Field equations

Any equation of the form

∂_ttu[t, x]  ∂_xxu[t, x] + f[u[t, x]]

\!\(\*SubscriptBox[\(\[PartialD]\),\(tt\)]\)u[t,x]==\!\(\*SubscriptBox[\(\[PartialD]\),\(xx\)]\)u[t,x]+f[u[t,x]]

can be thought of as a classical field equation for a scalar field. Defining

v[u] = -Integrate[f[u], u]

v[u] = -Integrate[f[u], u]

the field then has Lagrangian density

((∂_tu)² - (∂_xu)²)/2 - v[u]

(\!\(\*SuperscriptBox[\(\!\(\*SubscriptBox[\((\[PartialD]\),\(t\)]\)u)\),\(2\)]\)-\!\(\*SuperscriptBox[\(\!\(\*SubscriptBox[\((\[PartialD]\),\(x\)]\)u)\),\(2\)]\))/2-v[u]

and conserves the Hamiltonian (energy function)

Integrate[((∂_tu)² + (∂_tu)²)/2 + v[u], {x, -∞, ∞}]

Integrate[(\!\(\*SuperscriptBox[\((\!\(\*SubscriptBox[\(\[PartialD]\),\(t\)]\)u)\),\(2\)]\) + \!\(\*SuperscriptBox[\(\!\(\*SubscriptBox[\((\[PartialD]\),\(x\)]\)u)\),\(2\)]\))/2 + v[u], {x, -∞, ∞}]

With the choice for f[u]

f[u] made here (with a ≥ 0

✖

a ≥ 0), v[u]

✖

v[u] is bounded from below, and as a result it follows that no singularities ever occur in u[t, x]

✖

u[t, x].