Concentrating standing waves for Davey–Stewartson systems

2021 ◽  
pp. 1-40
Author(s):  
Yi He ◽  
Xiao Luo
Keyword(s):  
Water Waves ◽  
Elliptic Equations ◽  
Standing Waves ◽  
Regularity Theory ◽  
Local Minima ◽  
Shallow Water Waves ◽  
Penalization Method ◽  
Finite Set ◽  
New Ideas ◽  
Concentrating Solutions

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

scholarly journals Seiches and the slide/seiche dynamics; subcritical and supercritical subaquous mass flows and their deposits. Examples from Swiss Lakes

2021 ◽  
Vol 114 (1) ◽  
Author(s):  
Christoph Siegenthaler
Keyword(s):  
Shear Stress ◽  
Shallow Water ◽  
Mass Flow ◽  
Water Waves ◽  
Standing Waves ◽  
Shallow Water Waves ◽  
Flow Deposit ◽  
Mass Flows ◽  
Shear Energy ◽  
Small Shear

AbstractFour historically documented large and potentially dangerous lacustrine waves in Swiss lakes show that these waves have been seiches (standing waves) triggered by sublacustrine slides; a statement which is in accordance with the experience of seismologists who see earthquakes triggering seiches in lakes. Nevertheless, large historical waves in Switzerland have recently been modeled as progressive shallow water waves (tsunamis), probably because the slide/seiche dynamics are not known, and experiments with subaquatic slides fail to generate seiches in test–flumes. It appears that these tests exhibit a small shear–energy/slide–energy ratio ε, if compared with the situation in lakes. These facts incite a shear–stress lemma that states that ε is the constituent factor for the slide/seiche coupling. The structure of the subaqueous mass flow deposit (MFD) in lakes Lucerne and Geneva suggests the occurrence of subcritical and of supercritical slide flows. The former would generate a contortite, a MFD with contorted bedding, the latter a debrite (mudclast conglomerate). Potential slide energy considerations are used for an estimation of the amplitudes of large seiches produced by subaquatic slides, a proceeding that yields partly similar and partly very different results, as compared with numerical tsunami simulations.

RESPONSE CHARACTERISTICS OF GRAVEL BEACH FROM THE VIEWPOINT OF SHALLOW WATER WAVES AND MOVEMENT OF GRAVELS

2020 ◽  
Vol 76 (2) ◽  
pp. I_565-I_570
Author(s):  
Shin-ichi AOKI ◽  
Tomoki HAMANO ◽  
Taishi NAKAYAMA ◽  
Eiichi OKETANI ◽  
Takahiro HIRAMATSU ◽  
...  
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Shallow Water Waves ◽  
Response Characteristics ◽  
Gravel Beach

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

scholarly journals Active-subspace analysis of exceedance probability for shallow-water waves

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Kenan Šehić ◽  
Henrik Bredmose ◽  
John D. Sørensen ◽  
Mirza Karamehmedović
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Exceedance Probability ◽  
Subspace Analysis ◽  
Shallow Water Waves ◽  
Active Subspace

scholarly journals From metaphors to equations: How can we find the good ones?

2000 ◽  
Vol 23 (3) ◽  
pp. 409-410
Author(s):  
Gottfried Mayer-Kress
Keyword(s):  
Boundary Conditions ◽  
Forest Fire ◽  
Water Waves ◽  
Standing Waves ◽  
Musical Instruments ◽  
Oscillator Model ◽  
Mechanical Oscillator ◽  
Target Article ◽  
Mechanical Oscillators

Among the metaphors used in the target article are “musical instruments,” “water waves,” and other types of mechanical oscillators. The corresponding equations have inertial properties and lead to standing waves that depend on boundary conditions. Other, physiologically relevant quantities like refractory times are not contained in the mechanical oscillator model but occur naturally, for instance, in biological forest fire metaphors.

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