k_x^2+k_y^2<4$):
$$
\begin{equation}
\begin{aligned} \, \tau_1&=\frac{k_x+i k_y}{2} \biggl[-1 \pm i\sqrt{\frac{4-k_x^2 -k_y^2}{k_x^2+k_y^2}}\,\biggr], \\ \tau_2&=\frac{k_x+i k_y}{2} \biggl[1 \pm i\sqrt{\frac{4-k_x^2-k_y^2}{k_x^2+k_y^2}}\,\biggr], \end{aligned}\qquad |\tau_1|=|\tau_2|=1.
\end{equation}
\tag{68}
$$
By analogy with the NLS case [33], [35] it is convenient to parametrize the unstable modes by angles:
$$
\begin{equation}
k_x=2 \cos\phi \cos\theta \quad\text{and}\quad k_y=2 \cos\phi \sin\theta.
\end{equation}
\tag{69}
$$
Then
$$
\begin{equation}
\tau_1=-e^{i(\theta\mp\phi)}\quad\text{and} \quad \tau_2=e^{i(\theta\pm\phi)}.
\end{equation}
\tag{70}
$$
2. Resonant pairs corresponding to stable modes ($k_x^2+k_y^2>4$):
$$
\begin{equation}
\tau_1=\frac{k_x+ik_y}{2} \biggl[-1+\sqrt{\frac{k_x^2+k_y^2-4}{k_x^2+k_y^2}}\,\biggr], \qquad \tau_2=-\frac{1}{\overline\tau_1}\,.
\end{equation}
\tag{71}
$$
Remark 5. Because of the reality condition, the wave vectors $(k_x,k_y)$ and $(-k_x,-k_y)$ appear simultaneously, and they correspond to the same unstable mode. Following [33], we introduce a finite-gap approximation by neglecting all stable modes. Consider all resonant pairs $(\tau_{2j-1},\tau_{2j})$, $j=1,\dots,2N$, corresponding to unstable modes, where $N$ is the number of unstable modes. We also assume that $\operatorname{Im}(\tau_{2j-1}\tau_{2j}^{-1})>0$ for all $j$ and the points $\tau_1,\tau_3,\tau_5,\dots,\tau_{2j-1}$ are ordered clockwise. Example 1 (see Fig. 1). Let $L_x=2\pi/1.2$ and $L_y=2\pi/1.4$. Then $k_x=1.2 n_x$, $k_y=1.4 n_y$, and we have four unstable modes:  Figure 1.(a) The enumeration of the resonant points for Example 1; (b) the system of $a$- and $c$-cycles for Example 1; (c) the corresponding system of $a$- and $b$-cycles. We introduce a system of basic cycles for the unperturbed spectral curve. Let $a_j$ be a small cycle about the point $\tau_{2j}$ which is oriented counterclockwise, or equivalently, a small cycle about the point $\tau_{2j-1}$ which is oriented clockwise. Denote the line segment beginning at $ \tau_{2j-1}$ and ending at $\tau_{2j}$ by $c_j$. Of course, but $c_j\cdot c_k$ is not necessary zero. However, if we define $b_j$ by then we obtain a canonical basis of cycles on the unperturbed curve:
$$
\begin{equation*}
a_j \cdot a_k=b_j \cdot b_k=0, \quad a_j \cdot b_k=\delta_{j,k}.
