scholarly journals Orbital stability of periodic waves for the nonlinear Schrödinger equation

2007 ◽  
Vol 19 (4) ◽  
pp. 825-865 ◽  
Author(s):  
Thierry Gallay ◽  
Mariana Hǎrǎgus
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Periodic Waves ◽  
Nonlinear Schrödinger

scholarly journals Orbital stability of standing waves for a class of Schrödinger equations with unbounded potential

2006 ◽  
Vol 2006 ◽  
pp. 1-7
Author(s):  
Guanggan Chen ◽  
Jian Zhang ◽  
Yunyun Wei
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Variational Calculus ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Unbounded Potential

This paper is concerned with the nonlinear Schrödinger equation with an unbounded potential iϕt=−Δϕ+V(x)ϕ−μ|ϕ|p−1ϕ−λ|ϕ|q−1ϕ, x∈ℝN, t≥0, where μ>0, λ>0, and 1<p<q<1+4/N. The potential V(x) is bounded from below and satisfies V(x)→∞ as |x|→∞. From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.

scholarly journals Rogue periodic waves of the focusing nonlinear Schrödinger equation

2018 ◽  
Vol 474 (2210) ◽  
pp. 20170814 ◽  
Author(s):  
Jinbing Chen ◽  
Dmitry E. Pelinovsky
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Elliptic Functions ◽  
Nonlinear Schrodinger Equation ◽  
Periodic Waves ◽  
Jacobian Elliptic Functions ◽  
Nonlinear Schrödinger

Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrödinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn . Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov–Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine’s breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.

ORBITAL STABILITY OF STANDING WAVES FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH POTENTIAL

2001 ◽  
Vol 13 (12) ◽  
pp. 1529-1546 ◽  
Author(s):  
CARLOS CID ◽  
PATRICIO FELMER
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Local Minimum ◽  
Standing Waves ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

We prove existence and orbital stability of standing waves for the nonlinear Schrödinger equation [Formula: see text] concentrating near a possibly degenerate local minimum of the potential V, when the Plank's constant ℏ is small enough. Our method applies to general nonlinearities, including f(s)=sp - 1 with p ∈ (1,1 + 4/N), but does not require uniqueness nor non-degeneracy of the limiting equation.

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