Numerical analyses for spectral stability of solitary waves near bifurcation points
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Numerical analyses for spectral stability of solitary waves near bifurcation points Kazuyuki Yagasaki1 · Shotaro Yamazoe1 Received: 10 March 2020 / Revised: 23 June 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract We present a numerical approach for determination of the spectral stability of solitary waves by computing eigenvalues and eigenfunctions of the corresponding eigenvalue problems, along with their continuation, for nonlinear wave equations in one space dimension. We illustrate the approach for the nonlinear Schrödinger (NLS) equation with a small potential, and numerically determine the spectral stability of solitary waves near bifurcation points along with computations of eigenfunctions and eigenvalues. The numerical results demonstrate some theoretical ones the authors recently obtained for the example. Keywords Numerical analysis · Bifurcation · Nonlinear Schrödinger equations · Solitary wave · Spectral stability Mathematics Subject Classification Primary: 34B15 · 35J61 · Secondary: 35Q55 · 37D10
1 Introduction Bifurcations of solitary waves in nonlinear wave equations have been extensively investigated[13, 17, 23]. In particular, nonlinear Schrödinger (NLS) equations have been attracted special attention. For the NLS equation with a linear potential, when their frequencies are taken as a control parameter, saddle-node and pitchfork (symmetry-breaking) bifurcations of solitary waves were studied by many researchers This work was partially supported by JSPS KAKENHI Grant Number JP17H02859. * Kazuyuki Yagasaki [email protected]‑u.ac.jp Shotaro Yamazoe [email protected]‑u.ac.jp 1
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo‑ku, Kyoto 606‑8501, Japan
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(see [23–25] and references therein) but most of them were based on numerical computations and only a small number of analytical treatments were given (e.g., [9, 14, 15, 24–26]). Transcritical and Hamiltonian Hopf bifurcations were also discussed in [5, 24, 27]. Recently, the authors [19] developed a technique for analyzing saddle-node and pitchfork bifurcations of relative equilibria and their stability in infinite-dimensional perturbed Hamiltonian systems and applied it to solitary waves of the form u(t, x) = ei𝜔t 𝜑(x) with 𝜔 > 0 in the NLS equation with a small linear potential,
i𝜕t u = −𝜕x2 u + 𝜀V(x;𝜇)u − |u|2 u,
(1.1)
(t, x) ∈ ℝ × ℝ,
where 𝜀 is a parameter such that 0 < 𝜀 ≪ 1 and
(1.2)
V(x;𝜇) = −𝛼 sech (x + 𝜇) − sech (x − 𝜇)
with 𝜇 ∈ ℝ and 𝛼 > 0 . Here 𝜑 is an ℝ-valued function which decays at infinity, i.e.,
lim 𝜑(x) = 0,
x→±∞
and satisfies (1.3)
−𝜑�� + 𝜔𝜑 + 𝜀V(x;𝜇)𝜑 − 𝜑3 = 0.
The shapes of the linear potential (1.2) for 𝜇 = 3 and 𝛼 = 0.5, 1 are diplayed in Fig. 1. Saddle-node and pitchfork bifurcations of solitary waves were shown to occur when 𝜇 is taken as a control parameter for 𝛼 ≠ 1 and 𝛼 = 1 , respectively, and the spectral stability of the bifur
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