\end{equation*}
\notag
$$
Remark 6. The choice of $b$-cycles corresponding to a fixed system of $a$-cycles is not unique, since any integer symplectic transformation In this paper we use the same approximation as in [33]: the off-diagonal terms of the Riemann matrix and the $b$-periods of meromorphic differentials with zero $a$-periods are calculated for the unperturbed curve. The basic differentials on the unperturbed curve $\Gamma_0$ are
$$
\begin{equation}
\omega_j=\biggl[\frac{1}{\tau-\tau_{2j}}- \frac{1}{\tau-\tau_{2j-1}}\biggr]\,d\tau= d\log\biggl[\frac{\tau-\tau_{2j}}{\tau-\tau_{2j-1}}\biggr],\qquad j=1,\dots,g,
\end{equation}
\tag{75}
$$
$$
\begin{equation}
dp=d\biggl(\frac{1}{2}\biggl[\tau+\frac{1}{\tau}\biggr]\biggr)= \frac{iq}{\tau}\,d\tau, \qquad dq=-d\biggl(\frac{i}{2}\biggl[\tau-\frac{1}{\tau}\biggr]\biggr)= -\frac{i p}{\tau}\,d\tau,
\end{equation}
\tag{76}
$$
and
$$
\begin{equation}
\begin{alignedat}{2} \Omega_0&=\frac{d\tau}{\tau}=d\log\tau, &\qquad \Omega_{z}&=-\frac{i\,d\tau}{2\tau^2}=d\biggl(\frac{i}{2\tau}\biggr), \\ \Omega_{\overline z}&=\frac{i}{2}\,d\tau, &\qquad \Omega_t&=\frac{i}{2}\,d\biggl(\tau^2-\frac{1}{\tau^2}\biggr). \end{alignedat}
\end{equation}
\tag{77}
$$
Lemma 3. For the unperturbed curve the periods of differentials are as follows: Remark 7. The double ratio in (81) is always real, but its sign depends on the relative position of the points $\tau_{2j-1}$, $\tau_{2j}$, $\tau_{2k-1}$, and $\tau_{2k}$. The possible cases are shown in Fig. 3.  Figure 3.The possible configurations in formula (81): (a), (b) the double ratio is positive, and the period $b_{jk}$ is real; (c) the double ratio is negative, and $b_{jk}$ is a real number plus $\pi i$. 6.2. The perturbed spectral curve in the leading orderWe restrict ${\mathcal L}$ to the space of Bloch functions ${\mathcal F}(p,q)$ with fixed Bloch multipliers:
$$
\begin{equation}
\begin{gathered} \, \psi(x+L_x,y)=\varkappa_x \psi(x,y), \quad \psi(x,y+L_y)=\varkappa_y \psi(x,y), \\ \varkappa_x=\exp{[iL_x p]}, \quad \varkappa_y=\exp{[iL_yq]}, \qquad p,q\in\mathbb{C}. \end{gathered}
\end{equation}
\tag{83}
$$
The operator ${\mathcal L}_0$ has the following basis of eigenfunctions in ${\mathcal F}(p,q)$:
$$
\begin{equation}
\psi^{(\pm)}_{m,n}=\begin{bmatrix} 1 \\ -i p_m \pm \sqrt{1-p_m^2}\, \end{bmatrix} \exp\bigl(i[p_m x+q_n y]\bigr),
\end{equation}
\tag{84}
$$
where
$$
\begin{equation}
p_m=p+\frac{2\pi}{L_x}m \quad\text{and}\quad q_n=q+\frac{2\pi}{L_y}n, \qquad m,n\in\mathbb{Z},
\end{equation}
\tag{85}
$$
which solve the eigenvalue equation
$$
\begin{equation}
{\mathcal L}_0 \psi^{(\pm)}_{m,n}= \bigl(-q_n \pm \sqrt{1-p_m^2}\,\bigr)\psi^{(\pm)}_{m,n}.
\end{equation}
\tag{86}
$$
In (83)–(86) we do not assume that $p^2+q^2=1$. Consider a monochromatic unstable perturbation
$$
\begin{equation}
v(x,y)=c_{j} e^{i(k_x x+k_y y)}+c_{-j} e^{-i(k_x x+k_y y)}, \qquad k_x,k_y \in \mathbb{R}, \quad k_x^2+ k_y^2 < 4,
\end{equation}
\tag{87}
$$
and the corresponding resonant pair $(\tau_{2j-1},\tau_{2j})$:
$$
\begin{equation}
\begin{gathered} \, \tau_{2j-1}=p_{2j-1}+iq_{2j-1}, \qquad \tau_{2j}=p_{2j}+i q_{2j}, \\ p_{2j}-p_{2j-1}=k_x, \qquad q_{2j}-q_{2j-1}=k_y, \\ |\tau_{2j-1}|=|\tau_{2j}|=1. \end{gathered}
\end{equation}
\tag{88}
$$
(Recall that for each unstable mode we have two resonant pairs associated with it: $(\tau_{2j-1},\tau_{2j})$ and $(\tau_{2j+2N-1},\tau_{2j+2N})=(-\tau_{2j},-\tau_{2j})$.) The restriction of ${\mathcal L}_0$ to ${\mathcal F}(p_{2j-1},q_{2j-1})={\mathcal F}(p_{2j},q_{2j})$ has a two-dimensional zero subspace; it is spanned by the functions $f^{(+)}_{2j-1}$ and $f^{(+)}_{2j}$, where
$$
\begin{equation}
f^{(\pm)}_{2j-1}=\begin{bmatrix} 1 \\ -i p_{2j-1} \pm q_{2j-1} \end{bmatrix} \exp\bigl(i[p_{2j-1}x+q_{2j-1}y]\bigr)
\end{equation}
\tag{89}
$$
and
$$
\begin{equation}
f^{(\pm)}_{2j}=\begin{bmatrix} 1 \\ -i p_{2j} \pm q_{2j} \end{bmatrix} \exp\bigl(i[p_{2j}x+q_{2j}y]\bigr).
