The constant astigmatism equation. New exact solution

Manganaro, N.; Pavlov, Maxim V.

doi:10.1088/1751-8113/47/7/075203

Cited by 17 publications

(13 citation statements)

References 19 publications

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“…The difference of the radii of curvature is a critical quantity in what follows as it vanishes at, and only at, umbilic points. It is referred to as the astigmatism of the surface [20]. In the situation we consider in Theorem 1.2 below, the behaviour of the flow is determined by the vanishing of the astigmatism at the isolated umbilic points of the initial surface.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Evolving to non-round Weingarten spheres: integer linear Hopf flows

Guilfoyle

Klingenberg

2021

Partial Differ. Equ. Appl.

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Section: Introduction and Resultsmentioning

confidence: 99%

Evolving to non-round Weingarten spheres: integer linear Hopf flows

Guilfoyle

Klingenberg

2021

Partial Differ. Equ. Appl.

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“…The one-soliton solution of the CAE can be easily found by substituting λ 1 = 1 into Equations (18) and (19). Actually, since the solution does not depend on λ 1 (it disappears when eliminating parameters ξ, η from (18) and (19)), we obtain precisely the von Lilienthal solution (20).…”

Section: One-soliton Solutionsmentioning

confidence: 90%

“…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].…”

Section: Introductionmentioning

confidence: 99%

On multisoliton solutions of the constant astigmatism equation

Hlaváč¹

2015

J. Phys. A: Math. Theor.

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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.

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“…For the particular value g = 1/u, this factorization was indicated in [21] in terms of "first order reductions". Interpreting and developing the above results for (1+1)-dimensional nonlinear wave equations similarly to the consideration of the particular case G = const in [18, Section 6] will be the subject of a forthcoming paper.…”

Section: Singular Modules For Quasi-linear Second-order Pdesmentioning

confidence: 99%

Singular reduction modules of differential equations

Boyko

Kunzinger

Popovych

2016

Journal of Mathematical Physics

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The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in-depth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations. As examples, singular reduction modules of evolution equations and secondorder quasi-linear equations are studied. Reductions of differential equations to algebraic equations and to first-order ordinary differential equations are considered in detail within the framework proposed and are related to previous no-go results on nonclassical symmetries.

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The constant astigmatism equation. New exact solution

Abstract: In this paper we present a new solution for the Constant Astigmatism equation. This solution is parameterized by an arbitrary function of a single variable.

Cited by 17 publications

References 19 publications

Evolving to non-round Weingarten spheres: integer linear Hopf flows

Evolving to non-round Weingarten spheres: integer linear Hopf flows

On multisoliton solutions of the constant astigmatism equation

Singular reduction modules of differential equations

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