The Painlevé Handbook

Conte, Robert; Musette, Micheline

doi:10.1007/978-3-030-53340-3

Cited by 79 publications

(89 citation statements)

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“…Since A = (Q/δ + i)δe iϕ is single valued [18], [19], both terms Q/δ and δe iϕ are single valued, while Q, δ, and e iϕ can be multivalued. Because of the absence of methods to handle multivaluedness, the strategy is therefore to only consider δe iϕ , its complex conjugate, and Q/δ.…”

Section: The Generic Solutionmentioning

confidence: 99%

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Explicit breather solution of the nonlinear Schrödinger equation

Conte

2021

Theor Math Phys

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Section: The Generic Solutionmentioning

confidence: 99%

“…Since those singularities of A and A that depend on the initial conditions are simple poles [18], [19], the next constraint after (2) should be…”

Section: On Higher-degree Constraintsmentioning

confidence: 99%

Explicit breather solution of the nonlinear Schrödinger equation

Conte

2021

Theor Math Phys

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“…To find the exact solutions for non-integrable equations, many effective methods have been developed by now [22,23]. According to the popular truncated decomposition method, a solution with the order pole q shall be sought in the form of a complete polynomial according to some basic functions f (z) with the eigenorder of the pole q f :…”

Section: The First Asymptoticsmentioning

confidence: 99%

“…Having expressed Ψ ξ from (22) and substituted the result into the principal part of (20), we obtain the Klein-Gordon nonlinear/non-homogeneous equation…”

Section: The Second Asymptoticsmentioning

confidence: 99%

Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

et al. 2021

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The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity.

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“…Many nonintegrable evolutionary and quasi-hyperbolic equations have exact solitary-wave and periodic solutions [5]. In contrast to them, for nonintegrable nonlinear DDEs, exact solutions can be found only in rare cases.…”

Section: Introductionmentioning

confidence: 99%

Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

et al. 2020

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In this paper, stationary solitary and periodic waves of a nonlinear nonintegrable lattice are numerically constructed using a two-stage approach. First, as a result of continualization, a nonintegrable generalized Boussinesq—Ostrovsky equation is obtained, for which the solitary-wave and periodic solutions are numerically found by the Petviashvili method. In the second stage, discrete analogs of the obtained solutions are used as initial conditions in the numerical simulation of the original lattice. It is shown that the initial perturbations constructed in this way propagate along the lattice without changing their shape.

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The Painlevé Handbook

Cited by 79 publications

References 0 publications

Explicit breather solution of the nonlinear Schrödinger equation

Explicit breather solution of the nonlinear Schrödinger equation

Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

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