\end{equation}
\notag
$$
Following [48] and [49], we calculate the perturbation of the Riemann surface near this resonant pair. Denote by $\widehat{\mathcal F}(\delta p,\delta q)$ the Bloch space with multipliers
$$
\begin{equation}
\begin{gathered} \, \widetilde\varkappa_x=\exp{[iL_x(p_{2j-1}+\delta p)]}\quad\text{and} \quad \widetilde\varkappa_y=\exp{[iL_y (q_{2j-1}+\delta q)]}, \\ \notag |\delta p|\ll1, \quad |\delta q|\ll1. \end{gathered}
\end{equation}
\tag{90}
$$
In the leading-order approximation with respect to $\delta p$ and $\delta q$ we have
$$
\begin{equation}
\begin{aligned} \, \notag \widetilde f_k^{(\pm)}&=\begin{bmatrix} 1 \\ -i p_{2j-2+k} \pm q_{k}-i\delta p \mp \dfrac{p_{k}}{q_{k}} \delta p \end{bmatrix} \\ &\qquad\times \exp\bigl(i[(p_{k}+\delta p)x+(q_{k}+\delta q)y]\bigr),\qquad k=2j-1,2j, \end{aligned}
\end{equation}
\tag{91}
$$
and
$$
\begin{equation}
{\mathcal L}_0 \widetilde f_{k}^{(+)}= \biggl(-\delta q-\frac{p_{k}}{q_{k}}\delta p\biggr)\widetilde f_{k}^{(+)}.
\end{equation}
\tag{92}
$$
In the leading-order approximations for the matrix representation of the block of ${\mathcal L}_1$ corresponding to this subspace we have
$$
\begin{equation}
\begin{bmatrix} 0 & \langle f^{(+)}_{2j-1}|{\mathcal L}_1|f^{(+)}_{2j}\rangle \\ \langle f^{(+)}_{2j}|{\mathcal L}_1|f^{(+)}_{2j-1}\rangle & 0 \end{bmatrix}.
\end{equation}
\tag{93}
$$
Let us calculate these matrix elements in ${\mathcal L}_1$. The dual basic vectors $f_{k}^{(+)*}$, $k=2j-1,2j$, are defined by
$$
\begin{equation}
\begin{aligned} \, f^{(+)*}_{k}(x+L_x,y)&=\exp{[-iL_x p_{k}]} f^{(+)*}_{k}(x,y), \\ f^{(+)*}_{k}(x,y+L_y)&=\exp{[-iL_y q_{k}]} f^{(+)*}_{k}(x,y), \end{aligned}
\end{equation}
\tag{94}
$$
and
$$
\begin{equation}
\langle f^{(+)*}_{k},f_{l}^{(+)}\rangle=\delta_{k,l}, \quad \langle f^{(+)*}_{k},f_{l}^{(-)}\rangle=0, \qquad l=2j-1,2j;
\end{equation}
\tag{95}
$$
therefore,
$$
\begin{equation}
f^{(+)*}_{k}=\frac{1}{2 L_x L_y q_{k}}\begin{bmatrix} i \overline\tau_{k}, 1 \end{bmatrix} \exp(-i[ p_{k} x+q_{k} y])
\end{equation}
\tag{96}
$$
and
$$
\begin{equation}
\nonumber \langle f^{(+)}_{2j-1}|{\mathcal L}_1|f^{(+)}_{2j}\rangle = \frac{1}{2 q_{2j-1}}\begin{bmatrix} i \overline\tau_{2j-1}, 1 \end{bmatrix} \begin{bmatrix} 0 & c_{-j} \\ \overline c_j & 0 \end{bmatrix} \begin{bmatrix}1 \\ -i \tau_{2j} \end{bmatrix}
\end{equation}
\notag
$$
$$
\begin{equation}
=-\alpha_j=\frac{\overline{c}_j+\overline\tau_{2j-1}\tau_{2j} c_{-j}} {2q_{2j-1}}\,,
\end{equation}
\tag{97}
$$
$$
\begin{equation}
\nonumber \langle f^{(+)}_{2j}|{\mathcal L}_1|f^{(+)}_{2j-1}\rangle = \frac{1}{2 q_{2j}}\begin{bmatrix} i \overline\tau_{2j}, 1 \end{bmatrix} \begin{bmatrix} 0 & c_{j} \\ \overline c_{-j} & 0 \end{bmatrix} \begin{bmatrix}1 \\ -i \tau_{2j-1} \end{bmatrix}
\end{equation}
\notag
$$
$$
\begin{equation}
=\beta_j=\frac{\overline{c}_{-j}+\overline\tau_{2j}\tau_{2j-1} c_{j}} {2q_{2j}}\,.
