Quasiperiodic Solutions in Weakly Nonlinear Gas Dynamics. Part I. Numerical Results in the Inviscid Case

We exhibit and study a new class of solutions for the one-dimensional inviscid Euler equations of Gas Dynamics in a bounded domain with reflecting boundary conditions, in the weakly nonlinear regime. These solutions do not present the usual wave breaking leading to shock formation, even though they have nontrivial acoustic components and operate in the nonlinear regime. We also show that these 'Non Breaking for All Times' (NBAT) solutions are globally attracting for the long time evolution of the equations.
The Euler equations of Gas Dynamics (in the weakly nonlinear regime with reflecting boundary conditions) can be reduced to an inviscid Burgers-like equation for the acoustic component, with a linear integral self-coupling term and periodic boundary conditions. The integral term arises as a result of the nonlinear resonant interactions of the sound waves with the entropy variations in the flow. This integral term turns out to be weakly dispersive. The NBAT solutions arise as a result of the interplay of this dispersion with the 'standard' wave-breaking nonlinearity in the Burgers equation.
In addition to the previously known weakly nonlinear standing acoustic wave NBAT solutions, we found a family of new, never-breaking, attracting solutions by direct numerical simulation. These are quasiperiodic in time with two periods. In phase space these solutions lie on a surface 'centered' around the standing waves. Only two standing-wave solutions (the maximum amplitude and the trivial vanishing wave) are in the attracting set. All of the others are quasiperiodic in time with two periods.     

Cross-Diffusion Driven Instability in a Predator-Prey System with Cross-Diffusion

In this work we investigate the process of pattern formation induced by nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra predator-prey kinetics. We show that the cross-diffusion term is responsible of the destabilizing mechanism that leads to the emergence of spatial patterns. Near marginal stability we perform a weakly nonlinear analysis to predict the amplitude and the form of the pattern, deriving the Stuart-Landau amplitude equations. Moreover, in a large portion of the subcritical zone, numerical simulations show the emergence of oscillating patterns, which cannot be predicted by the weakly nonlinear analysis. Finally, when the pattern invades the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude equation which is able to describe the shape and the speed of the wave.     

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [K.R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R³-in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293-311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 × 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT.     

Traveling wavefronts for a reaction-diffusion-chemotaxis model with volume-filling effect

In this paper, we study the propagation of the pattern for a reaction-diffusionchemotaxis model. By using a weakly nonlinear analysis with multiple temporal and spatial scales, we establish the amplitude equations for the patterns, which show that a local perturbation at the constant steady state is spread over the whole domain in the form of a traveling wavefront. The simulations demonstrate that the amplitude equations capture the evolution of the exact patterns obtained by numerically solving the considered system.     

Formation of wavy nanostructures on the surface of flat substrates by ion bombardment

A popular mathematical model for the formation of an inhomogeneous topography on the surface of a plate (flat substrate) during ion bombardment was considered. The model is described by the Bradley-Harper equation, which is frequently referred to as the generalized Kuramoto-Sivashinsky equation. It was shown that inhomogeneous topography (nanostructures in the modern terminology) can arise when the stability of the plane incident wavefront changes. The problem was solved using the theory of dynamical systems with an infinite-dimensional phase space, in conjunction with the integral manifold method and Poincaré-Dulac normal forms. A normal form was constructed using a modified Krylov-Bogolyubov algorithm that applies to nonlinear evolutionary boundary value problems. As a result, asymptotic formulas for solutions of the given nonlinear boundary value problem were derived.     

关于奇摄动拟线性系统

本文目的在于对非线性两点边值问题(1.1)证明其解存在的充分条件。并利用这一结果研究拟线性弱耦合奇摄动系统(DP)q的边界层现象。     

Two-Wave Interactions for Hyperbolic Signaling Problems and the Application to Supersonic Gas Dynamics

The signaling problem for a system of conservation laws in a single space variable is treated through the deployment of a perturbation analysis. Our method of approach involves the direct use of two nonlinear phase variables making possible the study of weakly nonlinear interacting waves arising from a boundary disturbance consisting of two wave modes. As a result of our analysis, the asymptotic solution is derived, and the class of admissible boundary disturbances is distinguished as well. An application is then made to gas dynamics in one space dimension to investigate the propagation and interaction of two sound waves for which the base state is taken to be a steady supersonic flow.     

Criterion for the solvability of a class of nonlinear two-point boundary value problems on the plane

We study the solvability of a class of nonlinear two-point boundary value problems for systems of ordinary second-order differential equations on the plane. In these boundary value problems, we single out the leading nonlinear terms, which are positively homogeneous mappings. On the basis of properties of the leading nonlinear terms, we prove a criterion for the solvability of boundary value problems under arbitrary perturbations in a given set by using methods for the computation of the winding number of vector fields.     

Weakly Nonlinear Time-Adiabatic Theory

We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schrödinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections.     

