Ground States to the Generalized Nonlinear Schr?dinger Equations with Bernstein Symbols

This paper concerns with existence and qualitative properties of ground states to generalized nonlinear Schr?dinger equations(gNLS) with abstract symbols.Under some structural assumptions on the symbol, we prove a ground state exists and it satisfies several fundamental properties that the ground state to the standard NLS enjoys. Furthermore, by imposing additional assumptions, we construct, in small mass case, a nontrivial radially symmetric solution to gNLS with H~1-subcritical nonlinearity,even if the natural energy space does not control the H~1-subcritical nonlinearity.     

Parametric resonance of ground states in the nonlinear Schrödinger equation

We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t^-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.     

Ground state solution for a class fractional Hamiltonian systems

In this paper, we consider a class of Hamiltonian systems of the form $_tD_\infty^\alpha(_{-\infty} D_t^\alpha u(t))+L(t) u(t)-\nabla W(t,u(t))=0$ where $\alpha\in(\frac{1}{2},1)$, $_{-\infty}D_t^\alpha$ and $_{t}D_\infty^\alpha$ are left and right Liouville-Weyl fractional derivatives of order $\alpha$ on the whole axis $R$ respectively. Under weaker superquadratic conditions on the nonlinearity and asymptotically periodic assumptions, ground state solution is obtained by mainly using Local Mountain Pass Theorem, Concentration-Compactness Principle and a new form of Lions Lemma respect to fractional differential equations.     

On exterior Neumann problems with an asymptotically linear nonlinearity

In this paper, we study exterior Neumann problems with an asymptotically linear nonlinearity. We establish the existence of ground state solutions. Furthermore when the domain is a complement of a ball, we prove the ground state solutions are not radially symmetric. We also give asymptotic profiles of ground state solutions.     

Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics

We consider several models of networks of interacting particles and prove the existence of quasi-periodic equilibrium solutions. We assume (1) that the network and the interaction among particles are invariant under a group that satisfies some mild assumptions; (2) that the state of each particle is given by a real number; (3) that the interaction is invariant by adding an integer to the state of all the particles at the same time; (4) that the interaction is ferromagnetic and coercive (it favors local alignment and penalizes large local oscillations); and (5) some technical assumptions on the regularity speed of decay of the interaction with the distance. Note that the assumptions are mainly qualitative, so that they cover many of the models proposed in the literature. We conclude (A) that there are minimizing (ground states) quasi-periodic solutions of every frequency and that they satisfy several geometric properties; (B) if the minimizing solutions do not cover all possible values at a point, there is another equilibrium point which is not a ground state. These results generalize basic results of Aubry–Mather theory (take the network and the group to be ?). In particular, we provide with a generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls–Nabarro barrier vanishes.     

Minimal Blow-Up Solutions to the Mass-Critical Inhomogeneous NLS Equation

We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrödinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.     

STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations(SPDEs)with multiplicative noise.The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition.These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations(SODEs)under"minimum assumptions"were studied.As a result,the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs.There are several difficulties which need to be overcome for this generalization.First,obviously the spatial discretization,which does not appear in the SODE case,adds an extra layer of difficulty.It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix.In this paper we use a finite element interpolation technique to discretize the nonlinear drift term.Second,in order to prove the strong convergence of the proposed fully discrete finite element method,stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established,which are difficult and delicate.A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal.Finally,stability estimates for the second and higher order moments of the L2-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant.This is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical solution.After overcoming these difficulties,it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.     

带有临界增长的分数阶Kirchhoff方程的半经典解

本文研究如下带有临界增长的分数阶Kirchhoff方程ε^2s(ε^2s-3∫∫R³×R³|u(x)-u(y)/|²|x-y|^3+2s),x∈R³,其中M是一个连续正的Kirchhoff函数,λ>0是一个参数,3/40充分小和λ足够大时,我们首先证明了上述问题正基态解的存在性.其次,证明了基态解集中在一个由位势函数所刻画的特定集合中.最后,研究了基态解的衰减估计.     

Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity

In this paper, we study the existence of ground state solutions of nonlinear elliptic equation with logarithmic nonlinearity by the Linking theorem and logarithmic Sobolev inequality. Our results are quite different from those in the case of polynomial nonlinearity.     

On solutions of third order nonlinear differential equations

The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.     