\end{equation}
\tag{98}
$$
Let $(\tau_{2j-1},\tau_{2j})$ and $(\tau_{2j-1+2N},\tau_{2j+2N})=(-\tau_{2j-1},-\tau_{2j}) $ be two pairs of resonant points corresponding to the same monochromatic perturbation. Then we have the following symmetry:
$$
\begin{equation}
\alpha_{j+N}\beta_{j+N}=\overline{\alpha_{j}\beta_{j}}.
\end{equation}
\tag{99}
$$
Near the resonant pair $(\tau_{2j-1},\tau_{2j})$ the spectral curve is defined in the leading order by
$$
\begin{equation}
\det\begin{bmatrix} -\dfrac{p_{2j-1}}{q_{2j-1}} \delta p -\delta q & -\varepsilon\alpha_j \\ \varepsilon\beta_j & -\dfrac{p_{2j}}{q_{2j}} \delta p -\delta q \end{bmatrix}=0.
\end{equation}
\tag{100}
$$
Using equation (100) we define $\delta q$ locally as a two-valued function of $\delta p$. We observe that $\delta p$ is a well-defined local parameter near the points $\tau_{2j-1}$ and $\tau_{2j}$; it defines a local isomorphism between neighbourhoods of these points, and the map is locally a two-sheeted covering. To calculate the branch point of this covering set
$$
\begin{equation}
\delta q=\delta\widetilde q-\frac{1}{2}\biggl[\frac{p_{2j-1}}{q_{2j-1}}+ \frac{p_{2j}}{q_{2j}}\biggr]\delta p.
\end{equation}
\tag{102}
$$
Then equation (100) becomes
$$
\begin{equation}
\det\begin{bmatrix} -\dfrac{q_{2j}p_{2j-1}-q_{2j-1}p_{2j}}{2q_{2j-1}q_{2j}}\delta p- \delta\widetilde q & -\varepsilon\alpha_j \\ \varepsilon\beta_j & \dfrac{q_{2j}p_{2j-1}-q_{2j-1}p_{2j}}{2q_{2j-1}q_{2j}}- \delta\widetilde q \end{bmatrix}=0.
\end{equation}
\tag{103}
$$
Branch points correspond to double roots of (103) with respect to $\widetilde\delta q$, so that they correspond to the following values of $\delta p$:
$$
\begin{equation}
\begin{gathered} \, \delta p=\pm \frac{2q_{2j-1}q_{2j}}{q_{2j}p_{2j-1}-q_{2j-1}p_{2j}} \varepsilon\sqrt{\alpha_j \beta_j}\,, \\ \frac{p_{2j-1}}{q_{2j-1}} \delta p +\delta q= \pm\varepsilon\sqrt{\alpha_j \beta_j}\,. \end{gathered}
\end{equation}
\tag{104}
$$
Note that
$$
\begin{equation*}
q_{2j}p_{2j-1}-q_{2j-1}p_{2j}=\operatorname{Im} \frac{\tau_{2j}}{\tau_{2j-1}}= \operatorname{Im}(\tau_{2j}\overline\tau_{2j-1}).
