scholarly journals A Note on Sign-Changing Solutions to the NLS on the Double-Bridge Graph

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 161
Author(s):  
Diego Noja ◽  
Sergio Rolando ◽  
Simone Secchi
Keyword(s):  
Boundary Conditions ◽  
Standing Waves ◽  
Nls Equation ◽  
Sign Changing Solutions

We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on the circle, by proving some existing results of sign-changing solutions non-periodic on the circle.

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.

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

scholarly journals Variational properties and orbital stability of standing waves for NLS equation on a star graph

2014 ◽  
Vol 257 (10) ◽  
pp. 3738-3777 ◽  
Author(s):  
Riccardo Adami ◽  
Claudio Cacciapuoti ◽  
Domenico Finco ◽  
Diego Noja
Keyword(s):  
Standing Waves ◽  
Orbital Stability ◽  
Star Graph ◽  
Nls Equation

scholarly journals Standing Waves in One-Dimensional Resonator Containing an Ideal Isothermal Gas Affected by the Constant Mass Force

2017 ◽  
Vol 42 (2) ◽  
pp. 263-271
Author(s):  
Anna Perelomova
Keyword(s):  
Boundary Conditions ◽  
Acoustic Waves ◽  
Standing Waves ◽  
Ideal Gas ◽  
Excess Pressure ◽  
Mass Force ◽  
Constant Mass ◽  
Total Field ◽  
One Dimensional ◽  
Isothermal Gas

Abstract The study is devoted to standing acoustic waves in one-dimensional planar resonator which containing an ideal gas. A gas is affected by the constant mass force. Two types of physically justified boundary conditions are considered: zero velocity or zero excess pressure at both boundaries. The variety of nodal and antinodal points is determined. The conclusion is that the nodes of pressure and antinodes of velocity do not longer coincide, as well as antinodes of pressure and nodes of velocity. The entropy mode may contribute to the total field in a resonator. It is no longer isobaric, in contrast to the case when the external force is absent. Examples of perturbations inherent to the entropy mode in the volume of a resonator are discussed.

LEAST ENERGY NODAL SOLUTIONS OF THE BREZIS–NIRENBERG PROBLEM IN DIMENSION N = 5

2009 ◽  
Vol 11 (01) ◽  
pp. 59-69 ◽  
Author(s):  
PAOLO ROSELLI ◽  
MICHEL WILLEM
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
First Eigenvalue ◽  
Nodal Solutions ◽  
The First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Nirenberg Problem ◽  
Least Energy

We prove the existence of (a pair of) least energy sign changing solutions of [Formula: see text] when Ω is a bounded domain in ℝN, N = 5 and λ is slightly smaller than λ1, the first eigenvalue of -Δ with homogeneous Dirichlet boundary conditions on Ω.

scholarly journals Chladni Figures Simulation on a Rectangular Plate

2021 ◽  
Vol 26 (3) ◽  
Author(s):  
Pavlo Ihorovych Krysenko ◽  
Maksym Olehovych Zoziuk ◽  
Oleksandr Ivanovych Yurikov ◽  
Dmytro Volodymyrovych Koroliuk ◽  
Yurii Ivanovych Yakymenko
Keyword(s):  
Fourier Transform ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Standing Waves ◽  
Physical Nature ◽  
Practical Significance ◽  
The Fourier Transform ◽  
Further Development ◽  
Top View ◽  
Chladni Figures

An analytical model for creating flat Chladni figures is presented. The equation of a standing wave in the simplest boundary conditions and the Fourier transform are used. Top view images are shown at different frequencies. The practical significance of the results obtained for the further development of the field of creating Chladni figures based on standing waves of different physical nature has been determined.

scholarly journals On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

2019 ◽  
Vol 149 (5) ◽  
pp. 1163-1173
Author(s):  
Vladimir Bobkov ◽  
Sergey Kolonitskii
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Elliptic Equations ◽  
Nodal Set ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic ◽  
Least Energy

AbstractIn this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

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