Factorization of the nonlinear Schrödinger equation and applications

Bernstein, Swanhild

doi:10.1080/17476930500481400

Cited by 23 publications

(39 citation statements)

References 20 publications

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“…Next we recall that the fundamental solution to the diffusion equation of the full space ℝ n × ℝ + is given by where H (·) stands for the usual Heaviside function. See for instance 1, 2. Now we are able to prove…”

Section: K‐fold Periodic Solutions To the Diffusion Equation And Tsupporting

confidence: 70%

“…Suppose that U ⊂ℝ n × ℝ + is an open set. As in 1, 2 one can introduce a parabolic Dirac operator in ℝ n × ℝ + in terms of the Witt basis by whereis the usual Euclidean Dirac operator in ℝ n .…”

Section: Preliminariesmentioning

confidence: 79%

“…In the similar case of the Schr ödinger equation this method has also proven to be successful, cf. 2. Hence it was possible to obtain a Clifford algebra calculus for the diffusion equation that can be used to solve boundary‐value problems in Lipschitz domains in ℝ n + 1 .…”

Section: Introductionmentioning

confidence: 99%

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On the diffusion equation and diffusion wavelets on flat cylinders and the n-torus

Bernstein

Ebert

Kraußhar

2010

Math. Meth. Appl. Sci.

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In this paper we study the solutions to the diffusion equation on some conformally flat cylinders and on the n-torus. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the parabolic Dirac operator. We study their fundamental properties, give representation formulas of all these solutions and develop some integral representation formulas. In particular we set up a Green type formula for the solutions to the homogeneous diffusion equation on cylinders and tori. Then we also treat the inhomogeneous diffusion equation diffusion with prescribed boundary conditions in Lipschitz domains on these manifolds. As main application, we construct well localized diffusion wavelets on this class of cylinders and tori by means of multiperiodic eigensolutions to the parabolic Dirac operator. We round off with presenting some concrete numerical simulations for the three dimensional case.

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Section: K‐fold Periodic Solutions To the Diffusion Equation And Tsupporting

confidence: 70%

Section: Preliminariesmentioning

confidence: 79%

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On the diffusion equation and diffusion wavelets on flat cylinders and the n-torus

Bernstein

Ebert

Kraußhar

2010

Math. Meth. Appl. Sci.

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“…In the scientific literature the different denominations are used for functions satisfying equations of the form (18). For example, in the papers [17,18] they are called regular functions, and in the papers [19,20] they are called monogenic functions. We use the terminology of the papers [21,22].…”

Section: Examplesmentioning

confidence: 99%

Monogenic Functions in a Finite-Dimensional Semi-Simple Commutative Algebra

Plaksa

Pukhtaievych

2014

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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We obtain a constructive description of monogenic functions taking values in a finite-dimensional semi-simple commutative algebra by means of holomorphic functions of the complex variable. We prove that the mentioned monogenic functions have the Gateaux derivatives of all orders. For monogenic functions we prove also analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, the Morera theorem and the Cauchy integral formula.Introduction. William Hamilton (1843) constructed an algebra of noncommutative quaternions over the field of real numbers R, and developing the hypercomplex analysis began. C. Segre [1] constructed an algebra of commutative quaternions {x + iy + jz + kt : i 2 = j 2 = −1, ij = k, x, y, z, t ∈ R} over the field R that can be considered as a two-dimensional commutative semi-simple algebra of bicomplex numbers {z 1 + jz 2 : j 2 = −1, z 1 , z 2 ∈ C} over the field of complex numbers C.A theory of functions of a bicomplex variable was developed in papers of many authors (see, e.g., [2,3,4,5,6]). In particular, in the papers of F. Ringleb [2] and J. Riley [3], it is proved that any analytic function of a bicomplex variable can be constructed with an use of two holomorphic functions of complex variables. In addition, G. Price [4] considered multicomplex

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“…For a = 0, = 0, = 0 Theorem 1 was proved in [5] and was intensively used in [6,7,12,18,[23][24][25] and others.…”

Section: Remarkmentioning

confidence: 97%