Abstract
In this paper, we first compute the multiple non-trivial solutions of the Schrodinger equation on a unit disk, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with the mixed Fourier-Legendre spectral and pseudospectral methods. After that, we propose the extended systems, which can detect the symmetry-breaking bifurcation points on the branch of the O(2) symmetric positive solutions. We also compute the multiple positive solutions with various symmetries of the Schrodinger equation by the branch switching method based on the Liapunov-Schmidt reduction. Finally, the bifurcation diagrams are constructed, showing the symmetry/peak breaking phenomena of the Schr¨odinger equation. Numerical results demonstrate the effectiveness of these approaches.
Highlights
IntroductionWe shall find the multiple solutions to the following nonlinear Schrodinger equation (NLS):
In this paper, we shall find the multiple solutions to the following nonlinear Schrodinger equation (NLS):F (u(x), λ, l) := −∆u(x) + λu(x) + κ|x|l|u(x)|p−1u(x) = 0, x ∈ Ω, (1.1)u|∂Ω = 0, where Ω is a unit disk and p > 1, λ, κ, l are prescribed parameters
We shall find the multiple solutions to the following nonlinear Schrodinger equation (NLS): F (u(x), λ, l) := −∆u(x) + λu(x) + κ|x|l|u(x)|p−1u(x) = 0, x ∈ Ω, (1.1)
Summary
We shall find the multiple solutions to the following nonlinear Schrodinger equation (NLS):. For the Mtype with λ > −λ1, J is said to have a mountain pass structure and 0 is the only local minimum; for the W-type with k ≥ 1, J has two local minima In the literature, these two cases have to be treated by two very different types of variational methods. The previous algorithms usually need a good guess of solution, which seems to be a difficult task To overcome this disadvantage, we shall use the bifurcation method [16], [27], [29] to compute the multiple solutions of (1.1). We can compute the multiple solutions of the equation (1.1) for any Morse index by using the finite difference method.
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