Stability Study of Stationary Solutions of the Viscous Burgers Equation

<abstract><p>The article studied tsunami waves with consideration of important wave properties such as velocity, width, and collision through finding an approximate solution to the damped geophysical Korteweg-de Vries (dGKdV). The addition of the damping term in the GKdV is a result of studying the nonlinear waves in bounded nonplanar geometry. The properties of the wave in bounded nonplanar geometry are different than the unbounded planar geometry, as many experiments approved. Thus, this work reported for the first time the analytical solution for the dGKdV equation using the Ansatz method. The used method assumed a suitable hypothesis and the initial condition of the GKdV. The GKdV is an integrable equation and the solution can be found by several known methods either analytically or numerically. On the other hand, the dGKdV is a nonintegrable equation and does not have an initial exact solution, and this is the challenge. In this work, the novel Ansatz method proved its ability to reach the approximate solution of dGKdV and presented the effect of the damping term as well as the Coriolis effect term in the amplitude of the wave. The advantage of the Ansatz method was that the obtained solution was in a general solution form depending on the exact solution of GKdV. This means the variety of nonlinear wave structures like solitons, lumps, or cnoidal can be easily investigated by the obtained solution. We realized that the amplitude of a tsunami wave decreases if the Coriolis term or damping term increases, while it increases if wave speed increases.</p></abstract>

Asymptotically Minkowskian stationary axisymmetric solutions of the five-dimensional Einstein equations are generated from particular classes of stationary axisymmetric solutions of the four-dimensional Einstein-Maxwell equations. First, the five-dimensional electrostatic and magnetostatic solutions are generated from the four-dimensional electrostatic solutions and a harmonic scalar field. Then, a new class of five-dimensional nonstatic solutions are generated from the four-dimensional class ℰ=+1. As an example, a three-parameter family of regular asymptotically flat rotating solutions is constructed. These solutions can be interpreted, after dimensional reduction to four dimensions, as extended elementary particles with mass, spin, electric charge, and magnetic dipole moment.

Pulsate flow is an effective technique applied for cooling several engineering systems depending on their pulsate frequency. One very sound external flow pulsation application is heat transfer over heated bodies. In present work, an experimental design and numerical model of controlled pulsating flow according to generated pulsating frequency and wave shape around a heated cylinder were performed. The effects of pulsating frequency, amplitude, and mean velocity on the fluid flow and heat transfer characteristics over a heated cylinder were studied. The wave frequency varied from 2 to 12 Hz, and the amplitude varied from 0.2 to 0.8 m/s. Moreover, different waveforms were investigated to determine their effect on wall cooling. For constant wave frequency and amplitude, the most efficient wave in cooling was the sawtooth wave, with the average wall temperature after 30 s was 1.6 °C cooler than that of the forced convection case, followed by the triangular wave at 1.2 °C less. The heat transfer rate and the flow field were drastically influenced by the variations of these parameters. Optimization was conducted for each wave type to find the optimum wave frequency and amplitude. The optimizing showed that, the most efficient wave was the sawtooth with 12°C temperature reduction compared with that of the forced convection case, followed by the triangular. Furthermore, regression analysis was conducted to estimate the relationships between these variables and surface temperature. It was found that the wave amplitude had a greater role in cooling than that of the frequency

In the first part of this paper, we review the most important properties of the axisymmetric and stationary solutions of Einstein's vacuum field equations, the solution generating techniques developed in the last few years the relativistic definitions of multipole moments, and their use for obtaining and investigating static and stationary axisymmetric vacuum solutions. In the second part, we derive the general static axisymmetric vacuum solution in prolate spheroidal coordinates. Finally, this solution with all mass multipole moments is generalized to include a further parameter related to the rotation of the mass. This stationary solution can be used in order to describe the exterior gravitational field of a rotating deformed mass.

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions.

We investigate the Gross-Pitaevskii equation for a Bose-Einstein condensate in a PT symmetric double-well potential by means of the time-dependent variational principle and numerically exact solutions. A one-dimensional and a fully three-dimensional setup are used. Stationary states are determined and the propagation of wave function is investigated using the time-dependent Gross-Pitaevskii equation. Due to the nonlinearity of the Gross-Pitaevskii equation the potential dependson the wave function and its solutions decide whether or not the Hamiltonian itself is PT symmetric. Stationary solutions with real energy eigenvalues fulfilling exact PT symmetry are found as well as PT broken eigenstates with complex energies. The latter describe decaying or growing probability amplitudes and are not true stationary solutions of the time-dependent Gross-Pitaevskii equation. However, they still provide qualitative information about the time evolution of the wave functions.

