Periodic Initial Conditions Research Articles

CAPILLARY LEVELLING OF THIN LIQUID FILMS OF POWER-LAW RHEOLOGY

AbstractWe find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from periodic and localized initial conditions as separate cases. The particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist) at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localized initial condition, are shown to approach the appropriate particular solution.

Periodic boundary conditions and G 2 cosmology

In the standard concordance cosmology the spatial curvature is assumed to be constant and zero (or at least very small). In particular, in numerical computations of the structure of the Universe using N-body simulations, exact periodic boundary conditions are assumed which constrains the spatial curvature. In order to confirm this qualitatively, we numerically evolve a special class of spatially inhomogeneous G 2 models with both periodic initial data and non periodic initial data using zooming techniques. We consequently demonstrate that in these models periodic initial conditions do indeed suppress the growth of the spatial curvature as the models evolve away from their initial isotropic and spatially homogeneous state, thereby verifying that the spatial curvature is necessarily very small in standard cosmology.

Archipelagos, islands, necklaces, and other exotic structures in external force-driven chaotic dusty plasmas

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn’s magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system.

Singular effect of interfacial slip for an otherwise stable two-layer shear flow: analysis and computations

We consider instability of the flat interface in a two-layer Couette flow model developed earlier (Kalogirou & Papageorgiou, 2016, J. Fluid Mech. 802 , 5–36; Katsiavria & Papageorgiou, 2022, Wave Motion 114 , 103018. ( doi:10.1016/j.wavemoti.2022.103018 )) for a thin layer near one of the walls. For the case when the less viscous fluid resides next to the moving wall, we find that even a small slip effect at the interface can destabilize an otherwise highly stable flow to the Turing-type instability. The singular effect of small slip in an otherwise very stable configuration may have important ramifications in physical and technological applications. The neutral points of the dispersion relation give rise to travelling wave solutions that are continued to finite amplitude numerically and their linear stability properties identified for a set of parameter values for disturbances that include subharmonic modes with twice the wavelength of the nonlinear travelling wave. We determined Hopf and regular bifurcation points of travelling waves and rigorously justified their existence for some set of parameter values. Weakly nonlinear analysis close to bifurcation from a flat state is also presented for small amplitude waves in general. We also present global existence and regularity results for periodic initial conditions without any restriction on parameters.

On Hyperbolic Differential Equation with Periodic Control Initial Condition

The aim of this paper to study the existence solution of some types of hyperbolic differential equation with periodicity of some controls function as nonlocal initial condition for the equation and the technical and used depended on some interest iniquities.

The unified transform method to the high-order nonlinear Schrödinger equation with periodic initial condition

In this paper, we study the high-order nonlinear Schrödinger equation with periodic initial conditions via the unified transform method extended by Fokas and Lenells. For the high-order nonlinear Schrödinger equation, the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem. The related jump matrix can be explicitly expressed based on the initial data alone. Furthermore, we present the explicit solution, which corresponds to a one-gap solution.

Strong solutions of impulsive pseudoparabolic equations

We study the two-dimensional Cauchy problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term endowed with periodic initial data and periodicity conditions. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which models an instantaneous impulsive impact. We establish that the transition (shock) layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of strong solutions of the original problem converges to the strong solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter is derived based on the microstructure of the transition layer profile.

Using Leray-Schauder topological degree to solve a linear diffusion parabolic equation with periodic initial conditions

Throughout this manuscript, we show time periodic solutions to a linear diffusion parabolic equation with Diriclet condition. Based on the topological degree theorem, we prove a time periodic solutions of the system such that we found the fixed point when the domain of the solution is sufficiently small.

Discrete Dividend Payments in Continuous Time

We propose a model in which dividend payments occur at regular, deterministic intervals in an otherwise continuous model. This contrasts traditional models where either the payment of continuous dividends is controlled or the dynamics are given by discrete time processes. Moreover, between two dividend payments, the structure allows for other types of control; we consider the possibility of equity issuance at any point in time. The value is characterized as the fixed point of an optimal control problem with periodic initial and terminal conditions. We prove the regularity and uniqueness of the corresponding dynamic programming equation and the convergence of an efficient numerical algorithm that we use to study the problem. The model enables us to find the loss caused by infrequent dividend payments. We show that under realistic parameter values, this loss varies from around 1%–24% depending on the state of the system and that using the optimal policy from the continuous problem further increases the loss.

The implementation of the unified transform to the nonlinear Schr\xf6dinger equation with periodic initial conditions

The unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

Semiclassical dynamics and coherent soliton condensates in self‐focusing nonlinear media with periodic initial conditions

AbstractThe semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of ICs, referred to as “periodic single‐lobe” potentials, share the same qualitative features, which also coincide with those of solutions arising from localized ICs. The spectrum of the associated scattering problem in each of these cases is then numerically computed, and it is shown that such spectrum is confined to the real and imaginary axes of the spectral variable in the semiclassical limit. This implies that all nonlinear excitations emerging from the input have zero velocity, and form a coherent nonlinear condensate. Finally, by employing a formal Wentzel‐Kramers‐Brillouin expansion for the scattering eigenfunctions, asymptotic expressions for the number and location of the bands and gaps in the spectrum are obtained, as well as corresponding expressions for the relative band widths and the number of “effective solitons.” These results are shown to be in excellent agreement with those from direct numerical computation of the eigenfunctions. In particular, a scaling law is obtained showing that the number of effective solitons is inversely proportional to the small dispersion parameter.

