ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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A second-order normal system

Example

For analytic g,h, the equation $u_{t t} = u_{x x}$ with data $u (0, x) = g (x)$ , $u_{t} (0, x) = h (x)$ is equivalent to $u_{t} = v$ , $v_{t} = w_{x}$ , $w_{t} = v_{x}$ , with data $(u, v, w) (0, x) = (g, h, g^{'})$ .

Facts & Assumptions

Given: The equation $u_{t t} = u_{x x}$ , its two analytic Cauchy data, and the three proposed jet variables of the Example.

[F1]

The compatible first-order jet system has an analytic solution recovering the scalar equation. (Reduction of higher-order normal form with jet compatibility).

Verification

1.1

For an analytic scalar solution put $v = u_{t}, w = u_{x}$ . Then $u_{t} = v$ , $v_{t} = u_{t t} = u_{x x} = w_{x}$ and $w_{t} = u_{x t} = u_{t x} = v_{x}$ , with traces $(g, h, g^{'})$ . These are precisely the m=2 equations of F1.

givenF1algebra

2.1

Conversely F1 supplies an analytic system solution. Its error $e = w - u_{x}$ satisfies $e_{t} = v_{x} - \partial_{x} v = 0$ and $e (0, x) = g^{'} (x) - g^{'} (x) = 0$ , so e is identically zero. Thus $u_{t t} = v_{t} = w_{x} = u_{x x}$ and $u_{t} (0, x) = h$ . For $g = x^{2}, h = 1$ , the explicit solution is $u = x^{2} + t^{2} + t$ , $v = 2 t + 1, w = 2 x$ : both $v_{t}$ and $w_{x}$ equal 2, and both $w_{t}$ and $v_{x}$ equal 0.

step 1.1F1algebra∎

Source notes

Gantumur, §4 Corollary 20 and equations (55)–(57), printed p. 11; scalar wave specialization computed locally.

Depends on

Reduction of higher-order normal form with jet compatibility

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gantumur, Math 580 Lecture Notes 2: The Cauchy-Kovalevskaya Theorem (standard reference, not scraped)
Ageno, Part III: Analysis of Partial Differential Equations (standard reference, not scraped)