Existence result for fractional Schrödinger–Poisson systems involving a Bessel operator without Ambrosetti–Rabinowitz condition

The present study is concerned with the nontrivial solutions for fractional Schrödinger–Poisson system with the Bessel operator. Under certain assumptions on the nonlinearity $f$, a nontrivial nonnegative solution is obtained by perturbation method for the given problem. In particular, the Ambrosetti–Rabinowitz type condition or the monotone assumption on the nonlinearity is unnecessary.     

Multiple positive solutions for a Schrödinger–Newton system with sign-changing potential

In this paper, we study the Schrödinger–Newton systems with sign-changing potential in a bounded domain. By using the variational method and analytic techniques, the existence and multiplicity of positive solutions are established.     

Existence of the global solution for fractional logarithmic Schrödinger equation

In this paper we consider the initial boundary value problem for a class of fractional logarithmic Schrödinger equation. By using the fractional logarithmic Sobolev inequality and introducing a family of potential wells, we give some properties of the family of potential wells and obtain existence of global solution.     

Stability of standing waves for the fractional Schrödinger–Choquard equation

In this paper, we consider the stability of standing waves for the fractional Schrödinger–Choquard equation with an $L^2$-critical nonlinearity. By using the profile decomposition of bounded sequences in $H^s$ and variational methods, we prove that the standing waves are orbitally stable. We extend the study of Bhattarai for a single equation (Bhattarai, 2017) to the $L^2$-critical case.     

Lyapunov–Sylvester computational method for numerical solutions of a mixed cubic-superlinear Schrödinger system

Engineering with Computers - An upwind skewed radial basis function (USRBF)-based solution scheme is presented for stabilized solutions of convection-dominated problems over meshfree nodes. The...     

Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions

Systems of coupled non-linear Schrödinger equations with soliton solutions are integrated using the six-point scheme which is equivalent to the multi-symplectic Preissman scheme. The numerical dispersion relations are studied for the linearized equation. Numerical results for elastic and inelastic soliton collisions are presented. Numerical experiments confirm the excellent conservation of energy, momentum and norm in long-term computations and their relations to the qualitative behaviour of the soliton solutions.     

Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.     

The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator

The present work is mainly devoted to studying the fractional nonlinear Schrödinger equation with wave operator. We first derive two conserved quantities of the equation, and then develop a three-level linearly implicit difference scheme. This scheme is shown to be conserves the discrete version of conserved quantities. Using energy method, we prove that the difference scheme is unconditionally stable, and the difference solution converges to the exact one with second order accuracy in both the space and time dimensions. Numerical experiments are performed to support our theoretical analysis and demonstrate the accuracy, discrete conservation laws and effectiveness for long-time simulation.     

Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation

A linearized Crank–Nicolson Galerkin finite element method with bilinear element for nonlinear Schrödinger equation is studied. By splitting the error into two parts which are called the temporal error and the spatial error, the unconditional superconvergence result is deduced. On one hand, the regularity for a time-discrete system is presented based on the proof of the temporal error. On the other hand, the classical Ritz projection is applied to get the spatial error with order \(O(h^2)\) in \(L^2\)-norm, which plays an important role in getting rid of the restriction of \(\tau \). Then the superclose estimates of order \(O(h^2+\tau ^2)\) in \(H^1\)-norm is arrived at based on the relationship between the Ritz projection and the interpolated operator. At the same time, global superconvergence property is arrived at by the interpolated postprocessing technique. At last, three numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and \(\tau \) is the time step.     

Explicit approximate controllability of the Schrödinger equation with a polarizability term

We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts nonlinearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system with explicit controls. Numerical simulations are presented to illustrate those theoretical results.     

Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

A sinc-method computation for eigenvalues of Schrödinger operators with eigenparameter-dependent boundary conditions

We implement the sinc method to compute the eigenvalues of a second order boundary value problem with mixed type boundary conditions where the eigenparameter appears linearly in the boundary conditions. We investigate the behavior of the solutions as well as the characteristic determinant via successive iterations. The method is implemented by splitting the characteristic determinant into two parts, where it is proved that the unknown one lies in a Paley-Wiener space and it is approximated by an interpolation sampling theorem. Examples are illustrated numerically and graphically.