The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity

It is proved herein that certain smooth, global solutions of a class of quasi-linear, dissipative wave equations have precisely the same leading order, long-time, asymptotic behavior as the solutions with the same initial data of the corresponding linearized equations. The solutions of the nonlinear equations are shown to be asymptotically self-similar with explicitly determined profiles. The equations considered have homogeneous nonlinearities and homogeneous dispersive and dissipative symbols. By relating these degrees of homogeneous to the leading order asymptotic behavior of the Fourier transform of the initial data near k= 0, different classes of long-time asymptotic behavior are characterized. These results cover the case where dissipation is not asymptotically negligible in comparison with dispersion, and where nonlinear effect are asymptotically negligible in comparison with linear effect, i.e., dissipation and dispersion. They always hold for solutions with "small" initial data. In most circumstances however a new a priori bound on certain negative homogeneous Sobolev norms of solutions is obtained, which implies that any solution, even one which is initially "large" will eventually satisfy the smallness condition, and hence will have the above described asymptotic behavior     

The Interaction of Shocks with Dispersive Waves II. Incompressible-Integrable Limit

This is the second in a two-part series of articles in which we analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading-order shock equations and the leading-order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading-order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading-order shock equations.     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.     

Time-linearized,compact methods for the inviscid GRLW equation subject to initial Gaussian conditions

The objective of this paper is three-fold. First, four time-linearization methods that are second- and fourth-order accurate in time and space, respectively, are presented and used to study the dynamics of the modified and generalized regularized-long wave equations (mRLW and GRLW equations, respectively). Two of the methods use the conservation-law form of the equations and treat the wave amplitude and its second-order spatial derivative and the linear and nonlinear advection fluxes as unknowns, whereas the other two employ the non-conservation-law form of the equations and consider the wave amplitude and its first- and second-order spatial derivatives as unknowns. The methods employ three-point fourth-order accurate Padé discretizations for the first- and second-order derivatives, are second-order accurate in time, and yield linear systems of blocktridiagonal matrices. Second, the accuracy of these methods is assessed by comparing their results with those of the exact solution of the mRLW equation. It is reported that the four methods predict nearly the same values of the three invariants and have the same accuracy, and that an accurate prediction of the invariants may not correspond to small errors in space and time. Third, the dynamics of the inviscid GRLW equation is studied first qualitatively in terms of length and time scales and then numerically as a function of the linear advection speed, the exponent of the nonlinear advection flux, the dispersion coefficient and the amplitude and width of the initial bell-shaped or Gaussian conditions. It is shown that wide initial conditions result in wave steepening and breakup and the formation of solitary waves whose amplitude and speed decrease as the time for their formation increases. For narrow initial conditions, it is shown that only a single solitary wave may form. Behind this wave and depending on the parameters that characterized the inviscid GRLW equation, rarefaction or negative amplitude waves that propagate towards the upstream boundary or a train of localized oscillatory waves that do not emerge from the trailing edge of the leading solitary wave may be formed. These oscillatory waves exhibit the characteristics of, but are not dispersive shock waves and their amplitude and frequency increases as the width of the initial conditions is decreased. The results presented here do not only complement previous work by the authors, they also show that the dynamics of the inviscid GRLW equation undergoes new and interesting phenomena as the width of the initial conditions is decreased.     

On the breakdown of acceleration waves in dissociating gas flows

The present paper is devoted to the study of characteristic solution in the neighbourhood of the leading frozen characteristics in dissociating gas flows. It is found that at the cusp of the envelope of intersecting forward characteristics there occurs a breakdown of the wave after a finite critical timet _o. It is observed that there exists a critical value of the initial amplitude of the wave such that all compressive waves with an initial amplitude greater than the critical one will terminate into a shock wave due to non-linear steepening while an initial amplitude less than the critical one will result in a continuous decay. It is also concluded that the breakdown point moves forward along the leading characteristics due to dissociation effects.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.     

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.     

On the Riemann problem for 2D compressible Euler equations in three pieces

The Riemann problem for two-dimensional isentropic Euler equations is considered. The initial data are three constants in three fan domains forming different angles. Under the assumption that only a rarefaction wave, shock wave or contact discontinuity connects two neighboring constant initial states, it is proved that the cases involving three shock or rarefaction waves are impossible. For the cases involving one rarefaction (shock) wave and two shock (rarefaction) waves, only the combinations when the three elementary waves have the same sign are possible (impossible).     

Weak shock waves and its interaction with characteristic shocks in polyatomic gas

We present Lie symmetry analysis for investigating the shock‐wave structure of hyperbolic differential equations of polyatomic gases. With the application of symmetry analysis, we derive particular exact group invariant solutions for the governing system of partial differential equations (PDEs). In the next step, the evolutionary behavior of weak shock along with the characteristic shock and their interaction is investigated. Finally, the amplitudes of reflected wave, transmitted wave, and the jump in shock acceleration influenced by the incident wave after interaction are evaluated for the considered system of equations.     

