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A new higher order Schrödinger equation characterized by a position-dependent mass is introduced based on long-range spatial kernel effects and von Roos arguments. The extended Schrödinger equation depends on the sign of the moments M k , k = 0 , 1 , 2 , … and a stabilized quantum dynamic is realized for M 2 > 0 and M 4 > 0 . We have discussed its implications in several quantum mechanical systems where more than a few were raised, mainly the emergence of the quantum Pais-Uhlenbeck and the relativistic quantum harmonic oscillators besides the complex periodic potential characterized by a PT symmetry.

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev–Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

In this work, we study the physics of plasma waves and magnetohydrodynamic (MHD) equilibrium of sunspots based on the concept of non-standard Lagrangians which play an important role in several branches of science. We derived the modified fluid equations from the Maxwell–Vlasov equation using the moment conventional procedure. Several new interaction terms between physical quantities arise in the non-standard MHD (NS-MHD) equations that give rise to additional features in plasma MHD. A number of fundamental problems in plasma physics are discussed including the non-relativistic dynamics of inviscid fluid subject to the gravitational field, linear waves in plasma MHD and MHD equilibrium of sunspots. For the case of magnetoacoustic wave, it was observed that the NS-MHD equations modify the dispersion relation and its corresponding velocity depends on the sign (positive or negative) of the free parameters introduced in the theory. The non-standard Alfvén velocity is greater than the standard Alfvén velocity for the negative sign and smaller for the positive sign. Besides, in the MHD equilibrium of sunspots, non-standard MHD extends the conventional problem by adding several constraints that lead to an emergence of very low temperature inside the magnetic flux tube comparable to what is observed in low-temperature superconductors. Additional consequences are discussed accordingly.

Superconductivity is analysed based on the product-like fractal measure approach with fractal dimension α introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media. Our study shows the emergence of a massless state at the boundary of the superconductor and the simultaneous occurrence of isothermal and adiabatic processes in the superconductor depending on the position of the electrons. Several physical quantities were found to be position-dependent comparable with those arising in heavy doping and p–n junction. At the boundary of the superconductor, a shrinkage of the magnetic field was observed, leading to a scenario equivalent to the Meissner–Ochsenfeld effect. An enhancement of the London penetration depth is revealed and such an improvement was observed in pnictides at the onset of commensurate spin-density-wave order inside the superconducting phase at zero temperature. The Bardeen–Cooper–Schrieffer theory was also analysed and the appearance of zero-energy states is detected. Nucleation of superconductivity in a bulk was also studied. The system acts as a quantum damped harmonic oscillator and our analysis showed that type-I superconductivity occurs when κ < 2 / ( 1 + α ) , whereas type II occurs for κ > 2 / ( 1 + α ) , where κ is the Ginzburg–Landau parameter. The transition at the passage from the ‘genuine’ to the ‘intermediate’ type-I estimates 0.767767 < α ≤ 1 .

A new generalized uncertainty relation is constructed based on Li-Ostoja-Starzewski fractional gradient operator of order 0 α ≤ 1 introduced recently in literature which is motivated from dimensional regularization method. The new generalized uncertainty relation leads to a new form of Schrödinger equation and emergent position-dependent mass. Special forms of position-dependent mass were studied and the problem of a particle in a box was explored. Dissimilar forms of allowed quantum energies were obtained which lead to different scenarios. We have confronted the theory with observations by evaluating the maximum wavelength obtained in the 1,3-butadiene molecule. Several features were discussed accordingly.