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Stability of intrinsic localized modes on the lattice with competing power nonlinearities

Georgy L. Alfimov, Pavel A. Korchagin, Dmitry E. Pelinovsky

TL;DR

This work analyzes intrinsic localized modes in a DNLS lattice with competing power nonlinearities via the anticontinuum limit. It establishes existence and classification of ACL ILMs by sign-preserving codes, derives a truncated eigenvalue framework to predict spectral stability, and applies negative index theory to connect stability to energy minimization and VK-type slopes. The study reveals universal stable codes and a spectrum of stable stacked configurations, with stability depending on γ and the p,q exponents, and provides extensive numerical continuation results for small but finite coupling. The findings deepen understanding of multistability and mobility of localized modes in lattices with competing nonlinearities, and point to open questions in asymptotics near critical γ and variational characterizations of minimizers.

Abstract

We study the discrete nonlinear Schrodinger equation with competing powers (p,q) satisfying 2 <= p < q. The physically relevant cases are given by (p,q) = (2,3), (p,q) = (3,4), and (p,q) = (3,5). In the anticontinuum limit, all intrinsic localized modes are compact and can be classified by their codes, which record one of two nonzero (smaller and larger) states and their sign alternations. By using the spectral stability analysis, we prove that the codes for larger states of the same sign are spectrally and nonlinearly (orbitally) stable, whereas the codes for smaller states of the alternating signs are spectrally stable but have eigenvalues of negative Krein signature. We also identify numerically the spectrally stable codes which consist of stacked combinations of the sign-definite larger states and the sign-alternating smaller states.

Stability of intrinsic localized modes on the lattice with competing power nonlinearities

TL;DR

This work analyzes intrinsic localized modes in a DNLS lattice with competing power nonlinearities via the anticontinuum limit. It establishes existence and classification of ACL ILMs by sign-preserving codes, derives a truncated eigenvalue framework to predict spectral stability, and applies negative index theory to connect stability to energy minimization and VK-type slopes. The study reveals universal stable codes and a spectrum of stable stacked configurations, with stability depending on γ and the p,q exponents, and provides extensive numerical continuation results for small but finite coupling. The findings deepen understanding of multistability and mobility of localized modes in lattices with competing nonlinearities, and point to open questions in asymptotics near critical γ and variational characterizations of minimizers.

Abstract

We study the discrete nonlinear Schrodinger equation with competing powers (p,q) satisfying 2 <= p < q. The physically relevant cases are given by (p,q) = (2,3), (p,q) = (3,4), and (p,q) = (3,5). In the anticontinuum limit, all intrinsic localized modes are compact and can be classified by their codes, which record one of two nonzero (smaller and larger) states and their sign alternations. By using the spectral stability analysis, we prove that the codes for larger states of the same sign are spectrally and nonlinearly (orbitally) stable, whereas the codes for smaller states of the alternating signs are spectrally stable but have eigenvalues of negative Krein signature. We also identify numerically the spectrally stable codes which consist of stacked combinations of the sign-definite larger states and the sign-alternating smaller states.

Paper Structure

This paper contains 9 sections, 12 theorems, 76 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Existence of ILMs in the anticontinuum limit
  3. Stability of ILMs in the anticontinuum limit
  4. Truncated eigenvalue problem
  5. Stability results based on the truncated eigenvalue problem
  6. Stability results based on the negative index theory
  7. Numerical results on continuation of ILMs for $\varepsilon>0$
  8. Numerical results on stable ILMs for small $\varepsilon > 0$
  9. Conclusion

Key Result

Theorem 1

Let $p,q \in \mathbb{N}$ be fixed such that $2 \leq p < q$. For every $\gamma \in (0,\gamma_{{p,q}})$, where there exists $\varepsilon_0 > 0$ and $C_0 > 0$ such that for every $\varepsilon \in (0,\varepsilon_0)$, there exists a solution ${\bf u} \in \ell^2(\mathbb{Z})$ of the difference equation (Eq:u_gen_pq) such that where ${\bf u}^{(0)} = (\ldots,0,0,\tilde{\bf u},0,0,\ldots)$ with $\tilde{\b

Figures (4)

  • Figure 1: Bifurcations of the branch of ILM with code $(A_+a_-)$, $(p,q)=(3,4)$. Left panel: $\gamma=0.12$. At $\varepsilon\approx 0,105$ this branch merges with the branch with code $(a_+A_+a_-)$, the fold bifurcation. Right panel: $\gamma=0.22$. At $\varepsilon\approx0.099$ the pair of ILMs with codes $(A_+a_-)$ and $(a_+A_-)$ is connected to the branches of $-R$-symmetric ILMs with codes $(A_+A_-)$ and $(a_+a_-)$, the pitchfork bifurcation.
  • Figure 2: Bifurcations of branches of ILMs for the case $(p,q) = (3,4)$: Panel A: $\gamma \approx 0.30$; panel B: $\gamma \approx0.32$; panel C: $\gamma \approx0.33$. Values of $\varepsilon$ are shown versus $Q({\bf u}) = \sum_{n \in \mathbb{Z}} u_n^2$. Parts of the branches that correspond to stable (unstable) solutions are shown in blue (red).
  • Figure 3: Eigenvalues of the truncated spectral problem (\ref{['Gener-matrix']}) versus $\gamma$ in $(0,\gamma_{p,q})$ for the code $\mathcal{A}_{5,5}^+$ with $(p,q)=(3,4)$ (panel A), $(p,q)=(3,5)$ (panel B), and $(p,q)=(3,6)$ (panel C). The real parts of the eigenvalues are shown: orange for real negative eigenvalues, blue for real positive eigenvalues, and red (bold) for complex eigenvalues.
  • Figure 4: Eigenvalues of the truncated spectral problem (\ref{['Gener-matrix']}) versus $\gamma$ in $(0,\gamma_{3,4})$ with $(p,q) = (3,4)$ for the codes $\mathcal{A}_{5,4}^-$ (upper panel), $\mathcal{A}_{5,4}^+$ (middle panel), and $\mathcal{A}_{5,5}^-$ (lower panel). The color scheme is the same as in Fig. \ref{['Fig:5_5+']}.

Theorems & Definitions (42)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Example 1
  • Example 2
  • Proposition 1
  • ...and 32 more