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Abstract We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials as a Riemann–Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann–Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potentials evolve according to the Korteweg–de Vries Equation (KdV) equation, we use integrability to derive an associated Riemann–Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann–Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann–Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation. 

We consider the motion of ideal incompressible fluid with free surface. We analyzed the exact fluid dynamics through the time-dependent conformal mapping z=x+iy=z(w,t) of the lower complex half plane of the conformal variable w into the area occupied by fluid. We established the exact results on the existence vs. nonexistence of the pole and power law branch point solutions for 1/zw and the complex velocity. We also proved the nonexistence of the time-dependent rational solution of that problem for the second- and the first-order moving pole. 

We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and surface tension. A time-dependent conformal mapping $z(w,t)$ of the lower complex half-plane of the variable $$w$$ into the area filled with fluid is performed with the real line of $$w$$ mapped into the free fluid’s surface. We study the dynamics of singularities of both $z(w,t)$ and the complex fluid potential $$\unicode[STIX]{x1D6F1}(w,t)$$ in the upper complex half-plane of $$w$$ . We show the existence of solutions with an arbitrary finite number $$N$$ of complex poles in $$z_{w}(w,t)$$ and $$\unicode[STIX]{x1D6F1}_{w}(w,t)$$ which are the derivatives of $z(w,t)$ and $$\unicode[STIX]{x1D6F1}(w,t)$$ over $$w$$ . We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of $$z_{w}(w,t)$$ at these $$N$$ points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046 ) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of $$\unicode[STIX]{x1D6F1}_{w}(w,t)$$ are also the constants of motion while non-zero gravity $$g$$ ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both $$z_{w}(w,t)$$ and $$\unicode[STIX]{x1D6F1}_{w}(w,t)$$ at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is $4N$ for zero gravity and $4N-1$ for non-zero gravity. For the second-order poles we found $6N$ motion integrals for zero gravity and $6N-1$ for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics. 

We consider the Euler equations for the potential flow of an ideal incompressible fluid of infinite depth with a free surface in two-dimensional geometry. Both gravity and surface tension forces are taken into account. A time-dependent conformal mapping is used which maps the lower complex half-plane of the auxiliary complex variable $$w$$ into the fluid’s area, with the real line of $$w$$ mapped into the free fluid’s surface. We reformulate the exact Eulerian dynamics through a non-canonical non-local Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid’s velocity potential, both evaluated at the fluid’s free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. The new Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Zakharov ( J. Appl. Mech. Tech. Phys. , vol. 9(2), 1968, pp. 190–194) which is valid only for solutions for which the natural surface parametrization is single-valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, the new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to the Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations. In that case we identify powerful reductions that allow one to find general classes of particular solutions. 

Abstract The concept of a primitive potential for the Schrödinger operator on the line was introduced in Dyachenko et al. (2016, Phys. D, 333, 148–156), Zakharov, Dyachenko et al. (2016, Lett. Math. Phys., 106, 731–740) and Zakharov, Zakharov et al. (2016, Phys. Lett. A, 380, 3881–3885). Such a potential is determined by a pair of positive functions on a finite interval, called the dressing functions, which are not uniquely determined by the potential. The potential is constructed by solving a contour problem on the complex plane. In this article, we consider a reduction where the dressing functions are equal. We show that in this case, the resulting potential is symmetric, and describe how to analytically compute the potential as a power series. In addition, we establish that if the dressing functions are both equal to one, then the resulting primitive potential is the elliptic one-gap potential. 

We construct a broad class of bounded potentials of the one-dimensional Schroedinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hoelder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.