\end{equation*}
\notag
$$
Assume that we have fixed some value of $\sqrt{\alpha_j \beta_j}$; we use this value in all formulae below. For example, for generic data we may assume that $\operatorname{Re}\sqrt{\alpha_j \beta_j}>0$. Taking (76) into account we obtain the following formulae for the branch points of the map (101) in the leading order near $\tau_{2j-1}$ and $\tau_{2j}$, respectively:
$$
\begin{equation}
\begin{aligned} \, E_{4j-4+k}&=\tau_{2j-1}+(-1)^{k-1}\,\frac{2\tau_{2j-1}q_{2j}} {i\operatorname{Im}(\tau_{2j}\overline\tau_{2j-1})}\, \varepsilon\sqrt{\alpha_j \beta_j}\,, \\ E_{4j-2+k}&=\tau_{2j-1}+(-1)^{k-1}\,\frac{2\tau_{2j}q_{2j-1}} {i\operatorname{Im}(\tau_{2j}\overline\tau_{2j-1})}\, \varepsilon\sqrt{\alpha_j \beta_j}\,. \end{aligned}
\end{equation}
\tag{105}
$$
We assume here that
$$
\begin{equation}
\delta p=\begin{cases} \dfrac{2q_{2j-1}q_{2j}}{\operatorname{Im}(\tau_{2j}\overline\tau_{2j-1})}\, \varepsilon\sqrt{\alpha_j\beta_j} & \text{at}\ E_{4j-3} \sim E_{4j-1}, \\ -\dfrac{2q_{2j-1}q_{2j}}{\operatorname{Im}(\tau_{2j}\overline\tau_{2j-1})}\, \varepsilon\sqrt{\alpha_j\beta_j} & \text{at}\ E_{4j-2} \sim E_{4j}. \end{cases}
\end{equation}
\tag{106}
$$
For the perturbed operator the spectral curve is defined as follows (see Fig. 4). We cut the $\tau$-plane along the intervals $(E_{2j-1},E_{2j})$. For each resonant pair $(\tau_{2j-1},\tau_{2j})$ we glue together the sides of the cuts $(E_{4j-3},E_{4j-2})$ and $(E_{4j-1},E_{4j})$. The point $E_{4j-3}$ is glued to $E_{4j-1}$, and $E_{4j-2}$ to $E_{4j}$. Moreover, if we glue together a pair of points, then the corresponding values of the Bloch multipliers $\varkappa_x$ must be equal. The cycle $a_j$ is an oval surrounding the cut $(E_{4j-1},E_{4j})$ and oriented counterclockwise, the cycle $c_j$ is the union of the oriented intervals $[E_{4j-3},0]$ and $[0,E_{4j-1}]$, and the cycles $b_j$ are defined by (73). To calculate the basic differential $\omega_j$ in the leading order it is sufficient to know the positions of the points $E_k$, $k=4j-3,\dots,4j$, while the other branch points appear in higher-order corrections. On the corresponding elliptic curve we can use the following approximation:
$$
\begin{equation}
\begin{aligned} \, \notag \omega_j&=\frac{d\tau}{\sqrt{(\tau-E_{4j-1})(\tau-E_{4j})}}- \frac{d\tau}{\sqrt{(\tau-E_{4j-3})(\tau-E_{4j-2})}} \\ &=d\log\biggl[\frac{\tau-\tau_{2j}+\sqrt{(\tau-E_{4j-1})(\tau-E_{4j})}} {\tau-\tau_{2j-1}+ \sqrt{(\tau-E_{4j-3})(\tau-E_{4j-2})}}\biggr]. \end{aligned}
\end{equation}
\tag{107}
$$
In (107) we assume that if $\tau-\tau_{2j-1}$ is of order 1, then
$$
\begin{equation*}
\sqrt{(\tau-E_{4j-3})(\tau-E_{4j-2})}\sim \tau-\tau_{2j-1}.
\end{equation*}
\notag
$$
Analogously, if $\tau-\tau_{2j}$ is of order 1, then
$$
\begin{equation*}
\sqrt{(\tau-E_{4j-2})(\tau-E_{4j-1})}\sim \tau-\tau_{2j},
\end{equation*}
\notag
$$
and therefore outside a neighbourhood of this resonant pair formula (107) coincides with (75) in the leading order. Near the resonant points, in the leading order we have
$$
\begin{equation}
\omega_j=\begin{cases} d\log\bigl[\tau-\tau_{2j}+\sqrt{(\tau-E_{4j-1})(\tau-E_{4j})}\,\bigr], & \tau\sim\tau_{2j}, \\ -d\log\bigl[\tau-\tau_{2j-1}+\sqrt{(\tau-E_{4j-2})(\tau-E_{4j-2})}\,\bigr], & \tau\sim\tau_{2j-1}; \end{cases}
\end{equation}
\tag{108}
$$
therefore, in the leading-order approximation the basic holomorphic differential $\omega_j$ on the handle connecting $\tau_{2j-1}$ with $\tau_{2j}$ can be written as
$$
\begin{equation}
\omega_j=d\log\biggl[\frac{p_{2j-1}}{q_{2j-1}} \delta p+\delta q\biggr]= -d\log\biggl[\frac{p_{2j}}{q_{2j}} \delta p+\delta q\biggr].
