We perform complete group classification of the general class of quasi linear wave equations in two variables. This class may be seen as a broad generalization of the nonlinear d'Alembert, Liouville, sin/sinh-Gordon and Tzitzeica equations. In this way we derived a number of new genuinely nonlinear invariant models with high symmetry properties. In particular, we obtain four classes of nonlinear wave equations admitting five-dimensional invariance groups. Applying the symmetry reduction technique we construct multi-parameter families of exact solutions of these equations.
A controllable variation of the photoluminescence (PL) intensity of semiconductor quantum dots (QDs) via their coupling to plasmonic nanoparticles (NPs) is the potential basis for optoelectronic and sensing applications. In this work, the effect of Au NPs on the PL of colloidal CdTe QDs is investigated in solution and solid films. An PL enhancement for the QDs synthesized in water was observed in case of spectral overlap of the plasmon absorption band of and QD PL band. In case of Au NPs synthesized in dimethyl sulfoxyl the trend is to reduction of the PL intensity. For the reference samples prepared by mixing QDs not with Au NP solutions but with corresponding pure solvent, certain PL enhancement was observed and presumably attributed to reduction of self-absorption or non-radiative interparticle interaction in less concentrated QD solution. However, the contribution of this dilution-related enhancement is expected to be independent of the spectral properties of NPs and QDs. Therefore, the observed in this work different behavior of QD PL in certain combinations of QDs and NPs is attributed to interaction between electronic excitation in the QD and plasmon.
Transformation properties of a class of generalized Kawahara equations with time-dependent coefficients are studied. We construct the equivalence groupoid of the class and prove that this class is not normalized but can be presented as a union of two disjoint normalized subclasses. Using the obtained results and properly gauging the arbitrary elements of the class, we carry out its complete group classification, which covers gaps in the previous works on the subject.
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