Interactions of delta shock waves for the chromatography equations

This paper is devoted to the interactions of the delta shock waves with the shock waves and the rarefaction waves for the simplified chromatography equations. The global structures of solutions are constructed completely if the delta shock waves are included when the initial data are taken three piece constants and then the stability of Riemann solutions is also analyzed with the vanishing middle region. In particular, the strength of delta shock wave is expressed explicitly and the delta contact discontinuity is discovered during the process of wave interaction.     

Integrability and Linear Stability of Nonlinear Waves

It is well known that the linear stability of solutions of \(1+1\) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general \(N\times N\) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for \(N=3\) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.     

Delta shock waves with Dirac delta function in both components for systems of conservation laws

We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine–Hugoniot relation and entropy condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine–Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.     

New conservative difference schemes for a coupled nonlinear Schrödinger system

In this paper, two conservative difference schemes for solving a coupled nonlinear Schrödinger (CNLS) system are numerically analyzed. Firstly, a nonlinear implicit two-level finite difference scheme for CNLS system is studied, then a linear three-level difference scheme for CNLS system is presented. An induction argument and the discrete energy method are used to prove the second-order convergence and unconditional stability of the linear scheme. Numerical examples show the efficiency of the new scheme.     

On the stability of explicit finite difference methods for advection–diffusion equations

In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi-discretization of the ADE leads to a conditionally stable fully discretized method as long as the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.     

Resonant Interactions between Vortical Flows and Water Waves. Part I: Deep Water

Any weak, steady vortical flow is a solution to leading order of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic, resonant interactions can occur between the various components of the flow. The periodic vortical component of the flow is proposed as a model for more complicated vortical flows that would affect surface waves in the ocean, such as the turbulence in the wake of a ship. These resonant interactions are studied in two dimensions, both in the limit of deep water (Part I) and shallow water (Part II). For deep water, the resonant set of surface waves is governed by “triad-like” ordinary differential equations for the wave amplitudes, whose coefficients depend on the underlying rotational flow. These coefficients are calculated explicitly and the stability of various configurations of waves is discussed. The effect of three dimensionality is also briefly mentioned.     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

Interactions of delta shock waves and stability of Riemann solutions for nonlinear chromatography equations

This paper is devoted to studying the simplified nonlinear chromatography equations by introducing the change of state variables. The Riemann solutions containing delta shock waves are presented. In order to study wave interactions of delta shock waves with elementary waves, the global structure of solutions is constructed completely when the initial data are taken as three pieces of constants and the delta shock waves are included. In particular, the strength of delta shock wave is expressed explicitly and the delta contact discontinuity is discovered during the process of wave interactions. Moreover, by analyzing the limits of the solutions as the middle region vanishes, we observe that the Riemann solutions are stable for such a local small perturbation of the Riemann initial data.     

Stability of steady flows for multi-mode Maxwell fluids

We give a rigorous condition for stability of steady flows of multi-mode nonlinear Maxwell fluids. Specifically, we prove that a flow is linearly stable if and only if the spectrum of the linearization is in the left half plane and, in addition, a certain system of ordinary differential equations associated with short wave asymptotics is stable.     

Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically and the criteria for the stability are obtained. The long time evolution of the solutions to the coupled system is studied numerically for the unstable case emphasizing wave–wave interactions in nonlinear optics. Different kinds of evolution are observed depending on the coefficients of the system and the parameters of the unperturbed waves and perturbation. For certain ranges of parameters, the evolution appears to be periodic, while for some other ranges of parameters, solitary wave or solitary wave pairs can be excited among the irregular background although often the evolution is completely chaotic.     

Synchronization of two flow excited pendula

Two aerodynamically excited pendula are considered as a simple example of two linearly coupled, self sustained mechanical oscillators, modelled by two coupled Van-der-Pol equations. The considered mechanical application admits of a systematic survey of synchronized regimes within the framework of standard nonlinear stability analysis. Using normal form theory and the prevailing direct averaging approach the occurring Hopf bifurcation with two distinct pairs of purely imaginary eigenvalues is studied in the non-resonant case and in the 1:1-resonance corresponding, respectively, to strong and weak coupling. In particular, for the resonant case a graphical approach permits a comprehensive interpretation relating the stable stationary solutions of the averaged system with synchronized regimes and allows an analytical computation of the oscillation amplitudes and the synchronous frequency.     

