33D (Integrable Systems)

Equality of the mixed derivatives gives

- U_{y} Φ + U V Φ = - V_{x} Φ + V U Φ.

For a fundamental matrix solution this is equivalent to the zero-curvature condition

V_{x} - U_{y} + [U, V] = 0 .

b

Substituting the displayed matrices into

V_{x} - U_{y} + [U, V]

makes every off-diagonal entry vanish. The two nonzero diagonal entries are opposites, the first being

- u_{x y} + e^{u} - e^{- 2 u} .

Thus compatibility is equivalent to the Tzitzeica equation

u_{x y} = F (u) = e^{u} - e^{- 2 u} .

The flow of

x \partial_{x} - α y \partial_{y}

G_{ε} : (x, y, u) ⟼ (e^{ε} x, e^{- α ε} y, u) .

Under the induced action on solutions,

u_{x y}

acquires the factor

e^{(α - 1) ε}

, while

e^{u} - e^{- 2 u}

is unchanged. The equation is invariant for every

ε

exactly when

α = 1.

For

α = 1

, the invariant of the characteristic equations is

z = x y

, so an invariant solution has

u (x, y) = f (z)

. Then

u_{x y} = f^{'} (z) + z f^{''} (z) .

The invariant solutions are therefore characterized by

z f^{''} + f^{'} = e^{f} - e^{- 2 f} .

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