scholarly journals Nonexistence of Subcritical Solitary Waves

2021 ◽  
Author(s):  
Vladimir Kozlov ◽  
Evgeniy Lokharu ◽  
Miles H. Wheeler
Keyword(s):  
Solitary Waves ◽  
Open Problem ◽  
Water Waves ◽  
Complete Classification ◽  
Arbitrary Distribution ◽  
Two Dimensional ◽  
Wave Speeds ◽  
Nonexistence Result ◽  
Sign Conditions

AbstractWe prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.

scholarly journals Complete classification of trapping coins for quantum walks on the two-dimensional square lattice

2020 ◽  
Vol 102 (1) ◽  
Author(s):  
B. Kollár ◽  
A. Gilyén ◽  
I. Tkáčová ◽  
T. Kiss ◽  
I. Jex ◽  
...  
Keyword(s):  
Complete Classification ◽  
Quantum Walks ◽  
Square Lattice ◽  
Two Dimensional

Complete classification of second-order phase transitions in two-dimensional systems

1982 ◽  
Vol 44 (2) ◽  
pp. 163-165 ◽  
Author(s):  
L.A. Maksimov ◽  
I.Ya. Polishchuk ◽  
V.A. Somenkov
Keyword(s):  
Phase Transitions ◽  
Complete Classification ◽  
Second Order ◽  
Two Dimensional

scholarly journals The Froude number for solitary water waves with vorticity

2015 ◽  
Vol 768 ◽  
pp. 91-112 ◽  
Author(s):  
Miles H. Wheeler
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Froude Number ◽  
Simple Proof ◽  
Upper Bound ◽  
Water Waves ◽  
Wave Speed ◽  
Critical Value ◽  
Arbitrary Distribution ◽  
Two Dimensional

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

Homoclinic and Big Bang Bifurcations of an Embedding of 1D Allee’s Functions into a 2D Diffeomorphism

2017 ◽  
Vol 27 (09) ◽  
pp. 1730030 ◽  
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha ◽  
D. Fournier-Prunaret
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Big Bang ◽  
Complete Classification ◽  
Two Dimensional ◽  
Contour Lines ◽  
Dimension One ◽  
Necessary And Sufficient ◽  
Homoclinic Tangencies

In this work a thorough study is presented of the bifurcation structure of an embedding of one-dimensional Allee’s functions into a two-dimensional diffeomorphism. A complete classification of the nature and stability of the fixed points, on the contour lines of the two-dimensional diffeomorphism, is provided. A necessary and sufficient condition so that the Allee fixed point is a snapback repeller is established. Sufficient conditions for the occurrence of homoclinic tangencies of a saddle fixed point of the two-dimensional diffeomorphism are also established, associated to the snapback repeller bifurcation of the endomorphism defined by the Allee functions. The main results concern homoclinic and big bang bifurcations of the diffeomorphism as “germinal” bifurcations of the Allee functions. Our results confirm previous predictions of structures of homoclinic and big bang bifurcation curves in dimension one and extend these studies to “local” concepts of Allee effect and big bang bifurcations to this two-dimensional exponential diffeomorphism.

scholarly journals Solitary interfacial hydroelastic waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170099 ◽  
Author(s):  
Emilian I. Părău
Keyword(s):  
Dispersion Relation ◽  
Solitary Waves ◽  
Elastic Plate ◽  
Water Waves ◽  
Rigid Wall ◽  
Two Dimensional ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Two Fluids ◽  
Infinite Extent

Solitary waves travelling along an elastic plate present between two fluids with different densities are computed in this paper. Different two-dimensional configurations are considered: the upper fluid can be of infinite extent, bounded by a rigid wall or under a second elastic plate. The dispersion relation is obtained for each case and numerical codes based on integro-differential formulations for the full nonlinear problem are derived. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals On non-hyperbolic algebraic automorphisms of a two-dimensional torus

2021 ◽  
Vol 23 (3) ◽  
pp. 295-307
Author(s):  
Sergey V. Sidorov ◽  
Ekaterina E. Chilina
Keyword(s):  
Sufficient Conditions ◽  
Complete Classification ◽  
Orientable Surface ◽  
Topological Classification ◽  
Topological Conjugacy ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Algebraic Automorphism ◽  
Necessary And Sufficient

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

scholarly journals Interactions of large amplitude solitary waves in viscous fluid conduits

2014 ◽  
Vol 750 ◽  
pp. 372-384 ◽  
Author(s):  
Nicholas K. Lowman ◽  
M. A. Hoefer ◽  
G. A. El
Keyword(s):  
Solitary Wave ◽  
Viscous Fluid ◽  
Solitary Waves ◽  
Water Waves ◽  
Large Amplitude ◽  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Nonlinear Solitary Waves

AbstractThe free interface separating an exterior, viscous fluid from an intrusive conduit of buoyant, less viscous fluid is known to support strongly nonlinear solitary waves due to a balance between viscosity-induced dispersion and buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly nonlinear solitary waves has been classified theoretically for the Korteweg–de Vries equation and experimentally in the context of shallow water waves, but a theoretical and experimental classification of strongly nonlinear solitary wave interactions is lacking. The interactions of large amplitude solitary waves in viscous fluid conduits, a model physical system for the study of one-dimensional, truly dissipationless, dispersive nonlinear waves, are classified. Using a combined numerical and experimental approach, three classes of nonlinear interaction behaviour are identified: purely bimodal, purely unimodal, and a mixed type. The magnitude of the dispersive radiation due to solitary wave interactions is quantified numerically and observed to be beyond the sensitivity of our experiments, suggesting that conduit solitary waves behave as ‘physical solitons’. Experimental data are shown to be in excellent agreement with numerical simulations of the reduced model. Experimental movies are available with the online version of the paper.

scholarly journals Complete classification of two-dimensional algebras

2017 ◽  
Author(s):  
Houida Ahmed ◽  
U. Bekbaev ◽  
I. Rakhimov
Keyword(s):  
Complete Classification ◽  
Two Dimensional

scholarly journals Algebraic method for constructing singular steady solitary waves: a case study

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160194 ◽  
Author(s):  
Didier Clamond ◽  
Denys Dutykh ◽  
André Galligo
Keyword(s):  
Solitary Waves ◽  
Phase Plane ◽  
Algebraic Method ◽  
Travelling Wave ◽  
Complete Classification ◽  
Phase Plane Analysis ◽  
Localized Solutions ◽  
Plane Analysis

This article describes the use of algebraic methods in a phase plane analysis of ordinary differential equations. The method is illustrated by the study of capillary–gravity steady surface waves propagating in shallow water. We consider the (fully nonlinear, weakly dispersive) Serre–Green–Naghdi equation with surface tension, because it provides a tractable model that, at the same time, is not too simple, so interest in the method can be emphasized. In particular, we analyse a special class of solutions, the solitary waves, which play an important role in many fields of physics. In capillary–gravity regime, there are two kinds of localized infinitely smooth travelling wave solutions—solitary waves of elevation and of depression. However, if we allow the solitary waves to have an angular point, then the ‘zoology’ of solutions becomes much richer, and the main goal of this study is to provide a complete classification of such singular localized solutions using the methods of the effective algebraic geometry.

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