Stationary and quasi-stationary waves in even-order dissipative systems

A new equation was recently suggested by Rudenko and Robsman [1] for describing the nonlinear wave propagation in scattering media that are characterized by weak sound signal attenuation proportional to the fourth power of frequency. General self-similar properties of the solutions to this equation were studied. It was shown that stationary solutions to this equation in the form of a shock wave exhibit unusual oscillations around the shock front, as distinct from the classical Burgers equation. Here, similar solutions are studied in detail for nonlinear waves in even-order dissipative media; namely, the solutions are compared for the media with absorption proportional to the second, fourth, and sixth powers of frequency. Based on the numerical results and the self-similar properties of the solutions, the fine structure of the shock front of stationary waves is studied for different absorption laws and magnitudes. It is shown that the amplitude and number of oscillations appearing in the stationary wave profile increase with increasing power of the frequency-dependent absorption term. For initial disturbances in the form of a harmonic wave and a pulse, quasi-stationary solutions are obtained at the stage of fully developed discontinuities and the evolution of the profile and width of the shock wave front is studied. It is shown that the smoothening of the shock front in the course of wave propagation is more pronounced when the absorption law is quadratic in frequency.     

自相似Euler方程的数值方法

Euler方程某些问题的解具有自相似特点，可以使用更准确的方法求解.提出了两种数值方法，分别称为自相似和准自相似方法，新方法可以使用现有守恒律方程的数值格式，无须设计特殊方法.对一维激波管问题、二维Riemann问题、激波反射以及激波折射问题进行了数值计算.对自相似Euler方程，一维计算结果显示数值解基本等同于精确解，二维结果也比现有文献计算的结果有更高的分辨率.对准自相似Euler方程，新方法可以求解不具有自相似性但接近自相似的问题，并在计算时间足够长时可以取得自相似Euler方程的效果.数值求解自相似Euler方程对自相似问题的研究，高分辨率、高精度格式的设计乃至Euler方程的精确解都有重要启示.      

Global Shock Waves¶for the Supersonic Flow Past a Perturbed Cone

We prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone. The flow is governed by the potential equation, as well as the boundary conditions on the shock and the surface of the body. It is shown that the solution to this problem exists globally in the whole space with a pointed shock attached at the tip of the cone and tends to a self-similar solution under some suitable conditions. Our analysis is based on a global uniform weighted energy estimate for the linearized problem. Combining this with the local existence result of Chen–Li [1] we establish the global existence and decay rate of the solution to the nonlinear problem. Received: 1 August 2001 / Accepted: 14 January 2002     

Spatiotemporal Self-Similar Solutions of the Generalized (3+1)-dimensional Nonlinear Schrödinger Equation with Polynomial Nonlinearity of Arbitrary Order

We construct analytical self-similar solutions for the generalized (3+1)-dimensional nonlinear Schrödinger equation with polynomial nonlinearity of arbitrary order. As an example, we list self-similar solutions of quintic nonlinear Schrödinger equation with distributed dispersion and distributed linear gain, including bright similariton solution, fractional and combined Jacobian elliptic function solutions. Moreover, we discuss self-similar evolutional dynamic behaviors of these solutions in the dispersion decreasing fiber and the periodic distributed amplification system.     

On the Stability of Self-Similar Solutions to Nonlinear Wave Equations

We consider an explicit self-similar solution to an energy-supercritical Yang-Mills equation and prove its mode stability. Based on earlier work by one of the authors, we obtain a fully rigorous proof of the nonlinear stability of the self-similar blowup profile. This is a large-data result for a supercritical wave equation. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations.     

Spatiotemporal self-similar waves for the (3 + 1)-dimensional inhomogeneous cubic-quintic nonlinear medium

Spatiotemporal self-similar waves of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation, describing propagation of optical pulses in a cubic-quintic nonlinear medium with inhomogeneous dispersion and gain, are derived. A one-to-one correspondence between such self-similar waves and solutions of the constant-coefficient cubic-quintic nonlinear Schrödinger equation exists when two certain compatibility conditions are satisfied. Under these conditions, we discuss dynamical behaviors of self-similar waves in dispersion decreasing fiber.     

On primitive variable behaviour near shocks in ensemble-averaged methods

Models for averaged shock corrugation effects and the impact of turbulent entropy or acoustic modes on the energy equation are presented, for application in Reynolds-Averaged Navier Stokes(RANS) simulations of shock-turbulence interactions. Unlike previous work that has focused the modification of turbulent statistics by the shock, the proposed models are introduced to capture the effects of the turbulence on the profiles of primitive variables - mean density, velocity, and pressure. By producing accurate profiles for the primitive variables, it is shown that the proposed models improve numerical convergence behaviour with mesh refinement about a shock, and introduce the physical effects of shock asphericity in a converging shock geometry. These effects are achieved by local closures to turbulent statistics in the averaged Navier-Stokes equations, and can be applied in conjunction with existing Reynolds stress closures that have been constructed for broader applications beyond shock-turbulence interactions.     

