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Register a new userPhase-suppressed hydrodynamics of solitons on constant-background plane wave
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Soliton and breather solutions of the nonlinear Schrodinger equation (NLSE) are known to model localized structures in nonlinear dispersive media such as on the water surface. One of the conditions for an accurate propagation of such exact solutions is the proper generation of the exact initial phase-shift profile in the carrier wave, as defined by the NLSE envelope at a specific time or location. Here, we show experimentally the significance of such initial exact phase excitation during the hydrodynamic propagation of localized envelope solitons and breathers, which modulate a plane wave of constant amplitude (finite background). Using the example of stationary black solitons in intermediate water depth and pulsating Peregrine breathers in deep-water, we show how these localized envelopes disintegrate while they evolve over a long propagation distance when the initial phase shift is zero. By setting the envelope phases to zero, the dark solitons will disintegrate into two gray-type solitons and dispersive elements. In the case of the doubly-localized Peregrine breather the maximal amplification is considerably retarded; however locally, the shape of the maximal focused wave measured together with the respective signature phase-shift are almost identical to the exact analytical Peregrine characterization at its maximal compression location. The experiments, conducted in two large-scaled shallow-water as well as deep-water wave facilities, are in very good agreement with NLSE simulations for all cases.
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The dynamics of initially truncated and bent line solitons for the Kadomtsev-Petviashvili (KPII) equation modelling internal and surface gravity waves are analysed using modulation theory. In contrast to previous studies on obliquely interacting solitons that develop from acute incidence angles, this work focuses on initial value problems for the obtuse incidence of two or three partial line solitons, which propagate away from one another. Despite counterpropagation, significant residual soliton interactions are observed with novel physical consequences. The initial value problem for a truncated line soliton-describing the emergence of a quasi-one-dimensional soliton from a wide channel-is shown to be related to the interaction of oblique solitons. Analytical descriptions for the development of weak and strong interactions are obtained in terms of interacting simple wave solutions of modulation equations for the local soliton amplitude and slope. In the weak interaction case, the long-time evolution of truncated and large obtuse angle solitons exhibits a decaying, parabolic wave profile with temporally increasing focal length that asymptotes to a cylindrical Korteweg-de Vries soliton. In contrast, the strong interaction case of slightly obtuse interacting solitons evolves into a steady, one-dimensional line soliton with amplitude reduced by an amount proportional to the incidence slope. This strong interaction is identified with the Mach expansion of a soliton with an expansive corner, contrasting with the well-known Mach reflection of a soliton with a compressive corner. Interestingly, the critical angles for Mach expansion and reflection are the same. Numerical simulations of the KPII equation quantitatively support the analytical findings.
We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted nonuniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports traveling waves, governed by a novel modified Kuramoto--Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equations allows for the existence of dissipative solitons. These permanent traveling waves propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The traveling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the traveling waves. Interestingly, multiperiodic waves, which are a non-integrable analog of the double cnoidal wave, also found to propagate under the model long-wave equation. These multiperiodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified above.
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We consider the long-time evolution of pulses in the Korteweg-de Vries equation theory for initial distributions which produce no soliton, but instead lead to the formation of a dispersive shock wave and of a rarefaction wave. An approach based on Whitham modulation theory makes it possible to obtain an analytic description of the structure and to describe its self-similar behavior near the soliton edge of the shock. The results are compared with numerical simulations.
We study the evolution of nonlinear surface gravity water-wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrodinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing, and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approximation and validated by NLSE simulations. Our result will contribute to understand dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media.
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