Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger Equation with Complicated Potential

Tribelsky, Michael I.

doi:10.3390/physics3030043

Cited by 2 publications

(1 citation statement)

References 23 publications

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“…It is hardly possible to overrate the importance of the Schrödinger equation (SE). Being the keystone of entire non-relativistic quantum mechanics, it also arises in many quite classical problems, see, e.g., [1][2][3][4][5]. Among the vast diversity of problems related to the SE and its applications, there are several revealing its fundamental properties.…”

Section: Introductionmentioning

confidence: 99%

Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions

Tribelsky¹

2022

Preprint

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A fall of a particle to the center of a singular potential is one of a few fundamental problems of quantum mechanics. Nonetheless, its solution is not complete yet. The known results just indicate that if the singularity of the potential is strong enough, the spectrum of the Schrodinger equation is not bounded from below. However, the wave functions of the problem do not admit the limiting transition to the ground state. Therefore, the unboundedness of the spectrum is only a necessary condition. To prove that a quantum particle indeed can fall to the center, a wave function describing the fall should be obtained explicitly. This is done in the present paper. Firstly, classical collapse is inspected. Then, the quantum problem is analyzed. A family of exact self-similar solutions of the time-dependent Schrodinger equation corresponding to the fall is obtained and discussed. A law for the collapse of the region of the wave function localization to a single point is obtained explicitly. It is shown that the known necessary conditions for the particle to fall simultaneously are sufficient. Comparison of the classical and quantum collapse exhibits striking similarity of the two cases.

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Section: Introductionmentioning

confidence: 99%

Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions

Tribelsky¹

2022

Preprint

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Periodic and Axial Perturbations of Chaotic Solitons in the Realm of Complex Structured Quintic Swift-Hohenberg Equation

Iqbal,

Mohammed,

Alqudah

et al. 2024

MCA

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This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems that produce patterns. We have found that there exist numerous chaotic soliton solutions with periodic and axial perturbations to the intended CQSHE, provided that the coefficients are constrained by certain conditions. Furthermore, by applying a sophisticated transformation, the provided transformative approach RMESEM transforms CQSHE into a set of Nonlinear Ordinary Differential Equations (NODEs). The resulting set of NODEs is then transformed into an algebraic system of equations by incorporating the extended Riccati NODE to assume a series form solution. The soliton solutions to this system of equations can be found as periodic, hyperbolic, exponential, rational-hyperbolic, and rational families of functions. A variety of 3D and contour visuals are also provided to graphically illustrate the axially and periodically perturbed dynamics of these chaotic soliton solutions and the formation of fractals. Our findings are noteworthy because they shed light on the chaotic nature of the framework we are examining, enabling us to better understand the dynamics that underlie it.

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Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger Equation with Complicated Potential

Cited by 2 publications

References 23 publications

Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions

Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions

Periodic and Axial Perturbations of Chaotic Solitons in the Realm of Complex Structured Quintic Swift-Hohenberg Equation

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