Bifurcation analysis of a viscoelastic fluid heated from below

Convection of a viscoelastic fluid in a square domain heated from below is investigated for the case of nondeformable free surfaces. To describe the rheological behavior of the fluid the generalized Oldroyd model is used. A weakly nonlinear analysis is performed in order to determine the character of branching for both the monotonic and oscillatory modes. We also perform a reduction of the boundary value-problem to the set of nonlinear amplitude equations. The analysis of this dynamic system demonstrates the onset and competition of five convection modes.     

Modeling weakly nonlinear acoustic wave propagation

Three weakly nonlinear models of lossless, compressible fluidflow—a straightforward weakly nonlinear equation (WNE),the inviscid Kuznetsov equation (IKE) and the Lighthill–Westerveltequation (LWE)—are derived from first principles and theirrelationship to each other is established. Through a numericalstudy of the blow-up of acceleration waves, the weakly nonlinearequations are compared to the ‘exact’ Euler equations,and the ranges of applicability of the approximate models areassessed. By reformulating these equations as hyperbolic systemsof conservation laws, we are able to employ a Godunov-type finite-differencescheme to obtain numerical solutions of the approximate modelsfor times beyond the instant of blow-up (that is, shock formation),for both density and velocity boundary conditions. Our studyreveals that the straightforward WNE gives the best results,followed by the IKE, with the LWE's performance being the poorestoverall.     

GLOBAL EXISTENCE OF WEAKLY DISCONTINUOUS SOLUTIONS TO A KIND OF MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.     

Asymptotic Solution of the Weakly Nonlinear Schrödinger Equation with Variable Coefficients

A perturbation method based on Fourier analysis and multiple scales is introduced for solving weakly nonlinear, dispersive wave propagation problems with Fourier-transformable initial conditions. Asymptotic solutions are derived for the weakly nonlinear cubic Schrödinger equation with variable coefficients, and verified by comparison with numerical solutions. In the special case of constant coefficients, the asymptotic solution agrees to leading order with previously derived results in the literature; in general, this is not true to higher orders. Therefore previous asymptotic results for the strongly nonlinear Schrödinger equation can be valid only for restricted initial conditions.     

The fluid dynamic limit of the nonlinear boltzmann equation

Solutions of the nonlinear Boltzmann equation are constructed up to the first appearance of shocks in the corresponding fluid dynamics. This construction assumes the knowledge of solutions of the Euler equations for compressible gas flow. The Boltzmann solution is found as a truncated Hilbert expansion with a remainder, and the remainder term solves a weakly nonlinear equation which is solved by iteration. The solutions found have special initial values. They should serve as “outer expansions” to which initial layers, boundary layers and shock layers can be matched.     

A priori estimates and solvability of the third two-point boundary value problem

We study a priori estimates and solvability of a nonlinear two-point boundary value problem for systems of second-order ordinary differential equations with leading positively homogeneous nonlinearity of order > 1 vanishing on a single surface. Assuming that an a priori estimate holds, we prove the invariance of the solvability of the problem under a continuous change of the leading nonlinear homogeneous terms and under arbitrary perturbations that do not affect the behavior of the leading nonlinear homogeneous terms at infinity.     

Construction of solutions of the defocusing nonlinear Schrödinger equation with asymptotically time‐periodic boundary values

We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. We use the unified transform method, also known as the Fokas method, and the Deift‐Zhou nonlinear steepest descent method to construct solutions in a sector close to the boundary whose leading behavior is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long‐time asymptotics of the constructed solutions.     

Conservative solutions to a system of asymptotic variational wave equations

This paper is concerned with an asymptotic variational wave system which models weakly nonlinear waves for a system of variational wave equations arising in the theory of nematic liquid crystals and a few other physical contexts. By constructing a global semigroup, we establish the well-posedness of the initial–boundary value problem within the class of energy-conservative solutions for initial data of finite energy.     

Low-Frequency Multidimensional Instabilities for Reacting Shock Waves

This paper presents a weakly nonlinear analysis for one scenario for the development of transversal instabilities in detonation waves in two space dimensions. The theory proposed and developed here is most appropriate for understanding the behavior of regular and chaotically irregular pulsation instabilities that occur in detonation fronts in condensed phases and occasionally in gases. The theory involves low-frequency instabilities and through suitable asymptotics yields a complex Ginzburg-Landau equation that describes simultaneously the evolution of the detonation front and the nonlinear interactions behind this front. The asymptotic theory mimics the familiar theory of nonlinear hydrodynamic instability in outline; however, there are several novel technical aspects in the derivation because the phenomena studied here involve a complex free boundary problem for a system of nonlinear hyperbolic equations with source terms.     

Weakly Nonlinear Lower-Branch Stability of Fully Developed and Developing Free-Surface Flows

We examine the weakly nonlinear stability of both fully developedand of developing liquid layers. The study of these free-surfaceflows is more complicated than that of many other flows owingto the fully nonlinear boundary conditions present. The scalingsused for the two problems follow from the work of J. S. B. Gajjar,who described their linear stability properties. We use thetechnique given by F. T. Smith to derive amplitude equationsof the type presented by J. T. Stuart and J. Watson. Both flowsare found to be supercritically stable in general and a varietyof asymptotic cases are considered.