# Talk:Hyperbolic partial differential equation

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Field: Analysis

Hi !

Do you know what is the link between a hyperbolic partial differential equation and the hyperbolic system of conservation laws (I do not see any link between the definition with the determinant, and the one with the diagonalizable matrix ?

## Contents

Is it correct that these are called a hyperbolic equation because the quadratic form corresponding to the matrix is a hyperbolic surface? —Ben FrantzDale 17:00, 13 February 2007 (UTC)

## Formatting

Suggest boldface italic font used to signify vectors instead of the current arrow notation. — DIV (128.250.204.118 08:19, 27 June 2007 (UTC))

## "For" versus "for each"

$A^j:= \text{ blah blah blah blah} ,\text{ for each }j = 1, \ldots, d.$

That's clearly wrong. It should say either

$A^j:= \text{ blah blah blah blah} ,\text{ for each }j \in \{ 1, \ldots, d\}.$

or

$A^j:= \text{ blah blah blah blah} ,\text{ for }j = 1, \ldots, d.$

Mathematicians don't use the version that appeared in the article (which I've corrected). But I've come across it on Wikipedia maybe more a dozen times in the past year or so, I think. I edit dozens of math articles here every day, so maybe that's not a huge trend, but one starts to notice an odd usage after the dozenth time or so.

Where is this coming from? Is there some community among which this is conventional? Physicists, maybe?—they're often hostile to being precise about math. Michael Hardy (talk) 22:07, 28 August 2008 (UTC)

## First Sentence

Does anyone else have a problem with the first sentence of the article:

"In mathematics, a hyperbolic partial differential equation is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem."

My understanding is that hyperbolic partial differential equations are generalizations of the wave equation. But regardless, they are not characterized by being well posed. Does anyone mind if I change it? Paul Laroque (talk) 19:40, 29 March 2010 (UTC)

I'd be amenable to a different wording, as here "well-posed" should not be misinterpreted to mean "well-posed in the sense of Hadamard". However, the qualitative feature that distinguishes hyperbolic equations is that initial data for the Cauchy problem can be prescribed arbitrarily (on a "non-characteristic" hypersurface). What did you have in mind? Sławomir Biały (talk) 23:00, 29 March 2010 (UTC)

I would think that pretty much anyone who reads the article will think "well-posed in the sense of Hadamard" as I did. I have actually never seen the characterization of hyperbolic PDE that you state. What is your reference for that statement?

My understanding is that hyperbolic PDE are generalizations the wave equation. For example, linear 2nd order hyperbolic PDE are of the form $d^2u/dt^2 + Lu = f$ where $L$ is uniformly hyperbolic (the typical example being $L=-\Delta$ which yields the wave equation). Paul Laroque (talk) 01:43, 1 April 2010 (UTC)

A clear statement of this formulation of hyperbolicity can be found in Chapter 5 of Fritz John's textbook "Partial differential equations", at least in the case of linear constant-coefficient operators. This idea is also certainly implicit in Courant and Hilbert's formulation of hyperbolicity in general as well: in Volume 2, Chapter III, §3.7, they say that "One could equate [solvability of the Cauchy problem] with the concept of hyperbolicity", and earlier in Volume 2, Chapter III, §2.4, hyperbolicity of equations of higher order is defined in terms of the formal solvability of the Cauchy problem. This motivates what might be called the "usual" formulation of hyperbolicity, which is a condition on the zeros of the leading symbol. But the former is intuitive and qualitative, and is what motivates the latter technical definition. As it currently stands, the lead does say "roughly speaking", the purpose of which is to indicate that there are nuances not being discussed at the outset. Sławomir Biały (talk) 11:52, 1 April 2010 (UTC)
Compare also with the first sentence of Rozhdestvenskii, B.L. (2001), "Hyperbolic partial differential equation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4. Sławomir Biały (talk) 12:30, 1 April 2010 (UTC)

I would prefer something more along the lines of the first sentence of Rozhdestvenskii. That made perfect sense to me, it gave all the details required to understand the statement. I'm afraid that most people, including myself (I have a M.Sc. in Math) will completely misinterpret that first sentence of the Wikipedia article. It just is far to vague. To me it said "A hyperbolic PDE is a PDE which is well-posed" and I think that every layman and a good majority of Mathematicians who are not so familiar with partial differential equations will read it in the same way. I think it should be changed in one of two ways, either:

1) Add some more details along the lines of Rozhdestvenskii or,
2) Give the other hyperbolicity condition on the zeros of the leading symbol

I would advocate the second one and moving the discussion equating the concept of hyperbolicity to solvability of the Cauchy problem to another section. Although 1) is certainly more elegant and intuitive (once you understand it), I think it will be lost on a majority of the readers of the article (especially those who read only the first few sentences to get an idea of what a hyperbolic PDE is). Paul Laroque (talk) 12:52, 1 April 2010 (UTC)

I certainly agree with (2): hyperbolicity criteria deserve a section of their own. However, I don't think that the criterion given by Rozhdestvenskii, John, and Courant and Hilbert should be pushed down to a separate section. Instead I think some effort should be made to make it accessible—it shouldn't be all that difficult, since solvability of the Cauchy problem is a fairly basic feature. I suppose one could start the article by saying that hyperbolic equations are those possessing "wave-like" solutions (in the sense that the solutions "propagate" initial data), though I'm not sure how to write it in an accessible way: to me this seems like a less clear explanation, and probably one that requires more mathematical sophistication to grasp. Sławomir Biały (talk) 13:15, 1 April 2010 (UTC)