Bound Coherent Structures Propagating on the Free Surface of Deep Water

Abstract

<p><span>               The work presents the results of studying the bound coherent structures propagating on the free surface of ideal incompressible fluid of infinite depth. Examples of such structures are bi-solitons which are exact solutions of the known approximate model for deep water waves — the nonlinear Schrödinger equation (NLSE). Recently, when studying multiple breathers collisions, the occurrence of such objects was found in a more accurate model of the supercompact equation for unidirectional water waves [1]. The aim of this work is obtaining and further studying such structures with different parameters in the supercompact equation and in the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. </span><span>The algorithm used for finding the bound coherent objects was similar to the one described in [2]. As the initial conditions for obtaining such structures in the framework of the above models, the NLSE bi-soliton solutions were used, as well as two single breathers numerically found by the Petviashvili method and placed in a same point of the computational domain. During the evolution calculation the initial structures emitted incoherent waves which were filtered at the boundaries of the domain using the damping procedure. It is shown that after switching off the filtering of radiation, periodically oscillating coherent objects remain on the surface of the liquid, propagate stably during one hundred thousand characteristic wave periods and do not lose energy. The profiles of such structures at different parameters are compared.</span></p><p><span>This work was supported by RSF grant </span><span>19-72-30028</span><span> and </span><span>RFBR grant </span><span>20-31-90093</span><span>.</span></p><p><span>[1] Kachulin D., Dyachenko A., Dremov S. Multiple Soliton Interactions on the Surface of Deep Water //Fluids. – 2020. – Т. 5. – №. 2. – С. 65.</span></p><p><span>[2] Dyachenko A. I., Zakharov V. E. On the formation of freak waves on the surface of deep water //JETP letters. – 2008. – Т. 88. – №. 5. – С. 307.</span></p>