On multisoliton solutions of the constant astigmatism equation

Abstract

We introduce an algebraic formula producing infinitely many exact solutions of the constant astigmatism equation z yy +(1/z) xx +2 = 0 from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of the constant astigmatism equation defined as counterparts of famous multisoliton solutions of the sine-Gordon equation. A few particular examples are surveyed as well.

Nonlocal conservation laws of the constant astigmatism equation

“…As we know from Section 3, vectors Φ, H −1 Φ, S −1 Φ, P −1 Φ satisfy system (10). More generally, we can write…”

“…unless λ = 0, 2. Consequently, H −1 -Φ Φ satisfies system (10). Furthermore, the constant astigmatism equation is invariant under the reciprocal transformations [12] X (x, y, z) = (x ′ , y ′ , z ′ ) and Y(x, y, z) = (x * , y * , z * ), where…”

“…Known are the zero curvature representation, see [2] or eq. ( 7) below, the bi-Hamiltonian structure and hierarchies of higher order symmetries and conservation laws [27], as well as multisoliton solutions [10].…”

More exact solutions of the constant astigmatism equation

“…• Von Lilienthal solutions z = c − y 2 and z = 1/(c − x 2 ), where c is a constant. For details and corresponding surfaces of constant astigmatism see [2,13,16].…”

“…• Lipschitz solutions [11,Thm. 1] corresponding to Lipschitz surfaces of constant astigmatism [17], see also [13] for pictures of the surfaces.…”