15 research outputs found
Bright and Gap Solitons in Membrane-Type Acoustic Metamaterials
We study analytically and numerically envelope solitons (bright and gap
solitons) in a one-dimensional, nonlinear acoustic metamaterial, composed of an
air-filled waveguide periodically loaded by clamped elastic plates. Based on
the transmission line approach, we derive a nonlinear dynamical lattice model
which, in the continuum approximation, leads to a nonlinear, dispersive and
dissipative wave equation. Applying the multiple scales perturbation method, we
derive an effective lossy nonlinear Schr\"odinger equation and obtain
analytical expressions for bright and gap solitons. We also perform direct
numerical simulations to study the dissipation-induced dynamics of the bright
and gap solitons. Numerical and analytical results, relying on the analytical
approximations and perturbation theory for solions, are found to be in good
agreement
Higher-dimensional extended shallow water equations and resonant soliton radiation
The higher order corrections to the equations that describe nonlinear wave motion in shallow water are derived from the water wave equations. In particular, the extended cylindrical Korteweg-de Vries and Kadomtsev-Petviashvili equations—which include higher order nonlinear, dispersive, and nonlocal terms—are derived from the Euler system in (2+1) dimensions, using asymptotic expansions. It is thus found that the nonlocal terms are inherent only to the higher dimensional problem, both in cylindrical and Cartesian geometry. Asymptotic theory is used to study the resonant radiation generated by solitary waves governed by the extended equations, with an excellent comparison obtained between the theoretical predictions for the resonant radiation amplitude and the numerical solutions. In addition, resonant dispersive shock waves (undular bores) governed by the extended equations are studied. It is shown that the asymptotic theory, applicable for solitary waves, also provides an accurate estimate of the resonant radiation amplitude of the undular bore
Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach
The existence of stationary solitary waves in symmetric and non-symmetric
complex potentials is studied by means of Melnikov's perturbation method. The
latter provides analytical conditions for the existence of such waves that
bifurcate from the homogeneous nonlinear modes of the system and are located at
specific positions with respect to the underlying potential. It is shown that
the necessary conditions for the existence of continuous families of stationary
solitary waves, as they arise from Melnikov theory, provide general constraints
for the real and imaginary part of the potential, that are not restricted to
symmetry conditions or specific types of potentials. Direct simulations are
used to compare numerical results with the analytical predictions, as well as
to investigate the propagation dynamics of the solitary waves.Comment: 19 pages, 14 figure
Water Waves and Light: Two Unlikely Partners
We study a generic model governing optical beam propagation in media featuring a nonlocal nonlinear response, namely a two-dimensional defocusing nonlocal nonlinear Schrödinger (NLS) model. Using a framework of multiscale expansions, the NLS model is reduced first to a bidirectional model, namely a Boussinesq or a Benney-Luke-type equation, and then to the unidirectional Kadomtsev-Petviashvili (KP) equation – both in Cartesian and cylindrical geometry. All the above models arise in the description of shallow water waves, and their solutions are used for the construction of relevant soliton solutions of the nonlocal NLS. Thus, the connection between water wave and nonlinear optics models suggests that patterns of water may indeed exist in light. We show that the NLS model supports intricate patterns that emerge from interactions between soliton stripes, as well as lump and ring solitons, similarly to the situation occurring in shallow water
Extended shallow water wave equations
Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation, incorporating all appropriate dispersive and nonlinear terms. Specifically, first we derive the extended Korteweg–de Vries (KdV) equation, and then proceed with the extended Benjamin–Bona–Mahony and the extended Camassa–Holm equations in (1+1)-dimensions, the extended cylindrical KdV equation in the quasi-one dimensional setting, as well as the extended Kadomtsev–Petviashvili and its cylindrical counterpart in (2+1)-dimensions. We conclude with the case of the extended Green–Naghdi equations
Vector nematicons: Coupled spatial solitons in nematic liquid crystals
Families of soliton pairs, namely vector solitons, are found within the
context of a coupled nonlocal nonlinear Schrodinger system of equations,
as appropriate for modeling beam propagation in nematic liquid crystals.
In the focusing case, bright soliton pairs have been found to exist
provided their amplitudes satisfy a specific condition. In our
analytical approach, focused on the defocusing regime, we rely on a
multiscale expansion methods, which reveals the existence of dark-dark
and antidark-antidark solitons, obeying an effective Korteweg-de Vries
equation, as well as dark-bright solitons, obeying an effective
Mel'nikov system. These pairs are discriminated by the sign of a
constant that links all physical parameters of the system to the
amplitude of the stable continuous wave solutions, and, much like the
focusing case, the solitons' amplitudes are linked, leading to mutual
guiding
Extended shallow water wave equations
Extended shallow water wave equations are derived, using the method of
asymptotic expansions, from the Euler (or water wave) equations. These extended
models are valid one order beyond the usual weakly nonlinear, long wave
approximation, incorporating all appropriate dispersive and nonlinear terms.
Specifically, first we derive the extended Korteweg-de Vries (KdV) equation,
and then proceed with the extended Benjamin-Bona-Mahony and the extended
Camassa-Holm equations in (1+1)-dimensions, the extended cylindrical KdV
equation in the quasi-one dimensional setting, as well as the extended
Kadomtsev-Petviashvili and its cylindrical counterpart in (2+1)-dimensions. We
conclude with the case of the extended Green-Naghdi equations.Comment: To appear in Wave Motio
Dark Solitons in Acoustic Transmission Line Metamaterials
We study dark solitons, namely density dips with a phase jump across the density minimum, in a one-dimensional, weakly lossy nonlinear acoustic metamaterial, composed of a waveguide featuring a periodic array of side holes. Relying on the electroacoustic analogy and the transmission line approach, we derive a lattice model which, in the continuum approximation, leads to a nonlinear, dispersive and dissipative wave equation. The latter, using the method of multiple scales, is reduced to a defocusing nonlinear Schrödinger equation, which leads to dark soliton solutions. The dissipative dynamics of these structures is studied via soliton perturbation theory. We investigate the role—and interplay between—nonlinearity, dispersion and dissipation on the soliton formation and dynamics. Our analytical predictions are corroborated by direct numerical simulations