Hirota method for oblique solitons in two-dimensional supersonic nonlinear Schrödinger flow

Abstract

In a previous work[1] exact stable oblique soliton solutions were revealed in two dimensional nonlinear Schrödinger flow. In this work we show that single soliton solution can be expressed within the Hirota bilinear formalism. An attempt to build two-soliton solutions shows that the system is "close" to integrability provided that the angle between the solitons is small and/or we are in the hypersonic limit. PACS numbers: 47.40.Nm, 03.75.Kk, 05.45.Yv IntroductionThe nonlinear Schrödinger (NLS) flow is ubiquitous in many physical systems such as photorefractive crystals, and the superfluids Bose-Einstein condensates and exciton-polaritons. A fundamental problem is how a superfluid reacts to the presence of an obstacle. One can define a Mach velocity (M ) as the velocity of the obstacle relative to the sound velocity in the medium. Two-dimensional studies showed that when M > 0.37 the system loses superfluidity and start to emit pair of vortices [2][3][4][5]. Increasing the velocity showed that vortices merge in a "vortex street" [6], which were later understood as oblique solitons [7] and its exact single soliton solution determined [1,8]. Oblique solitons were long know to be unstable but it was showed that under the flow they are only convectively unstable provided that M > 1.44 [9,10]. Studies with extended obstacles also presented oblique solitons in the wake, and an analytical approach based on Whitham modulation theory was successfully applied [11]. Oblique solitons were realized experimentally in the system of exciton-polaritons [12], though for lower Mach number than originally predicted by theory. Corrections to the model including losses were able to match experimental observations [13]. Dynamics of formation and decay of oblique solitons were recently observed in [14].A key question about solitons is how they behave in collisions. As long as we know, there is no general proof about the non-integrability of the 2D-NLS. Numerical studies with two obstacles were able to generate collisions between these oblique solitons [15]. These collisions were shown to be practically elastic suggesting integrability or "close" to integrability in such system. In the same work, an analytical treatment was considered using hydrodynamical approach and the system was show to follow a 1D-NLS equation in the hypersonic limit, and collisions could be described by the well known phase shifts [16]. Numerical calculations were in good agreement with predicted phase shifts, considering that they were perturbed by previous interactions with linear waves. Since the exact single soliton (1SS) was already obtained, one might ask if an exact two-soliton solution (2SS) could be found. A possible framework to find multiple soliton solutions is the Hirota method [17,18]. In the following we build up a bilinear Hirota form of the 2D-NLS in the stationary frame relative to the obstacle. Then, we show that the single oblique soliton solution indeed satisfy this form. In the sequence we propose an ansatz to the exact solution of the two-...

Supersonic flow of a Bose-Einstein condensate past an oscillating attractive-repulsive obstacle

“…For greater radius more pairs of oblique solitons are generated. Interaction of solitons was studied in [15,16] where it was found that the angle between dark solitons decreases as the obstacle radius increases for a fixed supersonic velocity of the flow. In previous experimental works [13,17,18] the existence of such nonlinear structures were suggested.…”

Nonlinear interactions of two-kink-breather solution in Yu-Toda-Sasa-Fukuyama equation by modulated phase shift

“…However, it can be seen from figure 2(b) that the amplitude of the kink wave increases with the increase of parameter α 1 . 4), and equation (3) is two-kink-breather solution of the YTSF equation as follows: From equation (10), it is easy to know that the existence conditions of the nonsingular solution of equation (9…”

“…Many studies of nonlinear wave interactions have been widely used in the fields of optics, plasma physics, fluid dynamics and solid state physics, etc [1][2][3][4][5]. With the development of the soliton theory, it appears some nonlinear waves such as solitons, rouge waves, kink solitons, lumps, breathers, kink-breathers, etc [6][7][8][9][10][11]. Among which, kink solitons and breathers are two typical categories of solitons.…”

“…Equation(36) represents the parameter region near point o 1 and the line ν 1 in figure 3. According to equations(10) and(36), the amplitude of the breather in the two-kinkbreather solution is very small, we call such this solution the two-kink-breather with small amplitude. It means that the waveform of the two-kink-breather solution is approximate to two-kink solution.…”