Several Types of Similarity Solutions for the Hunter-Saxton Equation

We study separable and self-similar solutions to the HunterSaxton equation,a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal(among other applications).Essentially,we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the HunterSaxton equation.For each type of solution,we are able to obtain some simple exact solutions in closed-form,and more complicated solutions through an analytical approach.We find that there is a whole family of self-similar solutions,each of which depends on an arbitrary parameter.This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data.The simpler solutions found constitute exact solutions to a nonlinear partial differential equation,and hence are also useful in a mathematical sense.Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions.Both types of solutions cast light on self-similar phenomenon arising in the HunterSaxton equation.     

Focusing of noncircular self-similar shock waves

We study the focusing of noncircular shock waves in a perfect gas. We construct an explicit self-similar solution by combining three convergent plane waves with regular shock reflections between them. We then show, with a numerical Riemann solver, that there are initial conditions with smooth shocks whose intermediate asymptotic stage is described by the exact solution. Unlike the focusing of circular shocks, our self-similar shocks have bounded energy density.     

Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions

We consider some questions related to the self-similar asymptotics in the kinetic theory of both elastic and inelastic particles. In the second case we have in mind granular materials, when the model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of the colliding particles. We first discuss how to define the n-dimensional (n = 1,2,...) inelastic Maxwell model and its connection with the more basic Boltzmann equation for inelastic hard spheres. Then we consider both elastic and inelastic Maxwell models from a unified viewpoint. We prove the existence of (positive in the inelastic case) self-similar solutions with finite energy and investigate their role in large time asymptotics. It is proved that a recent conjecture by Ernst and Brito devoted to high energy tails for inelastic Maxwell particles is true for a certain class of initial data which includes Maxwellians. We also prove that the self-similar asymptotics for high energies is typical for some classes of solutions of the classical (elastic) Boltzmann equation for Maxwell molecules. New classes of (not necessarily positive) finite-energy eternal solutions of this equation are also studied.     

Squeezed states of light as self-similar structures

A new approach to investigating a broad class of dynamic states for a quantum oscillator is suggested. It is based on an invariant transformation of the equation to a new time determined by the quantum dispersion of the corresponding state. The squeezed states of a quantum system generated by the ground-state wave function are constructed. In coordinate representation, these states are described by a self-similar wave function localized near a classical trajectory. The statistics of the squeezed state of light is analyzed in the single-mode approximation. The parametric excitation of squeezed states for a quantum harmonic oscillator is considered.     

Self-Similar Solutions for a Fractional Thin Film Equation Governing Hydraulic Fractures

In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient K ≥ 0. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness (K = 0) and the finite toughness K ∈ (0, ∞) cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all K > 0 there exists an injection rate for the fluid such that self-similar solutions exist.     

色散渐减光纤中自相似脉冲传输区域的研究

从非线性薛定谔(NLS)方程出发,用分步傅里叶方法结合对数值解的波形分析,确定了色散渐减光纤(DDF)中能实现自相似脉冲传输的区域,并研究了初始脉冲和光纤参数对自相似区域和演化速度的影响。结果表明,初始脉冲能量的减小有利于扩宽自相似区域,但会使自相似演化进程略为减慢;初始脉宽有一个最佳值,在最佳值上自相似区域最宽,演化较快且输出脉冲和啁啾较为稳定;高斯脉冲比双曲正割脉冲更快转化为自相似脉冲,传输区域也更广。选择具有较小非线性参量的DDF可以获得较广的自相似区域,同时非线性参量的增大可以加快自相似演化,而群速度色散参量和增益系数必须选择在最佳值附近,才能获得最大自相似区域和最快演化速度。     

Riccati generalization of self-similar solutions of nonautonomous Gross-Pitaevskii equation

We present a systematic analytical approach to construct a family of self-similar waves, related through a free parameter, in quasi one-dimension Gross-Pitaevskii equation with time-varying parameters. This approach enables us to control the dynamics of dark and bright similaritons, and first- and second- order self-similar rogue waves in Bose-Einstein condensate through the modulation of time dependent trapping potential. The analysis is done for the sech²? type time-varying quadratic trapping potential for two different choices of linear potential.     

Controlled self-similar matter waves in PT-symmetric waveguide

We study the dynamics of Bose-Einstein condensate coupled to a waveguide with parity-time symmetric potential in the presence of quadratic-cubic nonlinearity modelled by Gross-Pitaevskii equation with external source. We employ the self-similar technique to obtain matter wave solutions, such as bright, kink-type, rational dark and Lorentzian-type self-similar waves for this model. The dynamical behavior of self-similar matter waves can be controlled through variation of trapping potential, external source and nature of nonlinearities present in the system.     

Analytical calculation of the energy and pressure of a body under arbitrary compression from its experimental shock adiabat

Analytic expressions for the pressure and energy of a body are derived on the basis of classical statistics, the Mie-Grüneisen equation, and the virial theorem for this body using its experimental shock adiabat. The uniform non-Coulomb repulsive potential of the nth degree of uniformity is chosen as the potential of interaction between structural particles of the body.     

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansiontechnique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrödinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.     

Exact self-similar solitary waves and collisions in nonlinear optical media

This paper analyses bright and dark spatial self-similar waves propagation and collision in graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities. It finds an appropriate transformation for the first time such that the nonlinear Schrodinger equation （NLSE） with varying coefficients transform into standard NLSE. It obtains one-solitonlike, two-solitonlike and multi-solitonlike self-similar wave solutions by using the transformation. Furthermore, it analyses the features of the self-similar waves and their collisions.