scholarly journals Orbital Stability of Solitary Waves for the Nonlinear Derivative Schrödinger Equation

1995 ◽  
Vol 123 (1) ◽  
pp. 35-55 ◽  
Author(s):  
B.L. Guo ◽  
Y.P. Wu
Keyword(s):  
Schrödinger Equation ◽  
Solitary Waves ◽  
Schrodinger Equation ◽  
Orbital Stability ◽  
Stability Of Solitary Waves

scholarly journals Orbital stability vs. scattering in the cubic-quintic Schrödinger equation

2020 ◽  
pp. 2150004
Author(s):  
Rémi Carles ◽  
Christof Sparber
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Orbital Stability ◽  
Cubic Nonlinearity ◽  
The Cauchy Problem ◽  
Stability Of Solitary Waves ◽  
Dimension One ◽  
Three Space ◽  
Quintic Nonlinear Schrödinger Equation

We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable.

scholarly journals Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1398
Author(s):  
Natalia Kolkovska ◽  
Milena Dimova ◽  
Nikolai Kutev
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Second Derivative ◽  
Cubic Nonlinearity ◽  
Abstract Theory ◽  
Power Type ◽  
Analytical Formulas ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Double Dispersion

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

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