A Reciprocal Transformation for the Constant Astigmatism Equation

Abstract

Abstract. We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation z yy + (1/z) xx + 2 = 0. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter λ = ±1. It is also a nonlocal symmetry.

On multisoliton solutions of the constant astigmatism equation

“…We have The results from this example coincide with ones obtainable using a reciprocal transformation for the constant astigmatism equation introduced in [15], albeit the parameterisation of the results is different. For instance, one can observe apparent similarity between the surfacer r {2} in Figure 8 and the surface [15, Sect.…”

“…In the paper, the surfaces gained their name and Equation (1) was obtained as well. Recently, the Equation (1) has been examined by several authors [13,19,14,18,15].…”

More exact solutions of the constant astigmatism equation

“…Let ω be a solution of (7) and let ω (λ) = 4 arctan δ (λ) −ω be its Bäcklund transformation with parameter λ, where δ (λ) is given by (10). Let a, f, c be defined by (11) and let b satisfy (12). Then the associated potentials x (λ) , y (λ) , g (λ) corresponding to the pair ω, ω (λ) are given by formulas…”

Nonlocal conservation laws of the constant astigmatism equation

“…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…”