Poisson System Research Articles

Solutions of a quasilinear Schrödinger–Poisson system with linearly bounded nonlinearities

Solutions of a quasilinear Schrödinger–Poisson system with linearly bounded nonlinearities

Mass concentration phenomenon in the 3D bipolar compressible Navier–Stokes–Poisson system

Mass concentration phenomenon in the 3D bipolar compressible Navier–Stokes–Poisson system

On the Existence of Global Solutions for the Vlasov–Poisson System in a Half-space and Plasma Confinement

We consider the Vlasov–Poisson system with external magnetic field in a half-space with the Neumann boundary condition for the electric potential and specular reflection on a boundary. For arbitrary compactly supported initial density distribution functions, we obtain sufficient conditions for external magnetic field, which provide global existence of density distribution functions with compact supports lying at some distance from a boundary.

Existence and asymptotic behavior of positive solutions to some logarithmic Schrödinger–Poisson system

Existence and asymptotic behavior of positive solutions to some logarithmic Schrödinger–Poisson system

Zero-electron-mass and quasi-neutral limits for bipolar Euler–Poisson systems

Zero-electron-mass and quasi-neutral limits for bipolar Euler–Poisson systems

Elucidating Super-Compressible States in Inner Structure of Vehicular Flow

The article deals with the open questions in the theory of vehicular headway modeling. Specifically, the question of the existence of anomalous constellations in vehicular traffic micro-structure, in which the rate of fluctuations (measured by the stochastic compressibility) exceeds the fluctuation level of systems with non-interacting elements. The solution to this open problem is converted into the mathematical format working with the so-called balance particle systems, where seeking relevant relations is more straightforward and thus significantly easier. Presented research has shown that unit compressibility represents (despite popular opinion) the upper limit only for particle systems, in which there is no attractive interaction between the particles. In the article, the specific system is constructed in which the presence of an attractive force component will cause higher fluctuations than in the Poisson systems of non-interacting elements. This means that traffic constellations with higher compressibility (so-called super-compressible constellations) can be explained either by a discrepancy between the empirical traffic flow and the mathematical model used, or by the presence of attractive forces acting between individual vehicles. Using empirical vehicular data (measured on two parallel freeway lanes under reconstruction), we show that super-compressible states occur even though overtaking is prohibited. This means, therefore, that these super-compressible states arose without a doubt due to the mutual attraction of successive vehicles. In addition, the article shows that the presence of the aforementioned attractive forces appears predominantly in the fast lane, and only in situations where the traffic density is relatively low. At higher densities, the two freeway lanes are markedly synchronized, the opportunity for a sporty style of driving vanishes and the reason for changing lanes disappears. Under these circumstances, the attractive force component vanishes, which finally leads to the transition of the entire traffic system back to standard sub-compressible states.

Semiclassical solutions for a critical Choquard–Poisson system with competitive potentials

In this article, we establish the existence, concentration behaviour and decay estimation of semiclassical ground state solutions for a critical Choquard–Poisson system with multiple competitive potentials by variational methods. Moreover, we study the multiplicity of solutions by using the Lusternik–Schnirelmann theory.

Existence and multiplicity of non-trivial solutions for fractional Schrödinger–Poisson systems with a combined nonlinearity

Existence and multiplicity of non-trivial solutions for fractional Schrödinger–Poisson systems with a combined nonlinearity

On the logarithmic fractional Schrödinger–Poisson system with saddle-like potential

In this paper, we use variational methods to prove the existence of a positive solution for the following class of logarithmic fractional Schrödinger–Poisson system: { ϵ 2 s ( − Δ ) s u + V ( x ) u − ϕ ( x ) u = u log ⁡ u 2 in R 3 , ϵ 2 t ( − Δ ) t ϕ = | u | 2 in R 3 , where ϵ > 0 , s , t ∈ ( 0 , 1 ) , ( − Δ ) α is the fractional Laplacian and V is a saddle-like potential.

Existence of solution for a generalized Schrödinger–Poisson system via bifurcation theory

Existence of solution for a generalized Schrödinger–Poisson system via bifurcation theory

Bifurcation analysis of fractional Kirchhoff–Schrödinger–Poisson systems in R 3

In this paper, we investigate the bifurcation results of the fractional Kirchhoff–Schrödinger–Poisson system { M ( [ u ] s 2 ) ( − Δ ) s u + V ( x ) u + ϕ ( x ) u = λ g ( x ) | u | p − 1 u + | u | 2 s ∗ − 2 u i n R 3 , ( − Δ ) t ϕ ( x ) = u 2 i n R 3 , where s , t ∈ ( 0 , 1 ) with 2 t + 4 s > 3 and the potential function V is a continuous function. We show that the existence of components of (weak) solutions of the above equation bifurcates out from the first eigenvalue λ 1 of the problem ( − Δ ) s u + V ( x ) u = λ g ( x ) u in R 3 . The main feature of this paper is the inclusion of a potentially degenerate Kirchhoff model, combined with the critical nonlinearity.

