ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Characteristic coordinates reduce a constant-coefficient hyperbolic equation to mixed form

Example

For the wave-type equation

$u_{x x} - u_{y y} = 0,$

the characteristic coordinates

$ξ = x + y, η = x - y$

turn the principal part into

$u_{ξ η} = 0$

up to the nonzero factor $4$ .

Facts & Assumptions

Given: The constant-coefficient hyperbolic operator $u_{x x} - u_{y y}$ .

[L1]

Constant-coefficient hyperbolic principal parts admit canonical linear coordinates (Constant-coefficient second-order equations in two variables have canonical principal forms).

[L2]

Characteristic directions are coordinate invariant and are determined by the characteristic families (In two variables, type and characteristic directions are coordinate invariant).

Verification

technique · direct

1.1

The principal quadratic form is $ξ_{x}^{2} - ξ_{y}^{2}$ , so the characteristic covectors are proportional to $d (x + y)$ and $d (x - y)$ , and [L2] shows that using $ξ = x + y$ and $η = x - y$ follows the two characteristic families.

2.1

In these coordinates, $\partial_{x} = \partial_{ξ} + \partial_{η}$ and $\partial_{y} = \partial_{ξ} - \partial_{η}$ , so $u_{x x} - u_{y y} = (\partial_{ξ} + \partial_{η})^{2} u - (\partial_{ξ} - \partial_{η})^{2} u = 4 u_{ξ η}$ ; this is the mixed canonical form, equivalent to the hyperbolic normal form from [L1] after a further linear recombination of $ξ$ and $η$ .

L1step 1.1∎

Depends on

Used by

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Sources

Victor Ivrii, Partial Differential Equations (standard reference, not scraped)