\end{equation}
\tag{109}
$$
The points in the divisor are defined by the following condition: the first component of the Bloch eigenfunction for the perturbed operator is equal to zero at $z=0$, or, equivalently
$$
\begin{equation}
\begin{bmatrix} -\dfrac{p_{2j-1}}{q_{2j-1}} \delta p -\delta q & -\varepsilon\alpha_j \\ \varepsilon\beta_j & -\dfrac{p_{2j}}{q_{2j}}-\delta q \end{bmatrix} \begin{bmatrix} 1 \\ -1 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \end{bmatrix};
\end{equation}
\tag{110}
$$
therefore, for the point $\gamma_j$ we have
$$
\begin{equation}
\frac{p_{2j-1}}{q_{2j-1}}\delta p+\delta q=\varepsilon\alpha_j.
\end{equation}
\tag{111}
$$
Combining (104) and (111) we obtain the following. Lemma 4. Let $E_{4j-3}$ be the branch point obtained by perturbing the resonant pair $(\tau_{2j-1},\tau_{2j})$ and defined by (106), let it be the base point of the Abel–Jacobi map Then the Abel–Jacobi map of the point $\gamma_j$ in the divisor is given in the leading order by the following formula: Remark 8. It is easy to check that for $j\leqslant N$ the point $\sigma\gamma_{j+N}$ can be defined by In this approximation, for the Abel-Jacobi map with base point $\tau=0$ we obtain
$$
\begin{equation}
A_j(E_{4j-3}) =-\log\biggl[\frac{\tau_{2j} q_{2j}} {i\operatorname{Im}(\tau_{2j}\tau_{2j-1}^{-1})(\tau_{2j-1}-\tau_{2j})} \varepsilon\sqrt{\alpha_j\beta_j}\,\biggr],
\end{equation}
\tag{115}
$$
$$
\begin{equation}
A_j(E_{4j-1}) =\log\biggl[\frac{\tau_{2j-1} q_{2j-1}} {i\operatorname{Im}(\tau_{2j}\tau_{2j-1}^{-1})(\tau_{2j}-\tau_{2j-1})} \varepsilon\sqrt{\alpha_j\beta_j}\,\biggr]
\end{equation}
\tag{116}
$$
and
$$
\begin{equation}
b_{jj}=A_j(E_{4j-1})-A_j(E_{4j-3})= \log\biggl[\frac{\tau_{2j-1}\tau_{2j} q_{2j-1}q_{2j}} {\operatorname{Im}^2(\tau_{2j}\tau_{2j-1}^{-1})(\tau_{2j-1}-\tau_{2j})^2} \varepsilon^2(\alpha_j\beta_j)\biggr].
\end{equation}
\tag{117}
$$
Remark 9. The asymptotics of the Abel differentials and the Riemann matrix for Riemann surfaces with pinched cycles were discussed in [27]; moreover, using the techniques from that book it is possible to calculate the second-order corrections. For surfaces close to degenerate ones, one can effectively use the Schottky parametrization (see [9] and the references there for more details). Remark 10. If one of the quantities $\alpha_j$ and $\beta_j$ in (106) is 0, then the corresponding resonant point becomes a double point in the leading-order approximation. Using the arguments analogous to [48] and [49], it is easy to check that a regular doubly- periodic perturbation can be chosen so that we obtain a double point in the exact theory. For the 2D Schrödinger operator at a fixed energy level the existence of singular spectral curves corresponding to regular doubly-periodic potentials was pointed out in [48]. Note that non-removable double points correspond to resonant pairs associated with unstable modes; resonant points associated with stable modes either become small handles after a perturbation or remain removable double points. The role of singular spectral curves in the theory of soliton equations, including the DS2 and modified Novikov–Veselov equation, was considered in [71] and [74]. 7. The vector of Riemann constantsDenote the base point of the Abel–Jacobi map by $P_0$. Then the vector of Riemann constants is given by the following formulae (see [22]):
$$
\begin{equation}
K_j=\frac{b_{jj}}{2}-\pi i-\frac{1}{2\pi i}\sum_{k\ne j}^{2N} \int_{\widetilde a^k} A_j(\gamma)\omega_k,
\end{equation}
\tag{118}
$$
where $A_j(\gamma)$ is the $j$th component of the Abel–Jacobi map of the point $\gamma$: In this formula it is important to have a proper realization of the basis of cycles. More precisely, it is necessary that the following hold. 1. All basis cycles start and end at the same starting point $Q_0$. 2. These cycles do not intersect away from the point $Q_0$. 3. Near $Q_0$ the curves are arranged in the following order (listed clockwise): the beginning of $a_1$, the end of $b_1$, the end of $a_1$, the beginning of $b_1$, the beginning of $a_2$, the end of $b_2$, the end of $a_2$, the beginning of $b_2$, …, the beginning of $b_g$ (see Fig. 5, left, for $g=8$). A basis of such cycles is presented at Figure 5, (c). It is clear that for $j \ne k$, in the $\epsilon$-neighbourhood of the cut $[E_{4k-3},E_{4k-2}]$ we have and therefore
$$
\begin{equation}
-\frac{1}{2\pi i}\int_{\widetilde a^k} A_j(\gamma)\omega_k= -A_j(E_{2k-1})+O(\epsilon).
