New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

For the spatially-distributed Fermi–Pasta–Ulam (FPU) equation, irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes. The main result of this paper is the construction of families of special nonlinear systems of the Schrödinger type—quasinormal forms—whose nonlocal dynamics determines the local behavior of solutions to the original problem, as t→∞. On their basis, results are obtained on the asymptotics in the main solution of the FPU equation and on the interaction of waves moving in opposite directions. The problem of “perturbing” the number of N elements of a chain is considered. In this case, instead of the differential operator, with respect to one spatial variable, a special differential operator, with respect to two spatial variables appears. This leads to a complication of the structure of an irregular solution.

Use of Evolutionary Computation to Improve Rock Slope Back Analysis

Generally, in geotechnical engineering, back analyses are used to investigate uncertain parameters. Back analyses can be undertaken by considering known conditions, such as failure surfaces, displacements, and structural performances. Many geotechnical problems have irregular solution domains, with the objective function being non-convex, and may not be continuous functions. As such, a complex non-linear optimization function is typically required for most geotechnical problems to attain a better understanding of these uncertainties. Therefore, particle swarm optimization (PSO) and a genetic algorithm (GA) are utilized in this study to facilitate in back analyses mainly based on upper bound finite element limit analysis method. These approaches are part of evolutionary computation, which is appropriate for solving non-linear global optimization problems. By using these techniques with upper-bound finite element limit analysis (UB-FELA), two case studies showed that the results obtained are reasonable and reliable while maintaining a balance between computational time and accuracy.

Irregular solution thermodynamics of wood pulp in the superbase ionic liquid [m-TBDH][AcO]

Thermochemical analysis of cellulose dissolution character in the superbase containing protic ionic liquid [m-TBDH][AcO] reveals lower critical solution temperature (LCST) behaviour.

Bessel Equation in the Semiunbounded Intervalx∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point

This study expresses the solution of the Bessel equation in the neighbourhood ofx=∞as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval[x0,∞]. In order to guarantee the series’ uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution atx→∞(not to mention a so-called formal solution), but a solution in the interval[x0,∞]with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal.

A Parameter Sweep Experiment on Quasiperiodic Variations of a Polar Vortex due to Wave–Wave Interaction in a Spherical Barotropic Model

Weakly nonlinear aspects of a barotropically unstable polar vortex in a forced–dissipative system with a zonally asymmetric surface topography are investigated in order to obtain a deeper understanding of rather periodic variations of the winter circumpolar vortex in the Southern Hemisphere stratosphere that are characterized by the wave–wave interaction between the stationary planetary wave of zonal wavenumber 1 (denoted as Wave 1) and the eastward traveling Wave 2 as studied by Hio and Yoden in 2004. The authors use a spherical barotropic model with a forcing of zonally symmetric jet, dissipation, and sinusoidal surface topography. A parameter sweep experiment is performed by changing the amplitude of the surface topography, which forces the stationary Wave 1, and the width of the prescribed zonally symmetric jet, which controls the barotropic instability, to generate the traveling Wave 2. Several types of solutions from a time-independent solution to a nonperiodic irregular solution are obtained for the combination of these external parameters, but the predominant solution obtained in a wide parameter space is periodic. Details of the wave–wave interactions are described for the transition from a quasiperiodic vacillation to a periodic solution as the increase of the amplitude of topography. Phase relationships are locked at the transition, and variations of zonal-mean zonal flow and topographically forced Wave 1 synchronize with periodic progression of Wave 2 in the periodic solution. A diagnosis with a low-order “empirical mode expansion” of the vorticity equation gives a limited number of dominant nonlinear triad interactions among the zonal-mean, Wave-1, and Wave-2 components around the transition point.

A Second Order Lagrangian Model for Irregular Ocean Waves

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

A Second Order Lagrangian Model for Irregular Ocean Waves

Synthetic Aperture Radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, the slope and the mean curvature are studied.