Periodic multiphase solutions of the Kadomtsev-Petviashvili equation

Bobenko, Alexander I.; Bordag, Ljudmila A.

doi:10.1088/0305-4470/22/9/016

Cited by 40 publications

(26 citation statements)

References 15 publications

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“…10). It is based on the Schottky uniformization of the corresponding Riernann surface C and was used in the papers [14,15] to draw the plots of the finite-gap solutions of the Korteweg-de Vries and Kadomsev-Petviashvili equations. As a matter of fact, the actual parameter in these formulae is a hyperelliptic Riemann surface C which makes a parametrization to be rather complicated.…”

Section: Introductionmentioning

confidence: 99%

All constant mean curvature tori inR 3,S 3,H 3 in terms of theta-functions

1991

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Let us consider a small domain H in the complex plane, z e/7 and an immersion F:/7-.-,R 3 of/-/into Euclidean space. It is well known that if F is a conformal parametrization of a surface with constant mean curvature (CMC) and/7 is a neighborhood of a nonumbilic point then the metric g = 4eUdzd~ induced on H via F satisfies the elliptic sinh-Gordon equation uz~ + sinhu = 0.(1.1)Conversely, every solution of the Eq. (1.1) gives rise to CMC immersion. It is very important that here the elliptic sinh-Gordon equation may be applied not only in differential geometry in the small but also in differential geometry in the large. In particular, special doubly periodic solutions of this equation describe all CMC tori.Wente was the first [48] to show the existence of compact surfaces of genus one in R 3 with CMC, he constructed a countable number of isometrically distinct examples using a special solution of (1.1). These examples solved the long standing problem of Hopf [31]: Is a compact CMC surface in R 3 necessarily a round sphere?Important previous results were due to Alexandrov [3] and Hopf [31]. Alexandrov showed that if a CMC surface is embedded then it must be a round sphere. Hopf showed that an immersion of S 2 into R a with CMC, must be a round sphere.Wente's paper was followed by the series [1,43,45,49] where Wente tori were investigated in detail and other examples of CMC toil were constructed. All these modern results were obtained with the help of analytical investigation of special solutions of (1.1). In particular, Abresch described [1] all CMC tori having one family of planar curvature lines in terms of elliptic integrals. Walter [45] obtained more explicit representation of these tori in terms of elliptic and theta-functions of Jacobi type.

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Section: Introductionmentioning

confidence: 99%

All constant mean curvature tori inR 3,S 3,H 3 in terms of theta-functions

1991

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“…In all other cases, one first needs a method for computing the parameters B, V0, and V~ on some Riemann surface of genus three and, secondly, a procedure of calculating effectively with sufficient precision the Pdemaml theta functions. The second problem was solved by investigation of the solutions of KdV and KP equations [17,18]. However, the calculation of the parameters of the Riemann surfacees Was performed in [17,18] using the Schottky uniformization and in [2] using the symmetric uniformization.…”

Section: The Description Of the Computing Proceduresmentioning

confidence: 99%

“…The second problem was solved by investigation of the solutions of KdV and KP equations [17,18]. However, the calculation of the parameters of the Riemann surfacees Was performed in [17,18] using the Schottky uniformization and in [2] using the symmetric uniformization. In our case, these methods are either ineffective or inapplicable.…”

Section: The Description Of the Computing Proceduresmentioning

confidence: 99%

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Qualitative investigation of the three-phase solutions of the sine-Laplace equation

Babich¹,

Bordag²

1999

J Math Sci

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is one of the most interesting real realizations of the sine-Gordon (SG) equationand these equations are connected by a change of the complex variable. The goal of our investigation is to give a complete description of all real three-phase solutions of the SL equation. For this investigation, we need the general formula for multiphase solutions of the SG equation presented in [1]. The study of three-phase solutions of the SG equation is carried out in [2]. The general procedure giving the real solutions of the SL and SG equations is described in [3]. By this procedure, a Class of surfaces with constant mean curvature H < 1 in the three-dimensional hyperbolic space is constructed in [4]. We apply this method for the description of all real three-phase solutions of the SL equation. Multiphase solutions of SL equation (two-phase solutions) were first studied in [5,6] by a different method. A preliminary investigation of three-phase solutions is presented in [19]. The set of our solutions contains smooth and singular solutions. All singularities of the solutions are vortexes. These solutions are interesting in connection with superconductivity and superfluidity [7,8]. It was found that, under certain conditions, physical systems can have solutions with chain (or lattice) structure of the vortexes [8,9,10]. The vortex structure is stable and appears when taking into account the boundary conditions [11,12].

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“…Such plots can already be found in [17,2] and [7]. We will show general situations as well as almost degenerate surfaces which are identical to their corresponding solitonic solution up to numerical accuracy.…”

Section: Hyperelliptic Solutions To Kdv and Kp Equationsmentioning

confidence: 51%

Hyperelliptic Theta Functions and Spectral Methods

Klein

Lecture Notes in Physics

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Abstract. This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristic quantities of the Riemann surface with high precision even in almost degenerate situations as in the solitonic limit where the branch points coincide pairwise. As an example we consider hyperelliptic solutions to the Kadomtsev-Petviashvili and the Korteweg-de Vries equation. Tests of the numerics using identities for periods on the Riemann surface and the differential equations are performed. It is shown that an accuracy of the order of machine precision can be achieved.

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Periodic multiphase solutions of the Kadomtsev-Petviashvili equation

Cited by 40 publications

References 15 publications

All constant mean curvature tori inR 3,S 3,H 3 in terms of theta-functions

All constant mean curvature tori inR 3,S 3,H 3 in terms of theta-functions

Qualitative investigation of the three-phase solutions of the sine-Laplace equation

Hyperelliptic Theta Functions and Spectral Methods

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