Using the well-known reductive perturbation technique, the three-dimensional (3D) Burgers equation and modified 3D Burgers equation have been derived for a plasma system comprising of non-thermal ions, Maxwellian electrons, and negatively charged fluctuating dust particles. The salient features of nonlinear propagation of shock waves in such plasmas have been investigated in detail. The different temperature non-thermal ions and Maxwellian electrons are found to play an important role in the shock waves solution. The analytical solution of the 3D Burgers equation and modified 3D Burgers equation ratifying the propagation of dust acoustic shock waves are derived using the well-known tanh method. On increasing the population of non-thermal ions, an enhancement in the amplitude of shock waves is seen for negatively charged dust particles. A striking dependence of amplitude and width of shock waves on the ratio of ion temperatures and densities are also reported. Finally we introduced a new stretching coordinate and perturbation for the nth-order nonlinear 3D Burgers equation and its solution by the use of the tanh method. We found that, due to higher nonlinearity, the amplitude of shock waves decreases while width remains constant for all plasma parameters considered in the present investigation. The features accounted here could be relevant in the case of different space and astrophysical plasmas and laboratory dusty plasma for negatively charged dust fluctuation.

In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of the solutions of an algebraic equation for a cusp case. This one-to-one correspondence is established essentially based on the treatment in the work of (Jiao et al. in J Diff Dyn Syst 34:239-269, 2022) for a different case, and the relation gives directly parameter ranges for existence of multiple stationary solutions; in particular, multiple stationary solutions are created through countably many saddle-node bifurcations for the algebraic equation at critical shear speeds. The main result of the paper is on the stability of stationary solutions associated to the bifurcations; more precisely, (i) for each critical shear speed, there is a unique stationary solution and, for smaller shear speed, the stationary solution disappears but, for larger shear speed, two stationary solutions nearby bifurcate; (ii) more importantly, under a generic condition, there is a simple zero eigenvalue for the linearization of the shear flow at the critical stationary solution and, for larger shear speed, the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution.

The equations describing flows of an incompressible viscoelastic polymer liquid are studied. Stationary solutions similar to the Poiseuille and Couette solutions for the system of the Navier-Stokes equations are constructed. Stationary discontinuous solutions of the polymer liquid equation are also considered.

Lagrangian coordinates in free boundary problems for parabolic equations

On the effectiveness of constant coefficients roll motion equation

Stationary Fokker-Planck equation applied to fission dynamics

We first present for the porous medium equation, a typical nonlinear degenerate diffusion equation, that its (forward) self-similar solution well describes asymptotic behavior of solutions, as is observed for the heat equation, without proof. We next explain that it is important to classify backward self-similar solutions in order to analyze behavior of solutions near singularities for the axisymmetric mean curvature flow equation as an example. In what follows, a self-similar solution is regarded as a stationary solution of the equation written with similarity variables. Convergence behavior of a solution of the equation to its stationary corresponds to the asymptotic behavior of the solution of the original equation near singularities. We give an outline of the proof of convergence and mention that a monotonicity formula plays a key role. Moreover, we give a simple proof of uniqueness of the stationary solutions, i.e., the backward self-similar solutions of the original equation. The proof is simpler and easier than that in the literature. We remark that the method using similarity variables is applicable, to some extent, to other diffusion equations such as semilinear heat equations and harmonic map flow equations. Finally, we note that the existence of forward self-similar solutions has also been proved for nonlinear equations of nondiffusion type.

On the transmission of external pressure fluctuations to the cerebrospinal fluid

An investigation is carried out for understanding the properties of ion–acoustic (IA) solitary waves in an inhomogeneous magnetized electron-ion plasma with field-aligned sheared flow under the impact of q-nonextensive trapped electrons. The Schamel equation and its stationary solution in the form of solitary waves are obtained for this inhomogeneous plasma. It is shown that the amplitude of IA solitary waves increases with higher trapping efficiency (β), while the width remains almost the same. Further, it is found that the amplitude of the solitary waves decreases with enhanced normalized drift speed, shear flow parameter and the population of the energetic particles. The size of the nonlinear solitary structures is calculated to be a few hundred meters and it is pointed out that the present results are useful to understand the solar wind plasma.