A method for constructing solutions of homogeneous partial differential equations: localized waves

Donnelly, R.; Ziolkowski, Richard W.

doi:10.1098/rspa.1992.0086

Cited by 60 publications

(12 citation statements)

References 7 publications

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“…involving the splash pulses (39). Evidently, the above for n = 0 allows one to relate the rule (3) in the case of the v 0,N 's to the symmetry operators L ± (rised to ± 1 2 ) of the wave equation.…”

Section: Hermite-lorentzian Wave Functions: Relation To the Laguerre-mentioning

confidence: 99%

“…r ] n , have been discussed in detail, and there related, as mentioned earlier, to the RLPs, which in fact act as modulating factors of the axisymmetric Lorentzian-like wave functions (39), whose exponent is further increased to N + 2n + In the same vein, we may ask whether the above introduced Hermite-Lorentzian solutions of the 3D wave equation u (±) n,N (x, y, z, t), being obtained through (3) from the higher-order Hermite-Lorentzian solutions φ n,N (x, z, t) of the 2D wave equation, might be understood as higher-order solutions relatively to certain fundamental ones, which could reasonably be guessed to be u N (r, z, t). In other words, we may ask whether it is possible to identify a sort of generation scheme for the u In this connection, it is easy to verify that 2m,N one can avoid the singularity associated with the even-order forms of (33) simply because so doing one picks up only the odd-order forms of (33).…”

Section: Hermite-lorentzian Wave Functions: Relation To the Laguerre-mentioning

confidence: 99%

“…This was functional to the possibility of conveniently relating some higher-order solutions v n,l,N (x, y, z, t) to the RLPs (20). Such solutions are generated by repeatedly acting on (39) by the operators L ±…”

Section: Hermite-lorentzian Wave Functions: Relation To the Laguerre-mentioning

confidence: 99%

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The Relativistic Hermite Polynomials and the Wave Equation

Torre¹

2009

PIER B

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Abstract-Solutions of the homogeneous 2D scalar wave equation of a type reminiscent of the "splash pulse" waveform are investigated in some detail. In particular, it is shown that the "higher-order" solutions relative to a given "fundamental" one, from which they are obtained through a definite "generation scheme", come to involve the relativistic Hermite polynomials. This parallels the results of a previous work, where solutions of the 3D wave equation involving the relativistic Laguerre polynomials have been suggested. Then, exploiting a well known rule, the obtained wave functions are used to construct further solutions of the 3D wave equation. The link of the resulting wave functions with those analyzed in the previous work is clarified, the pertinent generation scheme being indeed inferred. Finally, solutions of the Klein-Gordon equation which relate to such Lorentzian-like solutions of the scalar wave equation are deduced.

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Section: Hermite-lorentzian Wave Functions: Relation To the Laguerre-mentioning

confidence: 99%

Section: Hermite-lorentzian Wave Functions: Relation To the Laguerre-mentioning

confidence: 99%

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The Relativistic Hermite Polynomials and the Wave Equation

Torre¹

2009

PIER B

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“…The parameterization is based on a remarkable class of explicit solutions of the scalar wave equation found by Ziolkowski [11][12][13][14][15] together with the behaviour of charged particle-pulse interactions over a broad parameter range without recourse to expensive numerical computation. Finally, we argue that such parameterizations can be used to find compact finite energy solutions to other linear wave equations.…”

Section: Introductionmentioning

confidence: 99%

“…The parameterization is based on a remarkable class of explicit solutions of the scalar wave equation found by Ziolkowski [11][12][13][14][15] following pioneering work by Brittingham [16]. Such solutions can be used to construct classical Maxwell solutions with bounded total electromagnetic energy and fields bounded in all three spatial directions.…”

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confidence: 99%

Classical dynamics of free electromagnetic laser pulses

Goto

Tucker

Walton

2016

Nuclear Instruments and Methods in Physics Research Section B:

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We discuss a class of exact finite energy solutions to the vacuum source-free Maxwell field equations as models for multiand single cycle laser pulses in classical interaction with relativistic charged test particles. These solutions are classified in terms of their chiral content based on their influence on particular charge configurations in space. Such solutions offer a computationally efficient parameterization of compact laser pulses used in laser-matter simulations and provide a potential means for experimentally bounding the fundamental length scale in the generalized electrodynamics of Bopp, Landé and Podolsky.Keywords: Laser pulse, finite energy, Maxwell equations, Bopp-Landé-Podolsky Advances in laser technology have made possible the exploration of physical processes on unprecedented temporal and spatial scales. They have also opened up new possibilities for accelerating charged particles using lasermatter interactions. Multi-and single cycle high intensity (10 10 − 10 15 Watts/cm 2 ) laser pulses can be produced using Q-switching or mode-locking techniques [1]. Such pulses can accelerate charged particles such as electrons to relativistic speeds where radiation reaction and quantum effects may influence their dynamics. Lower intensity pulses have also been used as diagnostic tools for exploring the structure of plasmas in various states [2,3]. In order to interpret experimental data involving classical laser interactions with both charged and neutral matter, theoretical models [4][5][6][7] rely crucially on parameterizations of the electromagnetic fields in laser pulses, particularly in situations where traditional formulations using monochromatic or paraxial-beam approximations have limitations [8][9][10].In this Letter we discuss a viable methodology for parameterizing a particular class of propagating solutions to the source free classical Maxwell equations in vacuo that offers an efficient means to explore the classical effects of compact laser pulses on free electrons in dynamical regimes where quantum effects are absent. The parameterization is based on a remarkable class of explicit solutions of the scalar wave equation found by Ziolkowski [11][12][13][14][15] together with the behaviour of charged particle-pulse interactions over a broad parameter range without recourse to expensive numerical computation. Finally, we argue that such parameterizations can be used to find compact finite energy solutions to other linear wave equations. This is illustrated by showing that the generalized theory of Bopp [17], Landé [18] and Podolsky [19] admits such particular solutions that reduce to the Maxwell solutions when a fundamental length parameter in their theory tends to zero. Compact laser pulses in this theory might be used to explore properties of the theory by searching experimentally for bounds on this parameter.If a complex scalar field α satisfies α = 0 on spacetime and Π µν is any covariantly constant (degree 2) antisymmetric tensor field on spacetime (i.e. Π µν;δ = 0) for all µ, ν, δ...

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Extracting and Using Electromagnetic Energy from the Active Vacuum

Bearden¹

Modern Nonlinear Optics, Part II

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The first principles of things will never be adequately known.Science is an openended endeavor, it can never be closed. We do science without knowing the first principles. It does in fact not start from first principles, nor from the end principles, but from the middle. We not only change theories, but also the concepts and entities themselves, and what questions to ask. The foundations of science must be continuously examined and modified; it will always be full of mysteries and surprises." {1}

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A method for constructing solutions of homogeneous partial differential equations: localized waves

Cited by 60 publications

References 7 publications

The Relativistic Hermite Polynomials and the Wave Equation

The Relativistic Hermite Polynomials and the Wave Equation

Classical dynamics of free electromagnetic laser pulses

Extracting and Using Electromagnetic Energy from the Active Vacuum

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