On the absence of global solutions to two-times-fractional differential inequalities involving Hadamard-Caputo and Caputo fractional derivatives

<abstract><p>In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.</p></abstract>

Optimal Hardy-weights for the (p, A)-Laplacian with a potential term

We construct new optimal $L^{p}$ Hardy-type inequalities for elliptic Schrödinger-type operators with a potential term.

Linear and nonlinear low-frequency collisional quantum Buneman Instability in electron-positron-ion Plasmas

The linear and nonlinear low-frequency collisional quantum Buneman instability in electronpositron- ion plasmas have been studied. Buneman instability in low frequency three species quantum plasma has been investigated using the approach of the quantum hydrodynamic model. The one-dimensional low-frequency collisional model is revisited by introducing the Bohm potential term in the momentum equation along with the role of the positron. Low-frequency Buneman instability which arises by one stream of particles drifting over another is investigated in the presence of the positron. Different plasma configurations based on the relative velocities of streaming particles are analyzed and it is observed that positron content enhances the instability in classical limits. Further, we found that in pure quantum limits the instability growth rate is decreased by increasing the positron concentration. The present work is very useful for the nonlinear problems in Quantum Coulomb systems.

Motion of Charged Spinning Particles in a Unified Field

Using a geometry wider than Riemannian one, the parameterized absolute parallelism (PAP) geometry, we derived a new curve containing two parameters. In the context of the geometrization philosophy, this new curve can be used as a trajectory of charged spinning test particle in any unified field theory constructed in the PAP space. We show that imposing certain conditions on the two parameters, the new curve can be reduced to a geodesic curve giving the motion of a scalar test particle or/and a modified geodesic giving the motion of neutral spinning test particle in a gravitational field. The new method used for derivation, the Bazanski method, shows a new feature in the new curve equation. This feature is that the equation contains the electromagnetic potential term together with the Lorentz term. We show the importance of this feature in physical applications.

Spacetime estimates and scattering theory for quasilinear Schrodinger equations in arbitrary space dimension

In this paper, we consider Cauchy problem of a quasilinear Schrodinger equation which has general form containing potential term, power type nonlinearity and Hartree type nonlinearity. The space dimension is arbitrary, that is, it is larger than or equals to one. First, we establish the local wellposedness of the solution and discuss the condition on the global existence of the solution. Next, we establish some conservation laws such as mass conservation law, energy conservation law, pseudoconformal conservation law of the solution. Based on these conservation laws, we give Morawetz type estimates, spacetime bounds for the global solution. Last, we take two ideas to establish scattering theory for the global solution in different functional spaces. The first idea is that we take different admissible pairs in Strichartz estimates for different terms on the right side of Duhamel’s formula in order to keep each term independent, another one is that we factitiously let a continuous function be the sum of two piecewise functions and choose different admissible pairs in Strichartz estimates for the terms containing these functions.

Unifying dark matter and dark energy with non-canonical scalars

Non-canonical scalar fields with the Lagrangian $${{\mathcal {L}}} = X^\alpha - V(\phi )$$ L = X α - V ( ϕ ) , possess the attractive property that the speed of sound, $$c_s^{2} = (2\,\alpha - 1)^{-1}$$ c s 2 = ( 2 α - 1 ) - 1 , can be exceedingly small for large values of $$\alpha $$ α . This allows a non-canonical field to cluster and behave like warm/cold dark matter on small scales. We derive a general condition on the potential in order to facilitate the kinetic term $$X^\alpha $$ X α to play the role of dark matter, while the potential term $$V(\phi )$$ V ( ϕ ) playing the role of dark energy at late times. We demonstrate that simple potentials including $$V= V_0\coth ^2{\phi }$$ V = V 0 coth 2 ϕ and a Starobinsky-type potential can unify dark matter and dark energy. Cascading dark energy, in which the potential cascades to lower values in a series of discrete steps, can also work as a unified model.

BPS Skyrme submodels of the five-dimensional Skyrme model

In this paper, we search for the BPS skyrmions in some BPS submodels of the generalized Skyrme model in five-dimensional spacetime using the BPS Lagrangian method. We focus on the static solutions of the Bogomolny’s equations and their corresponding energies with topological charge B > 0 is an integer. We consider two main cases based on the symmetry of the effective Lagrangian of the BPS submodels, i.e. the spherically symmetric and non-spherically symmetric cases. For the spherically symmetric case, we find two BPS submodels. The first BPS submodels consist of a potential term and a term proportional to the square of the topological current. The second BPS submodels consist of only the Skyrme term. The second BPS submodel has BPS skyrmions with the same topological charge B > 1, but with different energies, that we shall call “topological degenerate” BPS skyrmions. It also has the usual BPS skyrmions with equal energies, if the topological charge is a prime number. Another interesting feature of the BPS skyrmions, with B > 1, in this BPS submodel, is that these BPS skyrmions have non-zero pressures in the angular direction. For the non-spherically symmetric case, there is only one BPS submodel, which is similar to the first BPS submodel in the spherically symmetric case. We find that the BPS skyrmions depend on a constant k and for a particular value of k we obtain the BPS skyrmions of the first BPS submodel in the spherically symmetric case. The total static energy and the topological charge of these BPS skyrmions also depend on this constant. We also show that all the results found in this paper satisfy the full field equations of motions of the corresponding BPS submodels.