On the gradual reflection of weakly nonlinear Stokes waves in regions with varying topography

1986 ◽  
Vol 162 (-1) ◽  
pp. 187 ◽  
Author(s):  
James T. Kirby
Keyword(s):  
Weakly Nonlinear ◽  
Stokes Waves ◽  
Weakly Nonlinear Stokes Waves

scholarly journals Weakly nonlinear theory for oscillating wave surge converters in a channel

2017 ◽  
Vol 834 ◽  
pp. 55-91 ◽  
Author(s):  
S. Michele ◽  
P. Sammarco ◽  
M. d’Errico
Keyword(s):  
Nonlinear Theory ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Side Band ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Natural Modes ◽  
Stokes Waves ◽  
Conversion Performance ◽  
Incident Waves

We present a weakly nonlinear theory on the natural modes’ resonance of an array of oscillating wave surge converters (OWSCs) in a channel. We first derive the evolution equation of the Stuart–Landau type for the gate oscillations in uniform and modulated incident waves and then evaluate the nonlinear effects on the energy conversion performance of the array. We show that the gates are unstable to side-band perturbations so that a Benjamin–Feir instability similar to the case of Stokes’ waves is possible. The non-autonomous dynamical system presents period doubling bifurcations and strange attractors. We also analyse the competition of two natural modes excited by one incident wave. For weak damping and power take-off coefficient, the dynamical effects on the generated power of the OWSCs are investigated. We show that the occurrence of subharmonic resonance significantly increases energy production.

Refraction-diffraction model for weakly nonlinear water waves

1984 ◽  
Vol 141 ◽  
pp. 265-274 ◽  
Author(s):  
Philip L.-F. Liu ◽  
Ting-Kuei Tsay
Keyword(s):  
Water Waves ◽  
Model Equation ◽  
Harmonic Wave ◽  
Nonlinear Effects ◽  
Variable Coefficients ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Diffraction Model ◽  
Stokes Waves ◽  
Higher Harmonic

A model equation is derived for calculating transformation and propagation of Stokes waves. With the assumption that the water depth is slowly varying, the model equation, which is a nonlinear Schrödinger equation with variable coefficients, describes the forward-scattering wavefield. The model equation is used to investigate the wave convergence over a semicircular shoal. Numerical results are compared with experimental data (Whalin 1971). Nonlinear effects, which generate higher-harmonic wave components, are definitely important in the focusing zone. Mean free-surface set-downs over the shoal are also computed.

Investigating Dynamics of One Weakly Nonlinear System with Delay Argument

2018 ◽  
Vol 50 (1) ◽  
pp. 20-38 ◽  
Author(s):  
Denis Ya. Khusainov ◽  
Jozef Diblik ◽  
Jaromir Bashtinec ◽  
Andrey V. Shatyrko
Keyword(s):  
Nonlinear System ◽  
Weakly Nonlinear ◽  
Delay Argument ◽  
System With Delay ◽  
Weakly Nonlinear System

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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