Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

Abstract

We address a problem of potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensional geometry. We admit a 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 Π(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 Π w (w, t) which are the derivatives of z(w, t) and Π(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 nonzero 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 Ref. V. E. Zakharov and A. I. Dyachenko, arXiv:1206.2046(2012 for the preliminary results. All these constants of motion commute with each other in the sense of underlying Hamiltonian dynamics. In absence of both gravity and surface tension, the residues of Π w (w, t) are also the constants of motion while nonzero gravity g ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both z w (w, t) and Π w (w, t) at each poles position reveals an 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 nonzero gravity. For the second order poles we found 6N motion integral for zero gravity and 6N − 1 for nonzero gravity. We suggest that the existence of these nontrivial 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.Potential motion means that a velocity v of fluid is determined by a velocity potential Φ(r, t) as v = ∇Φ with ∇ ≡ ( ∂ ∂x , ∂ ∂y ). The incompressibility condition ∇ · v = 0 implies the Laplace equation(1.5) inside fluid, i.e. Φ is the harmonic function inside fluid. Eq. (1.5) is supplemented with a decaying boundary condition (BC) at infinity, ∇Φ → 0 for |x| → ∞ or y → −∞.(1.6)

Short branch cut approximation in two-dimensional hydrodynamics with free surface

“…The short branch cut approximation requires that both |normalΔU|≪|U|1emand1em|normalΔB|≪|B|. Qualitatively it implies that singularities in R and V must not be too strong. For example, if a singularity in R is stronger than in V , as studied in [14], then these conditions require that |Imfalse(afalse)Vc′Rfalse~|≪|RcVfalse~|. We note that the limit of infinitely short branch cut recovers pole solutions of [14].…”

“…Many of these integrals commute with respect to the non-canonical Poisson bracket found in [11,12]. It was suggested in [14] that the existence of such commuting integrals of motion might be a sign of the Hamiltonian integrability of the free surface hydrodynamics. It is well established (e.g.…”

“…A complex conjugation ffalse¯false(wfalse) of f ( w ) in equations (1.17)–(1.19) and throughout this paper is understood as applied with the assumption that f ( w ) is the complex-valued function of the real argument w even if w takes the complex values so that ffalse¯false(wfalse)≡ffalse(wfalse¯false)¯. That definition ensures the analytical continuation of f ( w ) from the real axis w = u into the complex plane of w∈C. Following [14], we consider an analytical continuation of the functions R and V into the Riemann surfaces which we call by Γ R ( w ) and Γ V ( w ), respectively.…”

“…Reference [14] found that the system (1.14)–(1.19) has an arbitrary number of pole solutions for z w and Π w . These poles are located for w∈C+, i.e.…”

“…[12,15–32]) that the system of the type (1.17)–(1.19) and its different generalizations also have solutions with branch points located for w∈C+. Generally, we expect coexistence of poles and branch points at different locations of w∈C+, see [14] for numerical examples. Also [15] demonstrated that purely rational solutions of the system (1.17)–(1.19) are not very likely for the decaying BCs (1.23).…”