Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.     

A note on existence of antisymmetric solutions for a class of nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation

$$-\triangle u + V(x)u= f(u)\quad {\rm in}\quad \mathbb{R}^N.$$

We assume that V is invariant under an orthogonal involution and show the existence of a particular type of sign changing solution. The basic tool employed here is the Concentration–Compactness Principle.

     

Stability of nonlinear Schrödinger equations on modulation spaces

We consider the Cauchy problem for a family of SchrSdinger equations with initial data in modulation spaces Mp,1^s. We develop the existence, uniqueness, blowup criterion, stability of regularity, scattering theory, and stability theory.     

Remarks on some systems of nonlinear Schrödinger equations

We prove the existence of solutions of some nonautonomous systems of nonlinear Schr?dinger equations, by means of perturbation techniques. The work has been supported by M.U.R.S.T. under the national project “Variational methods and nonlinear differential equations”.     

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.     

Loss of regularity for supercritical nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functional à la Brenier.     

On the new critical exponent for the nonlinear Schrödinger equations

We consider the Cauchy problem for the pth order nonlinear Schrödinger equation in one space dimension $$\left\{\begin{array}{ll}iu_{t} + \frac{1}{2} u_{xx} = |u|^{p}, x \in {\bf R}, \, t > 0, \\ \qquad u(0, x) = u_{0} (x), \; x \in {\bf R},\end{array}\right.$$ where \({p > p_{s} = \frac{3 + \sqrt{17}}{2}}\) . We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if p _s < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4.     

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

We present a class of orthogonal functions on infinite domain based on Jacobi polynomials. These functions are generated by applying a tanh transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo-derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced tanh Jacobi functions, we develop an efficient spectral tanh Jacobi collocation method for the numerical simulation of nonlinear Schrödinger equations on the infinite domain without using artificial boundary conditions. The applicability and accuracy of the solution method are demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg–Landau equation.     

A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations

Numerical Algorithms - In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and a...     

Blow-up solutions of inhomogeneous nonlinear Schrödinger equations on torus

In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation${\mathrm{\partial }}_{t}u=i(f(x)\mathrm{\Delta }u+\nabla f(x)·\nabla u+k(x)|u{|}^{2}u)$on ${\mathbb{T}}^2$. We present the $L^2$-concentration property for general initial data and investigate the $L^2$-minimality.     

A variant of Newton’s method based on Simpson’s three-eighths rule for nonlinear equations

We propose a new variant of Newton’s method based on Simpson’s three-eighth rule. It can be shown that the new method is cubically convergent.     

Multidomain spectral method for Schrödinger equations

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.     

Classical solutions of nonlinear Schrödinger equations

We prove the existence of global classical solutions to the initial value problem for the nonlinear Schrödinger equation, iu_t–u+q(|u|²)u=0 in iu_t - u + (|u|²)u = in (t, x)xⁿ for 6n11.     

Levenberg–Marquardt method based on probabilistic Jacobian models for nonlinear equations

In this paper, we propose a Levenberg–Marquardt method based on probabilistic models for nonlinear equations for which the Jacobian cannot be computed accurately or the computation is very expensive. We introduce the definition of the first-order accurate probabilistic Jacobian model, and show how to construct such a model with sample points generated by standard Gaussian distribution. Under certain conditions, we prove that the proposed method converges to a first order stationary point with probability one. Numerical results show the efficiency of the method.

     

Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

Science China Mathematics - In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation $$left{ {matrix{ { -...