\end{equation}
\tag{121}
$$
Hence by analogy with [35] we can use the following modification of the formulae. Let the point $\gamma_k$ in the divisor lie near the contour $a_k$. Then we redefine the Abel–Jacobi map of the divisor by setting The requirement that the argument of the theta function remain unchanged implies that
$$
\begin{equation}
K_j=\frac{b_{jj}}{2}-\pi i+A_j(E_{4j-3})+O(\epsilon).
\end{equation}
\tag{123}
$$
8. Summary of the resultsThe results in this paper can be summarized as follows. Consider the focusing Davey–Stewartson 2 equation (16) in the space of spatially doubly-periodic functions with periods $L_x$ and $L_y$ in $x$ and $y$, respectively. The Cauchy problem for (16) is called the doubly-periodic Cauchy problem for anomalous waves if the Cauchy data are a small doubly-periodic perturbation of the constant background (17). Theorem 1. Assume that the Cauchy data (17) for (16) have the following properties. 1. The background is unstable, that is, the open disc $k_x^2+k_y^2 < 4 |a|^2$ contains a least one point $(k_x,k_y)$ of type (66) such that $(n_x,n_y)\ne(0,0)$. 2. The periods $L_x$ and $L_y$ are generic in the following sense:
3. The Cauchy data satisfy the following genericity conditions: $\alpha_j\beta_j\ne 0$ for each unstable mode, where the quantities $\alpha_j$ and $\beta_j$ are expressed in terms of the corresponding Fourier coefficient of the perturbation by (97) and (98), respectively. To simplify the final formulae also assume that $a=1$ and the zero Fourier harmonic of the perturbation $v_0(x,y)$ is equal to zero (65). (Because of the scaling invariance of the DS equation, these constraints are not restrictive.) Then for $|\varepsilon|\ll 1$ the leading-order solution of the Cauchy problem (17) for the focusing DS2 equation is provided by formula (53), where ${\mathcal C}_z= {\mathcal C}_{\overline z} = {\mathcal C}_t =0$, the Riemann $\theta$-function is defined by (54), $g=2N$, where $N$ is the number of unstable modes (in counting the unstable modes it is assumed that $(n_x,n_y)\ne(0,0)$), the Riemann matrix of periods $B=(b_{jl})$ is defined by (117) and (81), the vectors $\vec W_z$, $\vec W_{\overline z}$, and $\vec W_{t}$ are given by (79), $\vec A(\infty_1)=\vec 0$, $\vec A(\infty_2)$ is given by (80), $\vec A({\mathcal D})$ is given by (112), $\vec K$ is given by (123) and (115). Moreover, following the scheme in [35], it is sufficient to keep only the leading- order terms in (54), and the final formulae can be expressed in terms of elementary functions of the Cauchy data. The leading-order terms are different for different regions in the $(x,y,t)$-space; therefore, the approximation formulae depend on the region. Following [35], the boundaries of these regions can be calculated explicitly in terms of the Cauchy data. We acknowledge the useful discussions we had with A. Bogatyrev. 
Citation in format AMSBIB
\Bibitem{GriSan22} |
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