Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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Quasilinear characteristics can cross before the lifted ODE blows up

Statement refuted

If the lifted characteristic ODE continues, then the quasilinear PDE remains a single-valued classical graph.

Counterexample

Given: Burgers data u0(ξ)=−tanh⁡ξ.

Proof technique: direct.

1.1

The lifted characteristic is Z(t,ξ)=−tanh⁡ξ and X(t,ξ)=ξ−ttanh⁡ξ, both defined for every t≥0.

givenalgebra
2.1

Yet Xξ(1,0)=0, so the projected map has a caustic at t=1.

step 1.1algebra
3.1

Thus the lifted ODE persists while inverse projection to a graph fails, refuting the statement.

step 2.1given∎

Depends on

Used by

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Dependency tree · two levels

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Sources