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Nonradial normalized solutions for nonlinear scalar field equations
2018
We study the following nonlinear scalar field equation $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ Here $f\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and $\mu\in\mathbb{R}$ is a Lagrange multiplier. In a mass subcritical case but under general assumptions on the nonlinearity $f$, we show the existence of one nonradial solution for any $N\geq4$, and obtain multiple (sometimes infinitely many) nonradial solutions when $N=4$ or $N\geq6$. In particular, all these solutions are sign-changing.
More indefinite integrals from Riccati equations
2019
ABSTRACTTwo new methods for obtaining indefinite integrals of a special function using Riccati equations are presented. One method uses quadratic fragments of the Riccati equation, the solutions of...
Porosities and dimensions of measures
1999
We introduce a concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on $R^n$ which satisfy the doubling condition and strongly porous measures on $R$.
Nonlinear nonhomogeneous Neumann eigenvalue problems
2015
We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator with a reaction which is $(p-1)$-superlinear near $\pm\infty$ and exhibits concave terms near zero. We show that for all small values of the parameter, the problem has at least five solutions, four of constant sign and the fifth nodal. We also show the existence of extremal constant sign solutions.
Adaptive rational interpolation for point values
2019
Abstract G. Ramponi et al. introduced in Carrato et al. (1997,1998), Castagno and Ramponi (1996) and Ramponi (1995) a non linear rational interpolator of order two. In this paper we extend this result to get order four. We observe the Gibbs phenomenon that is obtained near discontinuities with its weights. With the weights we propose we obtain approximations of order four in smooth regions and three near discontinuities. We also introduce a rational nonlinear extrapolation which is also of order four in the smooth region of the given function. In the experiments we calculate numerically approximation orders for the different methods described in this paper and see that they coincide with th…
Numerical Study of Blow-Up Mechanisms for Davey-Stewartson II Systems
2018
We present a detailed numerical study of various blow-up issues in the context of the focusing Davey-Stewartson II equation. To this end we study Gaussian initial data and perturbations of the lump and the explicit blow-up solution due to Ozawa. Based on the numerical results it is conjectured that the blow-up in all cases is self similar, and that the time dependent scaling is as in the Ozawa solution and not as in the stable blow-up of standard $L^{2}$ critical nonlinear Schr\"odinger equations. The blow-up profile is given by a dynamically rescaled lump.
Normalized solutions to the mixed dispersion nonlinear Schr��dinger equation in the mass critical and supercritical regime
2019
In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation γΔ2u − Δu + αu =
On the number of solutions of a Duffing equation
1991
The exact number of solutions of a Duffing equation with small forcing term and homogeneous Neumann boundary conditions is given. Several bifurcation diagrams are shown.
Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture
1995
Abstract We have performed a molecular dynamics computer simulation study to investigate the dynamical behavior of a supercooled simple liquid for comparison with the predictions of mode-coupling theory (MCT). By scaling the intermediate scattering function by the α-relaxation time r we find that the correlators fall onto a master curve extending over several decades in time. Thus we find that the time temperature superposition principle holds. In the late β-relaxation regime this master curve can be fitted very well by a master curve predicted by the idealized version MCT. However, there is no evidence for the presence of the critical decay predicted by the theory for the early part of the…
Existence of fixed points and measures of weak noncompactness
2009
Abstract The purpose of this paper is to study the existence of fixed points by using measures of weak noncompactness. Later on, we provide an existence principle for solutions for a nonlinear integral equation.