ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Right- and left-travelling waves

Example

Fix $c > 0$ ; no choice principle is needed in this example. Let $F, G \in C^{2} (R)$ and put $u (x, t) : = F (x - c t) + G (x + c t)$ . Then $u$ is a $C^{2}$ solution of $u_{t t} = c^{2} u_{x x}$ on $R^{2}$ , with $u (x, 0) = F (x) + G (x), u_{t} (x, 0) = - c F^{'} (x) + c G^{'} (x) .$ Conversely every $C^{2}$ solution on a rectangle has this form (General solution of the one-dimensional wave equation). Two checks: (i) for $F$ a compactly supported bump and $G = 0$ , the profile translates to the right at speed $c$ without changing shape; (ii) the data $(u_{0}, u_{1}) = (F + G, - c F^{'} + c G^{'})$ are exactly those fed into d'Alembert's formula and uniqueness in one dimension, whose expression reproduces $u$ .

Facts & Assumptions

Given: a speed $c > 0$ , functions $F, G \in C^{2} (R)$ , and $u (x, t) = F (x - c t) + G (x + c t)$ .

[F1]

If $f$ is totally differentiable at $a$ and $g$ at $f (a)$ , then $g \circ f$ is totally differentiable at $a$ with $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). The required total differentiability follows from continuous coordinate partial derivatives (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

[F2]

Sums, constant multiples of differentiable functions are differentiable with the usual rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F3]

Every $C^{2}$ solution of $u_{t t} = c^{2} u_{x x}$ on a nonempty open rectangle is a sum $F (x - c t) + G (x + c t)$ with $F, G \in C^{2}$ on the projections, and conversely (General solution of the one-dimensional wave equation).

[F4]

With data $u_{0} = F + G$ , $u_{1} = - c F^{'} + c G^{'}$ the d'Alembert expression of d'Alembert's formula and uniqueness in one dimension equals $u$ ; the integrals of $F^{'}$ and $G^{'}$ are evaluated by the fundamental theorem of calculus (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ).

Verification

1.1F1F2algebra

Substitution. By [F1] and [F2], $u$ is $C^{2}$ , $\partial_{t} u = - c F^{'} (x - c t) + c G^{'} (x + c t)$ , $\partial_{t}^{2} u = c^{2} F^{''} (x - c t) + c^{2} G^{''} (x + c t)$ and $\partial_{x}^{2} u = F^{''} (x - c t) + G^{''} (x + c t)$ , so $\partial_{t}^{2} u = c^{2} \partial_{x}^{2} u$ on $R^{2}$ ; at $t = 0$ the displayed data are read off directly.

1.2F2algebra

Check (i). If $G = 0$ then $u (x, t) = F (x - c t)$ ; for each fixed $t$ the graph of $u (\cdot, t)$ is the graph of $F$ translated by $c t$ , so the profile moves to the right at speed $c$ with its shape unchanged.

1.3F4algebra

Check (ii). The d'Alembert expression with data $(u_{0}, u_{1}) = (F + G, - c F^{'} + c G^{'})$ is $\frac{1}{2} (F (x - c t) + G (x - c t) + F (x + c t) + G (x + c t)) + \frac{1}{2 c} [- c (F (x + c t) - F (x - c t)) + c (G (x + c t) - G (x - c t))]$ by [F4], and the $F$ -terms and $G$ -terms collapse to $F (x - c t) + G (x + c t) = u (x, t)$ .

2.1F3given∎

By [F3] the converse holds on every nonempty open rectangle, so the sum of a right- and a left-moving profile is exactly the general one-dimensional solution, and the d'Alembert formula returns it from its data.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)