Bubbles Bad; Ripples Good

16/05/2011

Decay of waves IIIa: nonlinear tails in Minkowski space redux

Filed under: Maths,partial differential equations,Requires upper level university maths,wave and Schroedinger equations — Willie Wong @ 11:27

Before we move on to the geometric case, I want to flesh out the nonlinear case mentioned in the end of the last post a bit more. Recall that it was shown for generic nonlinear (actually semilinear; for quasilinear and worse equations we cannot use Duhamel’s principle) wave equations, if we put in compact support for the initial data, we expect the first iterate to exhibit a tail. One may ask whether it is possible that, in fact, this is an artifact of the successive approximation scheme; that in fact somehow it always transpires that a conspiracy happens, and all the higher order iterates cancel out the tail coming from the first iterate. This is rather unlikely, owing to the fact that the convergence to \phi_\infty is dominated by a geometric series. But to just make double sure, here we give a nonlinear system of wave equations such that the successive approximation scheme converges after finitely many steps (in fact, after the first iterate), and so we can also explicitly compute the rate of decay for the nonlinear tail. While the decay rate is not claimed to be generic (though it is), the existence of one such example with a fixed decay rate shows that for a statement quantifying over all nonlinear wave equations, it would be impossible to demonstrate better decay rate than the one exhibited. (more…)

14/05/2011

Decay of waves IIb: Minkowski space, with right-hand side

Filed under: Maths,partial differential equations,Requires upper level university maths,wave and Schroedinger equations — Willie Wong @ 13:50

In the first half of this second part of the series, we considered solutions to the linear, homogeneous wave equation on flat Minkowski space, and showed that for compactly supported initial data, we have strong Huygens’ principle. We further made references to the fact that this behaviour is expected to be unstable. In this post, we will further illustrate this instability by looking at Equation 1 first with a fixed source F = F(t,x), and then with a nonlinearity F = F(t,x, \phi, \partial\phi).

Duhamel’s Principle

To study how one can incorporate inhomogeneous terms into a linear equation, and to get a qualitative grasp of how the source term contributes to the solution, we need to discuss the abstract method known as Duhamel’s Principle. We start by illustrating this for a very simple ordinary differential equation.

Consider the ODE satisfied by a scalar function \alpha:

Equation 13
\displaystyle \frac{d}{ds}\alpha(s) = k(s)\alpha(s) + \beta(s)

when \beta\equiv 0, we can easily solve the equation with integration factors

\displaystyle \alpha(s) = \alpha(0) e^{\int_0^s k(t) dt}

Using this as a sort of an ansatz, we can solve the inhomogeneous equation as follows. For convenience we denote by K(s) = \int_0^s k(t) dt the anti-derivative of k. Then multiplying Equation 13 through by \exp -K(s), we have that

Equation 14
\displaystyle \frac{d}{ds} \left( e^{-K(s)}\alpha(s)\right) = e^{-K(s)}\beta(s)

which we solve by integrating

Equation 15
\displaystyle \alpha(s) = e^{K(s)}\alpha(0) + e^{K(s)} \int_0^s e^{-K(t)}\beta(t) dt

If we write K(s;t) = \int_t^s k(u) du, then we can rewrite Equation 15 as given by an integral operator

Equation 15′
\displaystyle \alpha(s) = e^{K(s)}\alpha(0) + \int_0^s e^{K(s;t)}\beta(t) dt

(more…)

12/05/2011

Decay of waves IIa: Minkowski background, homogeneous case

Filed under: Maths,partial differential equations,Requires upper level university maths,wave and Schroedinger equations — Willie Wong @ 18:53

Now let us get into the mathematics. The wave equations that we will consider take the form

Equation 1
-\partial_t^2 \phi + \triangle \phi = F

where \phi:\mathbb{R}^{1+n}\to\mathbb{R} is a real valued function defined on (1+n)-dimensional Minkowski space that describes our solution, and F represents a “source” term. When F vanishes identically, we say that we are looking at the linear, homogeneous wave equation. When F is itself a function of \phi and its first derivatives, we say that the equation is a semilinear wave equation.

We first start with the homogeneous, linear case.

Homogeneous wave equation in one spatial dimension

One interesting aspect of the wave equation is that it only possesses the second, multidimensional, dispersive mechanism as described in my previous post. In physical parlance, the “phase velocity” and the “group velocity” of the wave equation are the same. And therefore, a solution of the wave equation, quite unlike a solution of the Schroedinger equation, will not exhibit decay when there is only one spatial dimension (mathematically this is one significant difference between relativistic and quantum mechanics). In this section we make a computation to demonstrate this, a fact that would also be useful later on when we look at higher (in particular, three) dimensions.

Use x\in\mathbb{R} for the variable representing spatial position. The wave equation can be written as

-\partial_t^2 \phi + \partial_x^2\phi = 0

Now we perform a change of variables: let u = \frac{1}{2}(t-x) and v = \frac{1}{2}(t+x) be the canonical null variables. The change of variable formula replaces

Equation 2
\displaystyle \partial_t \to \frac{\partial u}{\partial t} \partial_u + \frac{\partial v}{\partial t} \partial v = \frac{1}{2}\partial_u + \frac{1}{2}\partial_v
\displaystyle \partial_x \to \frac{\partial u}{\partial x} \partial_u + \frac{\partial v}{\partial x} \partial v = -\frac{1}{2}\partial_u + \frac{1}{2}\partial_v

and we get that in the (u,v) coordinate system,

Equation 3
-\partial_u \partial_v \phi = 0

(more…)

10/05/2011

Getting Google Buzz to work

Filed under: Uncategorized — Willie Wong @ 12:59
Tags:

Allegedly one can link a blog to Google Buzz. And I am pretty sure it works, since Terry’s Buzz gets also his What’s New feed. But for the life of me I cannot get Google to realise that this blog is affiliated to my google profile.

And I hate to admit defeat to technology.

05/05/2011

Decay of waves I: Introduction

Filed under: general relativity,Maths,partial differential equations,Require high school maths,wave and Schroedinger equations — Willie Wong @ 18:59

In the next week or so, I will compose a series of posts on the heuristics for the decay of the solutions of the wave equation on curved (and flat) backgrounds. (I have my fingers crossed that this does not end up aborted like my series of posts on compactness.) In this first post I will give some physical intuition of why waves decay. In the next post I will write about the case of linear and nonlinear waves on flat space-time, which will be used to motivate the construction, in post number three, of an example space-time which gives an upper bound on the best decay that can be generally expected for linear waves on non-flat backgrounds. This last argument, due to Mihalis Dafermos, shows that why the heuristics known as Price’s Law is as good as one can reasonably hope for in the linear case. (In the nonlinear case, things immediately get much much worse as we will see already in the next post.)

This first post will not be too heavily mathematical, indeed, the only realy foray into mathematics will be in the appendix; the next ones, however, requires some basic familiarity with partial differential equations and pseudo-Riemannian geometry. (more…)

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