Collision of two breathers at surface of deep water

We applied canonical transformation to water wave equation not only to remove cubic nonlinear terms but to simplify drastically fourth order terms in Hamiltonian. This transformation explicitly uses the fact of vanishing exact four waves interaction for water gravity waves for 2D potential fluid. After the transformation well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in compact way. This new equation is very suitable as for analytic study as for numerical simulation. Localized in space breather-type solution was found. Numerical simulation of collision of two such breathers strongly supports hypothesis of integrability of 2-D free surface hydrodynamics.


Introduction
The work described here is motivated by two remarkable facts regarding onedimensional free surface hydrodynamics: • In [1] it was shown that four-wave interaction coefficient vanishes on the resonant manifold k + k 1 = k 2 + k 3 , ω k + ω k 1 = ω k 2 + ω k 3 .
• In [2,3] it was demonstrated that giant breather, highly nonlinear, exists on the fluid surface without radiation. Moreover, space-time spectrum of the breather consists of waves propagating in the same direction.
These two facts bring the idea that fourth order wave interactions can be drastically simplified by some canonical transformation of the Hamiltonian. Below we show how this transformation looks like. Dynamic equation derived using this transformation is very elegant and simple. It can be easily generalized for "almost" one-dimensional waves.

Compact equation
A one-dimensional potential flow of an ideal incompressible fluid with a free surface in a gravity field fluid is described by the following set of equations: here η(x, t) -is the shape of a surface, φ(x, z, t) -is a potential function of the flow and g -is a gravitational acceleration. As was shown in [4], the variables η(x, t) and ψ(x, t) = φ(x, z, t) z=η are canonically conjugated, and satisfy the equations ∂ψ ∂t = − δH δη ∂η ∂t = δH δψ .
Here H = K + U is the total energy of the fluid with the following kinetic and potential energy terms: It is convenient to introduce normal complex variable a k : here ω k = √ gk -is the dispersion law for the gravity waves, and Fourier transformations ψ(x) → ψ k and η(x) → η k are defined as follows: Hamiltonian can be expanded in an infinite series in powers of a k (see [4,5]) This variable a k satisfies the equation L kk 2   L kk 1 = ( k k 1 ) + |k||k 1 | Fourth order part of Hamiltonian is the following: Here W k 3 k 4 k 1 k 2 , G k 4 k 1 k 2 k 3 and R k 1 k 2 k 3 k 4 are equal to: Here is an arbitrary function satisfying the following symmetry conditions:B Coefficients C k 3 kk 1 k 2 and S kk 1 k 2 k 3 provide vanishing corresponding forth-order terms in the new Hamiltonian.
Details of this transformation can be found in [8,9]. In K-space Hamiltonian has the form: In X-space it corresponds to: After integrating by parts Hamiltonian acquires very nice form: Corresponding equation of motion is the following:

Monochromatic wave and modulational instability
Monochromatic wave is the simplest solution of (2.4). Indeed, plugging (3.5) in to the equation (2.4) one can get the following relation Recalling transformation from a k to b k one can see that for waves with small amplitude ( a k ≃ b k ) and relation (3.6) coincides with well known Stokes correction to the frequency due to finite wave amplitude.

Modulational instability of monochromatic wave
Let us consider perturbation to the solution Perturbed solution has the following form: with the following condition: Plugging perturbed solution (3.8) in to the equation we get the sum of two independent equations: Expressions forT k 0 +kk 0 k 0 +kk 0 and T k 0 −kk 0 +k k 0 k 0 can be easily obtained: Looking at even and odd powers of k one can see that Then Suppose δb k 0 +k growth as δb k 0 +k ⇒ δb k 0 +k e γ k t one can easily obtain the following formula for γ k : If we introduce steepness of the carrier wave ω k 0 µ 2 = |B 0 | 2 k 2 0 and approximate d(k) as then for growth rate is The difference between this formula and well-known expression derived from the nonlinear Schrodinger equation is highlighted by two boldfaced terms.

Breathers and numerical sumulation of its collisions
Breather is the localized solution of (2.4) of the following type: where k 0 is the wavenumber of the carrier wave, V is the group velocity and ω 0 is the frequency close to ω k 0 . In the Fourier space breather can be written where Ω is close to For φ k the following equation is valid: One can treat φ k as pure real function of k. To solve equation (4.15) one can use Petviashvili iteration method: Petviashvili coefficient M is the following: Below we present typical numerical solution of (4.15). Calculation were made in the periodic domain 2π with carier wavenumber k 0 ∼ 25, V = 0.1 and Ω = 2.53. 1 . In the Figures 1, 2, 3 one can see real part ofb(x), modulus of b(x) and Fourier spectrum of b(x).
Very important question from the point of view of integrability of the equation (2.4) is the question about collision of two breathers. To study breathers collision we performed the following numerical simulation: • As initial condition we have used two beathers separated in space (distance was π.) • First breather has the following parameters: Ω 1 = 5.1, V 1 = 0.05. Carrier wave number appears to be ∼ 100. • For the second breather -Ω 2 = 2.53, V 2 = 0.1. Carrier wave number appears to be ∼ 25 This initial condition is show in Figure 4. Its Fourier spectrum is shown in Figure 5. After time π (V 2 −V 1 ) ≃ 62.8 breathers collide. In the Figures 6 and 7 one can see breathers at the time close to collision (t = 50) and at the moment of collision (t = 63). Fourier spectrum of two breathers at t = 63 is shown in Figure 8. And finally we show the picture of two breathers at t = 126 when they separated again at distance ≃ π. Real part of b(x) and Fourier spectrum of that is given in Figures 9, 10. So, the simulation demonstrates no interaction of breathers. It suggests that equation (2.4) is integrable.

Conclusion
Simple equation describing evolution of 1-D water waves is derived. Derivation of this equation is based on the important property of vanishing four-wave interaction for gravity water waves. This property allows to simplify drastically well-known Zakharov's equation for water waves, which is very cumbersome. Written in X-space instead of K-space, it allows further analytical and numer-   ical study. The equation has breather-type solution which was found numerically using Petviashvili iteration method. Numerical simulation of collision of two breathers shows behavior which is typical for integrable system. It can be considered as numerical proof of integrability.
This new equation can be generalized for the "almost" 2-D waves, or "almost" 3-D fluid. When considering waves slightly inhomogeneous in transverse direction, one can think in the spirit of Kadomtsev-Petviashvili equation for Korteveg-de-Vries equation, namely one can treat now frequency ω k as two dimensional, ω kx,ky , while leaving coefficientT kk 1 k 2 k 3 not dependent on y. b now depends on both x and y: