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This is a survey on Chaos in Partial Differential Equations. First we classify soliton equations into three categories: 1. (1+1)-dimensional soliton equations, 2. soliton lattices, 3. (1+n)-dimensional soliton equations (n ≥ 2). A systematic program has been established by the author and collaborators, for proving the existence of chaos in soliton equations(More)
In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an ‘‘unstable(More)
Catalysts for oxygen reduction and evolution reactions are at the heart of key renewable-energy technologies including fuel cells and water splitting. Despite tremendous efforts, developing oxygen electrode catalysts with high activity at low cost remains a great challenge. Here, we report a hybrid material consisting of Co₃O₄ nanocrystals grown on reduced(More)
Existence of homoclinic orbits in the cubic nonlinear Schrödinger equation under singular perturbations is proved. Emphasis is placed upon the regularity of the semigroup eǫt∂ 2 x at ǫ = 0. This article is a substantial generalization of [3], and motivated by the effort of Dr. Zeng [9] [8]. The mistake of Zeng in [8] is corrected with a normal form(More)
Advanced materials for electrocatalytic and photoelectrochemical water splitting are central to the area of renewable energy. In this work, we developed a selective solvothermal synthesis of MoS(2) nanoparticles on reduced graphene oxide (RGO) sheets suspended in solution. The resulting MoS(2)/RGO hybrid material possessed nanoscopic few-layer MoS(2)(More)
Electrocatalyst for oxygen reduction reaction (ORR) is crucial for a variety of renewable energy applications and energy-intensive industries. The design and synthesis of highly active ORR catalysts with strong durability at low cost is extremely desirable but remains challenging. Here, we used a simple two-step method to synthesize cobalt oxide/carbon(More)
The Bäcklund-Darboux transformations are successfully utilized to construct heteroclinic orbits of Davey-Stewartson II equations through an elegant iteration of the transformations. In [17], we successfully built Melnikov vectors with the gradients of Floquet discriminants. Since there is no Floquet discriminant for Davey-Stewartson equations (in contrast(More)
Let F be a C3 diffeomorphism on a Banach space B. F has a homoclinic tube asymptotic to an invariant manifold. Around the homoclinic tube, Bernoulli shift dynamics of submanifolds is established through shadowing lemma. This work removes an uncheckable condition of Silnikov [Equation (11), page 625 of L. P. Silnikov, Soviet Math. Dokl., vol.9, no.3, (1968),(More)
We report the synthesis of a graphene-sulfur composite material by wrapping poly(ethylene glycol) (PEG) coated submicrometer sulfur particles with mildly oxidized graphene oxide sheets decorated by carbon black nanoparticles. The PEG and graphene coating layers are important to accommodating volume expansion of the coated sulfur particles during discharge,(More)