Nonlinear Schr�dinger equation with variable coefficients: Concentrated solution and its breakdown

In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in grapheneand in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral twodimensionalSchrÃ¶dinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection isexactly solved. The solutions of SchrÃ¶dinger equation in momentum space in studied materials by projection the twodimensionalspace of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral twodimensionalSchrÃ¶dinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. Instudied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and lightabsorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole areseparated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materialsallows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone andhence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a widthof band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of theequation of motion for a microscopic dipole of graphene where >1 s r . The absorption spectrums for the zinc-blendeGaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with availableexperimental data shows good agreement.

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Convergence of Finite Fifference Method for Parabolic Problem

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2003-0005 ◽

2003 ◽

Vol 3 (1) ◽

pp. 45-58 ◽

Cited By ~ 2

Author(s):

Dejan Bojović

Keyword(s):

Convergence Rate ◽

Boundary Value Problem ◽

Heat Equation ◽

Parabolic Problem ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Variable Coefficients ◽

Rate Estimate ◽

Convergence Rate Estimate ◽

Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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General Trend of Negative Transference Number in Li Salt/Ionic Liquid Mixtures

10.26434/chemrxiv.7728545 ◽

2019 ◽

Cited By ~ 1

Author(s):

Nicola Molinari ◽

Jonathan P. Mailoa ◽

Boris Kozinsky

Keyword(s):

Ionic Liquid ◽

Transference Number ◽

Liquid Mixtures ◽

Single Linkage ◽

Self Diffusion ◽

Wide Range ◽

Single Linkage Cluster ◽

Concentrated Solution ◽

The One ◽

Dynamics Simulations

We show that strong cation-anion interactions in a wide range of lithium-salt/ionic liquid mixtures result in a negative lithium transference number, using molecular dynamics simulations and rigorous concentrated solution theory. This behavior fundamentally deviates from the one obtained using self-diffusion coefficient analysis and agrees well with experimental electrophoretic NMR measurements, which accounts for ion correlations. We extend these findings to several ionic liquid compositions. We investigate the degree of spatial ionic coordination employing single-linkage cluster analysis, unveiling asymmetrical anion-cation clusters. Additionally, we formulate a way to compute the effective lithium charge that corresponds to and agrees well with electrophoretic measurements and show that lithium effectively carries a negative charge in a remarkably wide range of chemistries and concentrations. The generality of our observation has significant implications for the energy storage community, emphasizing the need to reconsider the potential of these systems as next generation battery electrolytes.<br>

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