Nonlinear Waves in Inhomogeneous Cylindrical Shells: A New Evolution Equation

Zemlyanukhin, A. I.; Mogilevich, L. I.

doi:10.1007/bf03353584

Cited by 13 publications

(12 citation statements)

References 2 publications

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“…Equation (4.31) were considered before. Exact solutions of this equation were found in works [11,14,53,54].…”

Section: Nonlinear Odes With Exact Solutions Of the First Degree Singmentioning

confidence: 99%

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

Kudryashov

2004

Zeitschrift Für Naturforschung A

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New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption that nonlinear differential equations have exact solution which is general solution of the simplest integrable equation. The Riccati equation is shown to be a building block to find a lot of nonlinear differential equations with exact solutions. Nonlinear differential equations of the second, third and fourth order with special solutions are given. Most of these equations are used at the description of processes in physics and in theory of nonlinear waves.

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“…Equation (4.31) were considered before. Exact solutions of this equation were found in works [11,14,53,54].…”

Section: Nonlinear Odes With Exact Solutions Of the First Degree Singmentioning

confidence: 99%

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

Kudryashov

2004

Zeitschrift Für Naturforschung A

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“…The models of nonlinear wave dynamics of deformable media with microstructure can lead to equations with spatial derivatives of the highest order. The work [16] contains a non-integrable fifth-order evolution equation with cubic nonlinearity…”

Section: Introductionmentioning

confidence: 99%

“…with three constants A, B, C to be determined. Substitute the solution (17) into (16) and collect the terms proportional to tanh powers. Equating the coefficients in front of these collections to zero, we obtain a system of four equations, where the solution can be as follows:…”

Section: Introductionmentioning

confidence: 99%

“…At the first step of the GNLS derivation [20], we allow for function u to depend on the phase variable θ = kx − ωt, slow coordinate X = εx and slow time T = εt, where ε is a small parameter, so that equation (16) takes the following form…”

Section: Introductionmentioning

confidence: 99%

“…and l, δ are free parameters. Note that in general the solution (30) is not a steady solitary wave solution of the original equation (16). In [18] it was shown that the requirement c = c g needed for steady solitary wave solution leads to the parameters υ and l of (30) to be O (ε).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Steady solitary wave solutions of the generalized sixth-order Boussinesq–Ostrovsky equation

Zemlyanukhin¹,

Бочкарев²

2015

Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki

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An overview of models that lead to the nonintegrable Ostrovsky equation and its generalizations having no exact solitary-wave solutions is given. A brief derivation of the Ostrovsky equation for longitudinal waves in a geometrically nonlinear rod lying on an elastic foundation is performed. It is shown that in the case of axially symmetric propagation of longitudinal waves in a physically nonlinear cylindrical shell interacting with a nonlinear elastic medium the displacement component obeys the generalized sixth-order Boussinesq-Ostrovsky equation. We construct an exact kink-like solution of this equation, establish a connection with the generalized nonlinear Schrödinger (GNLS) equation and find the steady travelling wave solution of the GNLS in the form of simple soliton with monotonic or oscillating tails.

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Mathematical Modeling of Waves in a Non-linear Shell with Wiscous Liquid Inside It, Taking into Account Its Movement Inertia

Mogilevich

Blinkov

Kondratov

et al. 2019

Recent Research in Control Engineering and Decision Making

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Nonlinear Waves in Inhomogeneous Cylindrical Shells: A New Evolution Equation

Cited by 13 publications

References 2 publications

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

Steady solitary wave solutions of the generalized sixth-order Boussinesq–Ostrovsky equation

Mathematical Modeling of Waves in a Non-linear Shell with Wiscous Liquid Inside It, Taking into Account Its Movement Inertia

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