A Simple Illustration of a Weak Spectral Cascade

نویسنده

  • David J. Muraki
چکیده

The textbook first encounter with nonlinearity in a partial differential equation (PDE) is the first-order wave equation: ut + uux = 0. Often referred to as the inviscid Burgers equation, many are familiar with this equation in the theoretical contexts of characteristics, wavebreaking, or shock propagation. Another canonical behavior contained within this simplest of PDEs is the spectral cascade. Surprisingly, buried in a little-known 1964 article by G.W. Platzman is an elegant example of an exact Fourier series solution associated with a purely sinusoidal initial condition. This Fourier representation, valid prior to wavebreaking, is generalized to arbitrary continuous initial conditions on both the periodic and infinite domains. Within the specific example of Platzman’s original problem, it is also shown that a perturbation solution linearized about the initial condition provides a short-time representation that embodies the spectral cascade uniformly to large wavenumbers.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Facile synthesis of dihydropyrimidinone derivatives via Biginelli reaction using Brønsted acidic ionic liquid [H-NMP]+[CH3SO3]- as an efficient homogeneous catalyst

In this research, a green, operationally simple, highly efficient and facile cascade Biginelli reaction using various aromatic aldehydes, ethyl acetoacetate, and urea/thiourea has been developed. First, the ionic liquid was prepared from the reaction of 1-methyl-2-pyrolidone with methane sulfonic acid. Then, this ionic liquid was applied as an acidic catalyst in the synthesis of dihydropyrimidi...

متن کامل

Facile synthesis of dihydropyrimidinone derivatives via Biginelli reaction using Brønsted acidic ionic liquid [H-NMP]+[CH3SO3]- as an efficient homogeneous catalyst

In this research, a green, operationally simple, highly efficient and facile cascade Biginelli reaction using various aromatic aldehydes, ethyl acetoacetate, and urea/thiourea has been developed. First, the ionic liquid was prepared from the reaction of 1-methyl-2-pyrolidone with methane sulfonic acid. Then, this ionic liquid was applied as an acidic catalyst in the synthesis of dihydropyrimidi...

متن کامل

On Max-injective modules

$R$-module. In this paper, we explore more properties of $Max$-injective modules and we study some conditions under which the maximal spectrum of $M$ is a $Max$-spectral space for its Zariski topology.

متن کامل

Phase-I monitoring of standard deviations in multistage linear profiles

In most modern manufacturing systems, products are often the output of some multistage processes. In these processes, the stages are dependent on each other, where the output quality of each stage depends also on the output quality of the previous stages. This property is called the cascade property. Although there are many studies in multistage process monitoring, there are fewer works on prof...

متن کامل

Some result on simple hyper K-algebras

A simple method is described, to prove some theorems for simple hyper K-algebras and to study positive implicative hyper K-ideals, weak (implicative ) hyper K-ideals in simple hyper K-algebras . Beside, some results on positive implicative and (weak) implicative simple hyper K-algebras are presented. Finally classification of simple hyper Kalgebras of order 4, which are satisfied in conditions ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • SIAM Journal of Applied Mathematics

دوره 67  شماره 

صفحات  -

تاریخ انتشار 2007