Solitary-wave solutions of the fractional nonlinear Schrödinger equation. I. Existence and numerical generation

Abstract

The present paper is the first part of a project devoted to the fractional nonlinear Schrödinger (fNLS) equation. It is concerned with the existence and numerical generation of the solitary-wave solutions. For the first point, some conserved quantities of the problem are used to search for solitary-wave solutions as relative equilibria. From the relative equilibrium condition, a result of existence via the Concentration-Compactness theory is derived. Several properties of the waves, such as the regularity and the asymptotic decay in some cases, are derived from the existence result. Some other properties, such as the monotone behaviour and the speed-amplitude relation, will be explored computationally. To this end, a numerical procedure for the generation of the profiles is proposed. The method is based on a Fourier pseudospectral approximation of the differential system for the profiles and the use of Petviashvili's iteration with extrapolation.

Figures (7)

• Figure 1: Numerical generation of solitary waves. Iteration (\ref{['fnls_312']}) with $\sigma=1, s=3/4$ y $\lambda_0^1=1$. (a) $\rho$ numerical profiles for several speeds; (b) residual error vs. number of iterations for the case $c_{s}=1$ and several values of the width of extrapolation $mw$.
• Figure 2: Numerical approximation of (\ref{['fnls1d']}) with $\sigma=1, s=3/4$, initial condition given by the approximate profile from $\lambda_0^1=1, \lambda_{0}^{2}=1$. (a) Evolution of the error in the amplitude; (b) Hamiltonian error as function of time.
• Figure 3: (a) Approximate $\rho$ profiles for $\sigma=1, \lambda_{0}^{1}=1,\lambda_{0}^{2}=0.75, s=0.5+\epsilon$ for several values of $\epsilon$. (b) Approximate $\rho$ profiles for $s=3/4, \lambda_{0}^{1}=1, \lambda_{0}^{2}=1$ for several values of $\sigma$.
• Figure 4: Approximate profiles for $\sigma=1, s=3/4, \lambda_{0}^{1}=1$. (a) $\rho=\sqrt{v^{2}+w^{2}}$ with $(v,w)$ solution of (\ref{['fnls_311']}) with initial iteration (\ref{['fnls_331']}) and $\theta(x)=x^{2}$; (b) solution $\rho$ of (\ref{['fnls_312']}) with initial iteration (\ref{['fnls_331']}) and $\theta(x)=Ax$.
• Figure 5: Approximate profiles for $\sigma=1, s=3/4, \lambda_{0}^{1}=1$ in log-log scale. (a) Solution $\rho$ of (\ref{['fnls_312']}) with initial iteration (\ref{['fnls_331']}) and $\theta(x)=Ax$ (cf. Figure \ref{['NR_ADfig1']}(a)); (b) $\rho=\sqrt{v^{2}+w^{2}}$ with $(v,w)$ solution of (\ref{['fnls_311']}) with initial iteration (\ref{['fnls_331']}) and $\theta(x)=x^{2}$ (cf. Figure \ref{['NR_ADfig4']}(a)).

Theorems & Definitions (12)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Proposition 2.1
• proof
• Theorem 2.1
• proof