Simulation of the Spatial Action of a Medium on a Body of Conical Form

We consider amathematical model of the spatial action of a medium on the axisymmetric rigid body whose external surface has a part that is a circular cone.We present a complete system of equations of motion under the quasistationary conditions. The dynamical part forms an independent system of the sixth order in which the independent subsystems of lower order are distinguished. We study the problem of stability with respect to the part of variables of the key regime—the spatial rectilinear translational deceleration of the body. For a particular class of bodies, we show the inertial mass characteristics under which the key regime is stable. For a plane analog of the problem, we obtain a family of phase portraits in the space of quasivelocities.     

On Wave Interactions,I: Explosive Resonant Triads

This paper considers the phenomenon of explosive resonant triads in weakly nonlinear, dispersive wave systems. These nearly linear waves with slowly varying amplitudes become unbounded in finite time. It is shown that such interactions are much stronger than previously thought. These waves can be thought of as a nonlinear instability in the sense that a weakly nonlinear perturbation to some system grows to such a magnitude that the behavior of the system is governed by strongly nonlinear effects. This may occur for systems that are linearly or neutrally stable. This resolution is contrasted with previous resolutions of the problem, which assumed such perturbations remained large amplitude, nearly linear waves. Analytical and numerical evidence is presented to support these claims.     

Analytic method for solving coupled nonlinear Schroedinger equations with oscillating terms describing polarized soliton oscillations

Using a variational method, a new model of the nonlinear propagation of optical solitons generated from semiconductor lasers through lossy elliptically low birefringent fiber, is presented within the framework of a system of coupled nonlinear Schroedinger (CNLS) equations with oscillating terms. This analytic model demonstrates polarized soliton oscillations in a lossy elliptically low birefringent optical fiber.     

A model-coupling framework for nearshore waves,currents, sediment transport,and seabed morphology

This paper presents a framework for synchronously coupling wave, current, sediment transport, and seabed morphology for the accurate simulation of multi-physics coastal ocean processes. The governing equations, which represent models that are commonly adopted in practical simulations, are discretized using finite-difference methods. The resulting system is validated against analytical solutions. In order to test the performance of the proposed framework and the numerical methods, dam-break flow over a mobile-bed and evolution of a wave-driven sand dune are simulated. The interactions among waves, currents, and seabed morphology are illustrated.     

Numerical Analysis of Spatial Hydrodynamic Stability of Shear Flows in Ducts of Constant Cross Section

A technique for analyzing the spatial stability of viscous incompressible shear flows in ducts of constant cross section, i.e., a technique for the numerical analysis of the stability of such flows with respect to small time-harmonic disturbances propagating downstream is described and justified. According to this technique, the linearized equations for the disturbance amplitudes are approximated in space in the plane of the duct cross section and are reduced to a system of first-order ordinary differential equations in the streamwise variable in a way independent of the approximation method. This system is further reduced to a lower dimension one satisfied only by physically significant solutions of the original system. Most of the computations are based on standard matrix algorithms. This technique makes it possible to efficiently compute various characteristics of spatial stability, including finding optimal disturbances that play a crucial role in the subcritical laminar–turbulent transition scenario. The performance of the technique is illustrated as applied to the Poiseuille flow in a duct of square cross section.     

Vanishing Viscosity Limit for Riemann Solutions to a Class of Non-Strictly Hyperbolic Systems

By the vanishing viscosity approach, a class of non-strictly hyperbolic systems of conservation laws that contain the equations of geometrical optics as a prototype are studied. The existence, uniqueness and stability of solutions involving delta shock waves and generalized vacuum states are discussed completely.     

On resonant interactions of gravity‐capillary waves without energy exchange

We consider resonant triad interactions of gravity‐capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two‐dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two‐dimensional generalization of Wilton ripples.     

Resonant instability and nonlinear wave interactions in electrically forced jets

We investigate the problem of linear temporal instability of the modes that satisfy the dyad resonance conditions and the associated nonlinear wave interactions in jets driven by either a constant or a variable external electric field. A mathematical model, which is developed and used for the temporally growing modes with resonance and their nonlinear wave interactions in electrically driven jet flows, leads to equations for the unknown amplitudes of such waves. These equations are solved for both water and glycerol jet cases, and the expressions for the dependent variables of the corresponding modes are determined. The results of the generated data for these dependent variables versus time indicate, in particular, that the instability resulted from the nonlinear interactions of such modes is mostly quite strong but can also lead to significant reduction in the jet radius.     

Diffraction of a plane wave by an infinite elastic plate stiffened by a doubly periodic set of rigid ribs

The diffraction of a plane wave by an infinite elastic plate stiffened by a doubly periodic set of rigid ribs of moderate wave dimensions is studied. The problem is reduced to an infinite quasiregular system of linear algebraic equations, and their solution describes the amplitudes of the waves propagating from the plate into the fluid.