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A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations
Author(s) -
Alouini Brahim
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6709
Subject(s) - attractor , mathematics , dissipative system , fractal dimension , nonlinear system , mathematical analysis , type (biology) , fractal , dimension (graph theory) , lebesgue integration , space (punctuation) , infinity , minkowski–bouligand dimension , forcing (mathematics) , pure mathematics , physics , quantum mechanics , ecology , biology , linguistics , philosophy
In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue : consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equationu t + i u x x + i | u | 2 u + γ u = f ,f ∈ 2 ( ℝ ) . “How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?”

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