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Register a new userNonlocal phase-field systems with general potentials
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We consider a phase-field model where the internal energy depends on the order parameter in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for the order parameter. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, in the case of a potential defined on (-1,1) and singular at the endpoints, the existence of a finite-dimensional global attractor has been proven. Here we examine both the case of smooth potentials as well as the case of physically realistic (e.g., logarithmic) singular potentials. We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional attractors in the present cases as well.
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In present paper, we prove the existence of solutions $(lambda_1,lambda_2, u_1,u_2)in R^2times H^1(R^N, R^2)$ to systems of nonlinear Schrodinger equations with potentials $$begin{cases} -Delta u_1+V_1(x)u_1+lambda_1 u_1=partial_1 G(u_1,u_2);quad&hbox{in};R^N -Delta u_2+V_2(x)u_2+lambda_2 u_2=partial_2G(u_1,u_2);quad&hbox{in};R^N 0<u_1,u_2in H^1(R^N), Ngeq 1 end{cases}$$ satisfying the normalization constraints $int_{R^N}u_1^2dx=a_1$ and $int_{R^N}u_2^2dx=a_2$, which appear in mean-field models for binary mixtures of Bose-Einstein condensates or models for binary mixtures of ultracold quantum gases of fermion atoms. The potentials $V_iota(x) (iota=1,2)$ are given functions. The nonlinearities $G(u_1,u_2)$ are considered of the form $$ begin{cases} G(u_1, u_2):=sum_{i=1}^{ell}frac{mu_i}{p_i}|u_1|^{p_i}+sum_{j=1}^{m}frac{ u_j}{q_j}|u_2|^{q_j}+sum_{k=1}^{n}beta_k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}}, mu_i, u_j,beta_k>0, ~ p_i, q_j>2, ~r_{1,k}, r_{2,k}>1. end{cases} $$ Under some assumptions on $V_iota$ and the parameters, we can prove the strict binding inequality for the mass sub-critical problem and obtain the existence of ground state normalized solutions for any given $a_1>0,a_2>0$.
The Gross-Pitaevskii equation is a widely used model in physics, in particular in the context of Bose-Einstein condensates. However, it only takes into account local interactions between particles. This paper demonstrates the validity of using a nonlocal formulation as a generalization of the local model. In particular, the paper demonstrates that the solution of the nonlocal model approaches in norm the solution of the local model as the nonlocal model approaches the local model. The nonlocality and potential used for the Gross-Pitaevskii equation are quite general, thus this paper shows that one can easily add nonlocal effects to interesting classes of Bose-Einstein condensate models. Based on a particular choice of potential for the nonlocal Gross-Pitaevskii equation, we establish the orbital stability of a class of parameter-dependent solutions to the nonlocal problem for certain parameter regimes. Numerical results corroborate the analytical stability results and lead to predictions about the stability of the class of solutions for parameter values outside of the purview of the theory established in this paper.
In this paper we derive, starting from the basic principles of Thermodynamics, an extended version of the nonconserved Penrose-Fife phase transition model, in which dynamic boundary conditions are considered in order to take into account interactions with walls. Moreover, we study the well-posedness and the asymptotic behavior of the Cauchy problem for the PDE system associated to the model, allowing the phase configuration of the material to be described by a singular function.
We investigate the validity of a soliton dynamics behavior in the semi-relativistic limit for the nonlinear Schrodinger equation in $R^{N}, Nge 3$, in presence of a singular external potential.
We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.
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