Existence of Ground State Solutions for the Schrödinger–Poisson System in $\mathbb{R}^{2}$

Existence of Ground State Solutions for the Schrödinger–Poisson System in $\mathbb{R}^{2}$

Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth

Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth

On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field

We consider the first mixed problem for the Vlasov–Poisson system with an external magnetic field in a domain with piecewise smooth boundary. This problem describes the kinetics of a two-component high-temperature plasma under the influence of a self-consistent electric field and an external magnetic field. The existence of global weak solutions is proved. In the case of a cylindrical domain, sufficient conditions are obtained for the existence of global weak solutions with supports in a strictly internal cylinder; this corresponds to the confinement of high-temperature plasma in a mirror trap.

Global convergence of a two-fluid non-isentropic Euler–Poisson system in one space dimension

In this paper, we study the global in time quasi-neutral limit for a two-fluid non-isentropic Euler–Poisson system in one space dimension. We prove that the system converges to the non-isentropic Euler equations as the Debye length tends to zero. This problem is studied for smooth solutions near the constant equilibrium state. To prove this result, we establish uniform energy estimates and various dissipation estimates with respect to the Debye length and the time. These estimates allow to pass to the limit to obtain the limit system by compactness arguments. In addition, the global convergence rate is obtained by use of stream function technique.

Suppressing instability in a Vlasov–Poisson system by an external electric field through constrained optimization

Maintaining the stability and shape of a plasma is a crucial task in many technological applications ranging from beam shaping to fusion energy. This is often challenging as plasma systems tend to be naturally unstable and kinetic effects can play an important role in the behavior of the instabilities. Due to the large energies involved, the primary way to control plasma systems is the application of external electromagnetic fields.In this work, we deploy a PDE-constrained optimization formulation that uses a kinetic description for plasma dynamics as the constraint. This is to optimize, over all possible controllable external electric fields, the stability of the plasma dynamics under the condition that the Vlasov–Poisson (VP) equation is satisfied. Mathematically we formulate it as a PDE-constrained optimization. For computing the functional derivative with respect to the external field in the optimization updates, the adjoint equation is derived. Furthermore, in the discrete setting, where we employ the semi-Lagrangian method as the forward solver, we also explicitly formulate the corresponding adjoint solver and the gradient as the discrete analogy to the adjoint equation and the Fréchet derivative. A distinct feature we observed of this constrained optimization is the complex landscape of the objective function and the existence of numerous local minima, largely due to the hyperbolic nature of the VP system. To overcome this issue, we utilize a gradient-accelerated genetic algorithm, leveraging the advantages of the genetic algorithm's exploration feature to cover a broader search of the solution space and the fast local convergence aided by the gradient information. We show that our algorithm obtains good electric fields that are able to maintain a prescribed profile in a beam shaping problem and uses nonlinear effects to suppress plasma instability in a two-stream configuration.

On Global Classical Limit of the (Relativistic) Vlasov–Maxwell System

It is shown that solutions of the (relativistic) Vlasov–Maxwell system converge pointwise to solutions of the Vlasov–Poisson system globally in time at the asymptotic rate of c^{-1}, as the light speed c tends to infinity, which extends the results of Asano and Ukai (Stud Math Appl 18:369–383, 1986), Degond (Math Methods Appl Sci 8:533–558, 1986) and Schaeffer (Commun Math Phys 104:403–421, 1986). The analysis relies on the method of Glassey and Strauss (Commun Math Phys 113:191–208, 1987) and Schaeffer (Commun Math Phys 104:403–421, 1986).

Bifurcation and regularity of entire solutions for the planar nonlinear Schrödinger–Poisson system

Bifurcation and regularity of entire solutions for the planar nonlinear Schrödinger–Poisson system

Learning Poisson Systems and Trajectories of Autonomous Systems via Poisson Neural Networks.

We propose the Poisson neural networks (PNNs) to learn Poisson systems and trajectories of autonomous systems from data. Based on the Darboux-Lie theorem, the phase flow of a Poisson system can be written as the composition of: 1) a coordinate transformation; 2) an extended symplectic map; and 3) the inverse of the transformation. In this work, we extend this result to the unknotted trajectories of autonomous systems. We employ structured neural networks with physical priors to approximate the three aforementioned maps. We demonstrate through several simulations that PNNs are capable of handling very accurately several challenging tasks, including the motion of a particle in the electromagnetic potential, the nonlinear Schrödinger equation, and pixel observations of the two-body problem.