Quantum-Mechanical Problem with Periodic Initial Conditions in Classical Physics and Geometrodynamics

It is shown that in classical physics there is a time analog of a spatial quantum-mechanical problem with periodic boundary conditions. On this basis, it is shown that the corresponding energy quantity has a band structure.

Chessboard-like spatio-temporal interference patterns and their excitation

We discover new type of interference patterns generated in the focusing nonlinear Schr\"odinger equation (NLSE) with localised periodic initial conditions. At special conditions, found in the present work, these patterns exhibit novel chess-board-like spatio-temporal structures which can be observed as the outcome of collision of two breathers. The infinitely extended chess-board-like patterns correspond to the continuous spectrum bands of the NLSE theory. More complicated patterns can be observed when the initial condition contains several localised periodic swells. These patterns can be observed in a variety of physical situations ranging from optics and hydrodynamics to Bose-Einstein condensates and plasma.

Квантово-механическая задача с периодическими начальными условиями в классической физике и геометродинамике

Квантово-механическая задача с периодическими начальными условиями в классической физике и геометродинамике

Optimal H2‐error estimates of conservative compact difference scheme for the Zakharov equation in two‐space dimension

In this paper, a conservative compact difference scheme is proposed for the two‐dimensional nonlinear Zakharov equation with periodic boundary condition and initial condition. The proposed scheme not only conserve the mass and energy in the discrete level but also are efficient in practical experiments because the Fast Fourier transform (FFT) can be used to speed up the numerical computation. By using the standard energy method and induction argument, we can establish rigorously the unconditional and optimal H2‐error estimates. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the new scheme.

Spectral method for vorticity-streamfunction equations, with application to Rayleigh-Benard convection

The spectral method with an explicit time integration scheme is implemented to solve the vorticity-streamfunction equations. In spectral method, functions are approximated with finite sums of periodic sine and cosine functions, which makes this method suitable for periodic problems. Numerical experiments for several cases are discussed. In Rayleigh-Benard convection problem, a horizontal layer of incompressible fluid is kept within high and low temperatures along the bottom and top boundaries, respectively. Triggered by a periodic initial condition, we simulate the occurrence of a regular pattern of convection cells, known as Benard cells.

Existence results for systems of conformable fractional differential equations

In this article, we study the existence of solutions to systems of conformable fractional differential equations with periodic boundary value or initial value conditions. where the right member of the system is $L^{1}_{\alpha }$-caratheodory function. We employ the method of solution-tube and Schauder’s fixed-point theorem.

Mini-Workshop: Reflectionless Operators: The Deift and Simon Conjectures

Reflectionless operators in one dimension are particularly amenable to inverse scattering and are intimately related to integrable systems like KdV and Toda. Recent work has indicated a strong (but not equivalent) relationship between reflectionless operators and almost periodic potentials with absolutely continuous spectrum. This makes the realm of reflectionless operators a natural place to begin addressing Deift’s conjecture on integrable flows with almost periodic initial conditions and Simon’s conjecture on gems of spectral theory establishing correspondences between certain coefficient and spectral properties.

Application of modal filtering to a spectral difference method

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a Spectral Difference Method (SD) solving hyperbolic conservation laws. In the SD Method we use selections of different orthogonal polynomials (APK polynomials). Furthermore we obtain new error bounds for filtered APK extensions of smooth functions. We demonstrate that the modal filter also depends on the chosen polynomial basis in the SD Method. Spectral filtering stabilizes the scheme and leaves weaker oscillations. Hence, the selection of the family of orthogonal polynomials on triangles and their specific modal filter possesses a positive influence on the stability and accuracy of the SD Method. In the second part, we initiate a stability analysis for a linear scalar test case with periodic initial condition to find the best selection of APK polynomials and their specific modal filter. To the best of our knowledge, this work is the first that gives a stability analysis for a scheme with spectral filtering. Finally, we demonstrate the influence of the underlying basis of APK polynomials in a well-known test case.

Effect of configuration symmetry on synchronization in a Van der Pol ring with nonlocal interactions

This paper discusses the influence of configuration symmetry on synchronization of coupled Van der Pol oscillators in a ring, where the inherent symmetry in an even number ring is broken by the presence of an odd oscillator. The coupling is considered to be nonlocal represented with a particular scaling exponent which decays with distance. The effect of initial conditions on synchronization dynamics is also studied.Synchronization in such ring networks is tracked from sets of initial conditions having periodic solutions. The set of these typical initial conditions is obtained by implementing a new generalized shooting strategy in the ring model. The robustness of this new method will bypass all the transients and give a periodic orbit, if any, for a given arbitrary initial condition. From these periodic initial conditions, interesting dynamics in odd and even number ring such as waking time of odd oscillator, shifting from antiphase to inphase clusters, resurrection of the oscillators from amplitude death, etc., are discussed. Important quantitative effects due to nonlocal interactions and the presence of initial condition-based amplitude death are also detailed.