Internal solitons in laboratory experiments: comparison with theoretical models

New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.

Experimental Investigation on the Vertical Structure Characteristics of Internal Solitary Waves

An experimental investigation of the vertical structure characteristics of internal solitary waves (ISWs) was systematically carried out in a large gravitationally stratified fluid flume. Four different stratifications were established, and basic elements of ISWs were measured by a conductivity probe array. The vertical distributions of the amplitude, characteristic frequency and waveform of two types of ISWs under different stratifications were obtained, and the experimental results were compared with the theoretical model. The study shows that most vertical structures of the amplitude under different stratifications agree with those of the theoretical model, while there are some deviations for ISWs with large amplitudes. Neither the two-layer model nor the continuously stratified model can effectively describe the variation in the characteristic frequency at different depths with amplitude. For a single small-amplitude ISW, the characteristic frequency first increases and then decreases with increasing depth. The characteristic frequency is largest at the depth of the maximum buoyancy frequency. For an ISW with a relatively large amplitude, there is likely to be a local minimum of the characteristic frequency near the depth where the maximum buoyancy frequency lies. In different stratifications, the sech2 function of KdV theory can describe the waveforms of ISWs at different depths well.

Generation of nonlinear internal waves by flow over topography: Rotational effects

We use the forced Ostrovsky equation to investigate the generation of internal waves excited by a constant background current flowing over localized topography in the presence of background rotation. As is now well known in the absence of background rotation, the evolution scenarios fall into three cases, namely subcritical, transcritical, and supercritical. Here an analysis of the linearized response divides the waves into steady and unsteady waves. In all three cases, steady waves occur downstream but no steady waves can occur upstream, while unsteady waves can arise upstream only when there is a negative minimum of the group velocity. The regions occupied by the steady and unsteady waves are determined by their respective group velocities. When the background current is increased, the wave number of the steady waves decreases. In addition, the concavity (canyon or sill), the topographic width, and the relative strength of the rotation play an important role in the generation mechanism. Nonlinear effects modulate the wave amplitude and lead to the emergence of coherent wave packets. All these findings are confirmed by numerical simulations.

Comprehensive vibrational dynamics of half-open fluid-filled shells

Fluid-filled shells are near-ubiquitous in natural and engineered structures-a familiar example is that of glass harps comprising partially filled wineglasses or glass bowls, whose acoustic properties are readily noticeable. Existing theories modelling the mechanical properties of such systems under vibrational load either vastly simplify shell geometry and oscillatory modal shapes to admit analytical solutions or rely on finite-element black-box computations for general cases, the former yielding poor accuracy and the latter offering limited tractability and physical insight. In the present study, we derive a theoretical framework encompassing elastic shell deformation with structural and viscous dissipation, accommodating arbitrary axisymmetric shell geometries and fluid levels; reductions to closed-form solutions under specific assumptions are shown to be possible. The theory is extensively verified against a range of geometries, fluid levels and fluid viscosities in experiments; an extension of the model encompassing additional solid objects within the fluid-filled shell is also considered and verified. The presented theoretical advance in describing vibrational response is relevant in performance evaluation for engineered structures and quality validation in manufacturing.

Adiabatic decay of internal solitons due to Earth's rotation within the framework of the Gardner-Ostrovsky equation

The adiabatic decay of different types of internal wave solitons caused by the Earth's rotation is studied within the framework of the Gardner-Ostrovsky equation. The governing equation describing such processes includes quadratic and cubic nonlinear terms, as well as the Boussinesq and Coriolis dispersions: (u_t + c u_x + α u u_x + α₁ u² u_x + β u_xxx)_x = γ u. It is shown that at the early stage of evolution solitons gradually decay under the influence of weak Earth's rotation described by the parameter γ. The characteristic decay time is derived for different types of solitons for positive and negative coefficients of cubic nonlinearity α₁ (both signs of that parameter may occur in the oceans). The coefficient of quadratic nonlinearity α determines only a polarity of solitary wave when α₁ < 0 or the asymmetry of solitary waves of opposite polarity when α₁ > 0. It is found that the adiabatic theory describes well the decay of solitons having bell-shaped profiles. In contrast to that, large amplitude table-top solitons, which can exist when α₁ is negative, are structurally unstable. Under the influence of Earth's rotation, they transfer first to the bell-shaped solitons, which decay then adiabatically. Estimates of the characteristic decay time of internal solitons are presented for the real oceanographic conditions.

The inverse problem for the Gross-Pitaevskii equation

Two different methods are proposed for the generation of wide classes of exact solutions to the stationary Gross-Pitaevskii equation (GPE). The first method, suggested by the work of Kondrat'ev and Miller [Izv. Vyssh. Uchebn. Zaved., Radiofiz IX, 910 (1966)], applies to one-dimensional (1D) GPE. It is based on the similarity between the GPE and the integrable Gardner equation, all solutions of the latter equation (both stationary and nonstationary ones) generating exact solutions to the GPE. The second method is based on the "inverse problem" for the GPE, i.e., construction of a potential function which provides a desirable solution to the equation. Systematic results are presented for one- and two-dimensional cases. Both methods are illustrated by a variety of localized solutions, including solitary vortices, for both attractive and repulsive nonlinearity in the GPE. The stability of the 1D solutions is tested by direct simulations of the time-dependent GPE.

Generation of solitons and breathers in the extended Korteweg-de Vries equation with positive cubic nonlinearity

The initial-value problem for box-like initial disturbances is studied within the framework of an extended Korteweg-de Vries equation with both quadratic and cubic nonlinear terms, also known as the Gardner equation, for the case when the cubic nonlinear coefficient has the same sign as the linear dispersion coefficient. The discrete spectrum of the associated scattering problem is found, which is used to describe the asymptotic solution of the initial-value problem. It is found that while initial disturbances of the same sign as the quadratic nonlinear coefficient result in generation of only solitons, the case of the opposite polarity of the initial disturbance has a variety of possible outcomes. In this case solitons of different polarities as well as breathers may occur. The bifurcation point when two eigenvalues corresponding to solitons merge to the eigenvalues associated with breathers is considered in more detail. Direct numerical simulations show that breathers and soliton pairs of different polarities can appear from a simple box-like initial disturbance.

Internal solitons in the ocean and their effect on underwater sound

Nonlinear internal waves in the ocean are discussed (a) from the standpoint of soliton theory and (b) from the viewpoint of experimental measurements. First, theoretical models for internal solitary waves in the ocean are briefly described. Various nonlinear analytical solutions are treated, commencing with the well-known Boussinesq and Korteweg-de Vries equations. Then certain generalizations are considered, including effects of cubic nonlinearity, Earth's rotation, cylindrical divergence, dissipation, shear flows, and others. Recent theoretical models for strongly nonlinear internal waves are outlined. Second, examples of experimental evidence for the existence of solitons in the upper ocean are presented; the data include radar and optical images and in situ measurements of wave forms, propagation speeds, and dispersion characteristics. Third, and finally, action of internal solitons on sound wave propagation is discussed. This review paper is intended for researchers from diverse backgrounds, including acousticians, who may not be familiar in detail with soliton theory. Thus, it includes an outline of the basics of soliton theory. At the same time, recent theoretical and observational results are described which can also make this review useful for mainstream oceanographers and theoreticians.

Solitons in nonintegrable systems

We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brief overview of some recent developments is provided, and the various contributions are described. The topics covered in this focus issue are the modulation of solitons, bores, and shocks, the dynamical evolution of solitary waves, and existence and stability of solitary waves and embedded solitons.