Stability of viscous shock wave under periodic perturbation for compressible Navier–Stokes–Korteweg system

In this paper, we study the stability of a viscous shock wave for the isentropic Navier–Stokes–Korteweg (N-S-K) equations under space-periodic perturbation. It is shown that if the initial perturbation around the shock and the amplitude of the shock are small, then the solution of the N-S-K equations tends to the viscous shock.     

Continuous compression waves in the two-dimensional Riemann problem

The interaction between a plane shock wave in a plate and a wedge is considered within the framework of the nondissipative compressible fluid dynamic equations. The wedge is filled with a material that may differ from that of the plate. Based on the numerical solution of the original equations, self-similar solutions are obtained for several versions of the problem with an iron plate and a wedge filled with aluminum and for the interaction of a shock wave in air with a rigid wedge. The behavior of the solids at high pressures is approximately described by a two-term equation of state. In all the problems, a two-dimensional continuous compression wave develops as a wave reflected from the wedge or as a wave adjacent to the reflected shock. In contrast to a gradient catastrophe typical of one-dimensional continuous compression waves, the spatial gradient of a two-dimensional compression wave decreases over time due to the self-similarity of the solution. It is conjectured that a phenomenon opposite to the gradient catastrophe can occur in an actual flow with dissipative processes like viscosity and heat conduction. Specifically, an initial shock wave is transformed over time into a continuous compression wave of the same amplitude.     

On the Whitham Equations for the Defocusing Nonlinear Schrodinger Equation with Step Initial Data

The behavior of solutions of the finite-genus Whitham equations for the weak dispersion limit of the defocusing nonlinear Schrodinger equation is investigated analytically and numerically for piecewise-constant initial data. In particular, the dynamics of constant-amplitude initial conditions with one or more frequency jumps (i.e., piecewise linear phase) are considered. It is shown analytically and numerically that, for finite times, regions of arbitrarily high genus can be produced; asymptotically with time, however, the solution can be divided into expanding regions which are either of genus-zero, genus-one, or genus-two type, their precise arrangement depending on the specifics of the initial datum given. This behavior should be compared to that of the Korteweg-de Vries equation, where the solution is divided into regions which are either genus-zero or genus-one asymptotically. Finally, the potential application of these results to the generation of short optical pulses is discussed: The method proposed takes advantage of nonlinear compression via appropriate frequency modulation, and allows control of both the pulse amplitude and its width, as well as the distance along the fiber at which the pulse is produced     

Breakdown of the Whitham Modulation Theory and the Emergence of Dispersion

The Whitham modulation theory for periodic traveling waves of PDEs generated by a Lagrangian produces first‐order dispersionless PDEs that are, generically, either hyperbolic or elliptic. In this paper, degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero. In this case, the Whitham equations are no longer valid. Reformulation and rescaling show that conservation of wave action morphs into the Korteweg–de Vries (KdV) equation on a longer time scale thereby generating dispersion in the Whitham modulation equations even for finite amplitude waves.     

The Interaction of Shocks with Dispersive Waves I. Weak Coupling Limit

We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of shocks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time-dependent Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Schrodinger equation that are steady in a frame of reference moving with the shock. Then we add a viscous diffusion term to the shock equation and by explicitly constructing asymptotic expansions in the (small) diffusion coefficient, we show that these solutions are zero diffusion limits of the regularized problem. The expansions are unusual in that it is necessary to keep track of exponentially small terms to obtain algebraically small terms. The expansions are compared to numerical solutions. We then construct a family of time-dependent solutions in the case that the initial data for the nonlinear wave equation evolves to a shock as t → t* < ∞. We prove that the shock formation drives a finite time blow-up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient blows up logarithmically in time. We construct several explicit time-dependent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states with phase discontinuities (which we call phase kinked steady states), (c) do not evolve to steady states.     

一类KdV-Burgers方程的奇摄动解与孤子解

讨论了一类具有大Reynolds数且弱频散性的KdV-Burgers方程, 在数学上表示为一类奇摄动KdV-Burgers方程.KdV-Burgers方程中含有的非线性项与频散项互补作用形成稳定向前传播的孤立子.通过数学分析, 描述了孤立子的传播途径和传播速度等物理量的发展变化规律.通过奇摄动展开方法, 构造了该问题的渐近解.首先,用Riemann-Earnshaw方法求得退化解, 得到了简单波, 该简单波波形中的任意一点与初始点都存在一个传播速度差, 这使得波在传播过程中波形不断畸变, 最终形成冲击波面, 即间断面, 在它的两侧质点的速度有一个跳跃, 且随时间不断变化;其次, 在退化解的间断曲面处做变量替换, 构造一种修正的行波变换, 得到了内解展开式的孤子解, 并证明了内外解的存在性与唯一性;最后,通过一致有界逆算子的存在性做了余项估计, 并得到渐近解的一致有效性.结果表明, KdV-Burgers方程在大Reynolds数且弱频散性的性质下,扰动集中在退化解的间断面附近,孤立子链接两侧质点,其传播途径不是时间与空间的线性形式,而是沿着退化解的间断面附近传播,形成稳定的波形.