Adiabatic approximation for the evolution generated by an <Emphasis Type="Italic">A</Emphasis>-uniformly pseudo-Hermitian Hamiltonian

We discuss an adiabatic approximation for the evolution generated by an A-uniformly pseudo-Hermitian Hamiltonian H(t). Such a Hamiltonian is a time-dependent operator H(t) similar to a time-dependent Hermitian Hamiltonian G(t) under a time-independent invertible operator A. Using the relation between the solutions of the evolution equations H(t) and G(t), we prove that H(t) and H^? (t) have the same real eigenvalues and the corresponding eigenvectors form two biorthogonal Riesz bases for the state space. For the adiabatic approximate solution in case of the minimum eigenvalue and the ground state of the operator H(t), we prove that this solution coincides with the system state at every instant if and only if the ground eigenvector is time-independent. We also find two upper bounds for the adiabatic approximation error in terms of the norm distance and in terms of the generalized fidelity. We illustrate the obtained results with several examples.     

Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part

Using the method of the Nehari manifold, we study the existence of ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part and superlinear nonlinearity. Copyright © 2014 John Wiley & Sons, Ltd.     

Semilinear wave models with friction and viscoelastic damping

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.     

R2中非局部椭圆型方程基态解的存在性

本文考虑了一类非局部椭圆型方程-△u+V(x)u=(1/|x|μ*Q(x)F(u)/|x|β)Q(x)f(u)|x|β,x∈R^x,其中V是正的连续位势函数,0<μ<2,0≤β<1/2,2β+μ≤2,F(s)是f(s)的原函数.假设非线性项f(s)满足Trudinger-Moser型次临界指数增长,利用变分方法证明了该方程基态解的存在性.     

Existence of solutions for a nonperiodic semilinear Schrödinger equation

In this paper we consider a class of semilinear Schrödinger equation which terms are asymptotically periodic at infinity. Under a weaker superquadratic condition on the nonlinearity, the existence of a ground state solution is established. The main tools employed here to overcome the new difficulties are the concentration-compactness principle and the Local Mountain Pass Theorem.     

The virial theorem and ground state energy estimates of nonlinear Schrödinger equations in $$\mathbb {R}^2$$ with square root and saturable nonlinearities in nonlinear optics

The virial theorem is a nice property for the linear Schrödinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrödinger equation with power-law nonlinearity, then a similar ratio can be obtained but there seems to be no way of getting any eigenvalue estimates. It is surprising as far as we are concerned that when the nonlinearity is either square-root or saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue estimates of nonlinear Schrödinger (NLS) equations in \({{\mathbb {R}}^{2}}\) with square-root and saturable nonlinearity, respectively. Furthermore, we show here that the eigenvalue estimates can be used to obtain the 2nd order term (which is of order \(\ln \Gamma \)) of the lower bound of the ground state energy as the coefficient \(\Gamma \) of the nonlinear term tends to infinity.     

Global dynamics above the ground state energy for the one-dimensional NLKG equation

In this paper we obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein–Gordon (NLKG) equation on the line with focusing nonlinearity ${|u|^{p-1}u, p >5 }$ , provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case (Nakanishi and Schlag in Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, preprint, 2010), the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, we establish local dispersive estimates for the perturbed linear Klein–Gordon equation, similar to those of Mizumachi (J Math Kyoto Univ 48(3):471–497, 2008). The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.     

Bogoliubov Spectrum of Interacting Bose Gases

We study the large‐N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and nondegenerate, and that there is complete Bose‐Einstein condensation on this state. Using our method we then treat two applications: atoms with “bosonic” electrons on one hand, and trapped two‐dimensional and three‐dimensional Coulomb gases on the other hand. © 2015 Wiley Periodicals, Inc.     

Instability of the finite‐difference split‐step method applied to the generalized nonlinear Schrödinger equation. III. external potential and oscillating pulse solutions

This is the final part of a series of articles where we have studied numerical instability (NI) of localized solutions of the generalized nonlinear Schrödinger equation (gNLS). It extends our earlier studies of this topic in two ways. First, it examines differences in the development of the NI between the case of the purely cubic NLS and the case where the gNLS has an external bounded potential. Second, it investigates how the NI is affected by the oscillatory dynamics of the simulated pulse. The latter situation is common when the initial condition is not an exact stationary soliton. We have found that in this case, the NI may remain weak when the time step exceeds the threshold quite significantly. This means that the corresponding numerical solution, while formally numerically unstable, can remain sufficiently accurate over long times, because the numerical noise will stay small. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 633–650, 2017     

Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators

We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.