Piotr Rozmej

Study of the potential energy of “octupole”-deformed nuclei in a multidimensional deformation space

Abstract The collective potential energy of even-even “octupule”-deformed nuclei is studied in a multidimensional deformation space in both radium and barium regions. This energy is calculated by the macroscopic-microscopic method, with the Yukawa-plus-exponential model taken for the macroscopic part and the Strutinski shell correction (based on the Woods-Saxon single-particle potential) used for the microscopic part of the energy. The deformations βλ of all multipolarity degrees: λ = 2, 3, …, 7 (or even 8) are treated as independent variables. The multipolarities: λ = 5, 6 and 7, usually omitted or treated in an average way up to now, are found to be important for the properties of the nuclei.

Energy invariant for shallow-water waves and the Korteweg-de Vries equation: Doubts about the invariance of energy.

It is well known that the Korteweg-de Vries (KdV) equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum, and energy. Here we try to answer the question of how this comes about and also how these KdV quantities relate to those of the Euler shallow-water equations. Here Lukes Lagrangian is helpful. We also consider higher-order extensions of KdV. Though in general not integrable, in some sense they are almost so within the accuracy of the expansion.

A new nonlinear equation in the shallow water wave problem

In this paper, a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid for weakly nonlinear, dispersive, and long wavelength limits. Some examples of soliton motion for various bottom shapes obtained in numerical simulations according to the derived equation are presented.In the paper a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid in the weakly nonlinear, dispersive and long wavelength limit. Some examples of soliton motion for various bottom shapes obtained in numerical simulations according to the derived equation are presented.

Numerical solutions to integral equations equivalent to differential equations with fractional time

Numerical solutions to integral equations equivalent to differential equations with fractional time This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

Stability and instability of a hot and dilute nuclear droplet

The diabatic approach to collective nuclear motion is reformulated in the local-density approximation in order to treat the normal modes of a spherical nuclear droplet analytically. In a first application the adiabatic isoscalar modes are studied and results for the eigenvalues of compressional (bulk) and pure surface modes are presented as function of density and temperature inside the droplet, as well as for different mass numbers and for soft and stiff equations of state. We find that the region of bulk instabilities (spinodal regime) is substantially smaller for nuclear droplets than for infinite nuclear matter. For small densities below 30% of normal nuclear matter density and for temperatures below 5 MeV all relevant bulk modes become unstable with the same growth rates. The surface modes have a larger spinodal region, reaching out to densities and temperatures way beyond the spinodal line for bulk instabilities. Essential experimental features of multifragmentation, like fragmentation temperatures and fragment-mass distributions (in particular the power-law behavior) are consistent with the instability properties of an expanding nuclear droplet, and hence with a dynamical fragmentation process within the spinodal regime of bulk and surface modes (spinodal decomposition).

A Finite Element Method for Extended KdV Equations

Abstract The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov–Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.

Sharing of excitation energy in dissipative nucleus-nucleus collisions

Abstract The sharing of the total excitation energy on both fragments in dissipative nucleus-nucleus collisions is studied on the basis of the dissipative diabatic dynamics. Numerical calculations are performed for the systems 98 Mo + 147 Sm and 98 Mo + 238 U. In agreement with experimental results the excitation energy per particle is larger for the lighter fragment and increases with increasing asymmetry of the system. The dependence on the total excitation energy is qualitatively different from that for the one-body dissipation model.

Adiabatic invariants of the extended KdV equation

Abstract When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present a direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from constant values are indeed small.

Numerical Solutions to Fractional Perturbed Volterra Equations

In the paper, a class of perturbed Volterra equations of convolution type with three kernel functions is considered. The kernel functions , , , correspond to the class of equations interpolating heat and wave equations. The results obtained generalize our previous results from 2010.

NON-AXIAL OCTUPOLE DEFORMATION OF A HEAVY NUCLEUS

The effect of the non-axial octupole deformation a32(Y32+Y3-2) of heavy nuclei on their potential energy is studied. The study is performed within a macroscopic-microscopic approach. It is found that the largest effect appears for the nucleus 238Fm and consists in a lowering of the energy by more than 3 MeV with respect to the energy at the spherical configuration.