A model for one-dimensional bistable systems characterized by a deformable double-well energy landscape is introduced in order to investigate the effect of shape deformability on the order of phase transition in quantum tunneling and on the quasi-exact integrability of the classical statistical mechanics of these systems. The deformable double-well energy landscape is modeled by a parameterized double-well potential possessing two fixed degenerate minima and a constant barrier height, but a tunable shape of its walls which affects the confinement of the two wells. It is found that unlike bistable models involving the standard 4 -field model for which the transition in quantum tunneling is predicted to be strictly of second order, a parameterization of the double-well potential also favors a first-order transition occurring above a universal critical value of the shape deformability parameter. The partition function of the model is constructed within the framework of the transfer integral formalism, with emphasis on low-lying eigenstates of the transfer integral operator. Criteria for quasi-exact integrability of the partition function were formulated, in terms of the condition for possible existence of exact eigenstates of the transfer integral operator. The quasi-exact solvability condition is obtained analytically, and from this, some exact eigenstates are derived at several temperatures. The exact probability densities obtained from the analytical expressions of the ground state wavefunctions at different temperatures are found to be in excellent agreement with the probability density obtained from numerical simulations of the Fokker-Planck equation.
Recent studies have emphasized the important role that a shape deformability of scalar-field models pertaining to the same class with the standard [Formula: see text] field, can play in controlling the production of a specific type of breathing bound states so-called oscillons. In the context of cosmology, the built-in mechanism of oscillons suggests that they can affect the standard picture of scalar ultra-light dark matter. In this paper, kink scatterings are investigated in a parametrized model of bistable system admitting the classical [Formula: see text] field as an asymptotic limit, with focus on the formation of long-lived low-amplitude almost harmonic oscillations of the scalar field around a vacuum. The parametrized model is characterized by a double-well potential with a shape-deformation parameter that changes only the steepness of the potential walls, and hence the flatness of the hump of the potential barrier, leaving unaffected the two degenerate minima and the barrier height. It is found that the variation of the deformability parameter promotes several additional vibrational modes in the kink-phonon scattering potential, leading to suppression of the two-bounce windows in kink–antikink scatterings and the production of oscillons. Numerical results suggest that the anharmonicity of the potential barrier, characterized by a flat barrier hump, is the main determinant factor for the production of oscillons in double-well systems.
We consider a one-dimensional system of interacting particles (which can be atoms, molecules, ions, etc.), in which particles are subjected to a bistable potential the double-well shape of which is tunable via a shape deformability parameter. Our objective is to examine the impact of shape deformability on the order of transition in quantum tunneling in the bistable system, and on the possible existence of exact solutions to the transfer-integral operator associated with the partition function of the system. The bistable potential is represented by a class composed of three families of parametrized double-well potentials, whose minima and barrier height can be tuned distinctly. It is found that the extra degree of freedom, introduced by the shape deformability parameter, favors a first-order transition in quantum tunneling, in addition to the second-order transition predicted with the $$\phi ^4$$ ϕ 4 model. This first-order transition in quantum tunneling, which is consistent with Chudnovsky’s conjecture of the influence of the shape of the potential barrier on the order of thermally assisted transitions in bistable systems, is shown to occur at a critical value of the shape-deformability parameter which is the same for the three families of parametrized double-well potentials. Concerning the statistical mechanics of the system, the associate partition function is mapped onto a spectral problem by means of the transfer-integral formalism. The condition that the partition function can be exactly integrable, is determined by a criterion enabling exact eigenvalues and eigenfunctions for the transfer-integral operator. Analytical expressions of some of these exact eigenvalues and eigenfunctions are given, and the corresponding ground-state wavefunctions are used to compute the probability density which is relevant for calculations of thermodynamic quantities such as the correlation functions and the correlation lengths. Graphic Abstract
The modulational instability properties regarding the evolution of interfacial disturbances of the flow of a thin liquid film down an inclined uniformly heated plate subject to thermal Marangoni (thermocapillary) effects are investigated under the framework of linear stability analysis. The investigation has been performed both analytically and numerically using a complex cubic Ginzburg–Landau equation without the driving term to provide comprehensive pictures of the influence of nonlinearity, dissipation, and dispersion on interfacial disturbance generation and evolution. It is shown that when the interplay between linear and nonlinear effects is relatively important, the disturbances evolve as a superposition of groups of traveling periodic waves with different amplitudes, and the interfacial disturbances evolve as smooth modulations. Furthermore, the dynamic modes of these disturbances become aperiodic. Conversely, when the evolution of instabilities is influenced by strong nonlinearity, the flow saturates, and different situations lead to different possible modulated wavy structures, caused by the interplay between nonlinear and linear dispersive and dissipative effects. Moreover, the appearance and the spatial and temporal evolution of the modulated disturbance profiles are influenced by both the amplitude of the disturbances and the linear dissipative term. Here, based on our investigation, two cases are highlighted. In the first case, which corresponds to very small amplitude of the disturbances, the dynamic modes of the disturbances evolve from periodic traveling waves to spatial and temporal modulated periodic solitary wave patterns. In the second case, by increasing the amplitude of the disturbances, the appearance of modulational modes is rapid, and therefore, we can observe the development of modulationally marginal-like stable patterns or spatial and temporal modulated patterns with nonuniform profiles.
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