scholarly journals Analytic smoothing effect for nonlinear Schrödinger equation with quintic nonlinearity

2014 ◽  
Vol 419 (1) ◽  
pp. 285-297 ◽  
Author(s):  
Gaku Hoshino ◽  
Tohru Ozawa
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Smoothing Effect ◽  
Nonlinear Schrödinger ◽  
Quintic Nonlinearity ◽  
Analytic Smoothing Effect

scholarly journals Growth of Solutions to NLS on Irrational Tori

2017 ◽  
Vol 2019 (9) ◽  
pp. 2919-2950 ◽  
Author(s):  
Yu Deng ◽  
Pierre Germain
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Strichartz Estimates ◽  
Nonlinear Schrödinger ◽  
Quintic Nonlinearity ◽  
Polynomial Bounds ◽  
Growth Of Solutions

Abstract We prove polynomial bounds on the $H^s$ growth for the nonlinear Schrödinger equation set on a torus, in dimension 3, with super-cubic and sub-quintic nonlinearity. Due to improved Strichartz estimates, these bounds are better for irrational tori than they are for rational tori.

scholarly journals Pseudospectral Renormalization Method for Solitons in Quasicrystal Lattice with the Cubic-Quintic Nonlinearity

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Nalan Antar
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Optical Lattice ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
The Self ◽  
Nonlinear Schrödinger ◽  
Quintic Nonlinearity ◽  
Self Focusing ◽  
Renormalization Method

We modified the spectral renormalization method as the pseudospectral renormalization method in order to find the localized solutions. The pseudospectral renormalization method can be applied to a large class of problems including different homogeneities. Using this computational method, we demonstrate the existence of two different solitons in optical media described by the self-focusing cubic and the self-defocusing quintic nonlinear Schrödinger equation with quasicrystal lattice. It is shown that there are two different lattice solitons corresponding to the first and the second renormalization factors for the self-focusing cubic and the self-defocusing quintic model. However, the self-focusing quintic nonlinearity without optical lattice does not support two different solitons. We showed that the lattice solitons corresponding to the first and the second renormalization factors have the same powers and amplitudes. We also demonstrate that quintic nonlinearity supports bistable solitons by adding the optical lattice such as a quasicrystal lattice. The linear and nonlinear stabilities of these solitons are investigated using direct simulation of the nonlinear Schrödinger equation with the cubic-quintic nonlinearity and its linearized equation.

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