nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

existence of the global solution for fractional logarithmic schrodinger equation abstract in this paper we consider the initial boundary value problem for a class of fractional logarithmic schrodinger 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

the l2 boundedness of pseudodifferential operators on donne une nouvelle demonstration du theoreme de calderon vaillancourt on obtient la l 2 continuite de a x d si le symbole a x satisfait a certaines conditions

multiplicity and concentration results for fractional schrodinger system with steep potential wells abstract this paper is concerned with the fractional coupled schrodinger system by using the nehari manifold and fibering map we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells where some new estimates will be established in particular although there exist concave convex nonlinearities in the coupled system it is not necessary to assume that the corresponding lebesgue norms of the weight functions of the convex terms need to be small enough

smooth solutions to a quasi linear system of diffusion equations for a certain population model abstract the system of diffusion equations proposed by kawasaki shigesada and teramoto describes a population model of two competing species with self and cross population pressures the densities of the two species are denoted by u and v in this paper the author studies the initial boundary value problem associated with the neumann boundary condition corresponds to the case where the flux is zero at the boundary many investigators have considered nonlinear diffusion systems arising from various physical and biological problems these equations however have a special structure the highest order derivatives are not coupled or at least the coefficient matrix for the highest order derivatives is positive definite this is not the case for the system and hence some of the techniques which are effective for the system are no longer applicable to our case nevertheless we can still use sobolevski s method see reference 2 to establish the local existence of solutions under the special assumption c1 c2 in we can also prove the global existence of solutions by energy estimates the unusual structure of seems to make it difficult to settle the question of asymptotic stability of solutions

compact embeddings and proper mappings abstract sufficient conditions are given to assert that a c 1 proper mapping defined on a compact embedding between banach spaces over k has a fixed point the proof of the result is based upon continuation methods

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

dynamic theory of quasilinear parabolic equations i abstracts evolution equations on etablit les fondements abstraits pour une theorie dynamique generale pour des problemes aux valeurs limites et initiales quasilineaires

semiclassical states for weakly coupled fractional schrodinger systems abstract in this paper we consider the nonlinear system of two weakly coupled fractional schrodinger equations with nonconstant potentials e 2 s s u p 1 x u u 2 p b u p 1 v p 1 u x r n e 2 s s v p 2 x v v 2 p b v p 1 u p 1 v x r n we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

comment on on the consistency of the solutions of the space fractional schrodinger equation j math phys 53 042105 2012 recently we have reanalyzed the consistency of the solutions of the space fractional schrodinger equation found in a piecewise manner and showed that an exact and a proper treatment of the relevant integrals prove that they are consistent in this comment for clarity we present additional information about the critical integrals and describe how their analytic continuation is accomplished

time dependent schrodinger like equation with nonlocal term we investigate a time dependent schrodinger like equation in presence of a nonlocal term by using the method of variable separation and the green function approach we analyze the green function for different forms of the memory kernel and the nonlocal term results for delta potential energy function are presented distributed order memory kernels are also considered and the asymptotic behaviors of the green function are derived by using tauberian theorem the obtained results for the green function for the considered schrodinger like equation may be transformed to those for the probability distribution function of a diffusion like equation with memory kernel and can be used to explain various anomalous diffusive behaviors

on the existence of zero points we prove the existence of at least one zero of a continuous function under one of two smoothness conditions the proof is constructive numerical methods can be used to follow a solution curve with a computer a global implicit function theorem is given

existence of zeros for bounded perturbations of proper mappings sufficient conditions asserting the existence of at least one zero of bounded perturbations of proper mappings are given the conditions are easily tested the proof of this theorem is constructive showing the existence of a solution curve that can be followed by numerical methods

continuous embeddings and continuation methods abstract sufficient conditions are given to assert that a compact mapping defined on a continuous embedding between hilbert spaces over k has a fixed point the proof of the result is based upon continuation methods

on a time dependent bio reactor model with chemotaxis long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio reactors with chemotaxis will be studied if the parameters of the system are periodic sufficient conditions for positive periodic solutions will be derived in terms of parabolic eigenvalue problems

existence and computation of zeros of perturbed mappings sufficient conditions are given to assert that a perturbed mapping has only one zero in a specific open ball this zero is estimated by means of a sequence of newton s iterations the proof of the result is constructive and is based upon continuation methods

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

profile decompositions and blowup phenomena of mass critical fractional schrodinger equations abstract we study under the radial symmetry assumption the solutions to the fractional schrodinger equations of critical nonlinearity in r 1 d d 2 with levy index 2 d 2 d 1 2 we first prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass critical hartree type nonlinearity

existence of solutions for a bi nonlocal fractional p kirchhoff type problem in this paper we are concerned with the existence of nonnegative solutions for a p kirchhoff type problem driven by a non local integro differential operator with homogeneous dirichlet boundary data as a particular case we study the following problem m x u s p p p s u f x u u s p p in u 0 in r n u s p p r 2 n u x u y p x y n p s d x d y where p s is a fractional p laplace operator is an open bounded subset of r n with lipschitz boundary m r 0 r is a continuous function and f r r 0 r is a continuous function satisfying the ambrosetti rabinowitz type condition the existence of nonnegative solutions is obtained by using the mountain pass theorem and an iterative scheme the main feature of this paper lies in the fact that the kirchhoff function m depends on x and the nonlinearity f depends on the energy of solutions

a preconditioned numerical solver for stiff nonlinear reaction diffusion equations with fractional laplacians that avoids dense matrices the numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the spatial nonlocality of the fractional differential operators the dense coefficient matrices that arise from spatial discretisation of these operators mean that even one dimensional problems can be difficult to solve using standard methods on grids comprising thousands of nodes or more in this work we address this issue of efficiency for one dimensional nonlinear space fractional reaction diffusion equations with fractional laplacian operators we apply variable order variable stepsize backward differentiation formulas in a jacobian free newton krylov framework to advance the solution in time a key advantage of this approach is the elimination of any requirement to form the dense matrix representation of the fractional laplacian operator we show how a banded approximation to this matrix which can be formed and factorised efficiently can be used as part of an effective preconditioner that accelerates convergence of the krylov subspace iterative solver our approach also captures the full contribution from the nonlinear reaction term in the preconditioner which is crucial for problems that exhibit stiff reactions numerical examples are presented to illustrate the overall effectiveness of the solver

nonlinear fractional schrodinger equations in one dimension abstract we consider the question of global existence of small smooth and localized solutions of a certain fractional semilinear cubic nls in one dimension i t u u c 0 u 2 u c 1 u 3 c 2 u u 2 c 3 u 3 x x 1 2 where c 0 r and c 1 c 2 c 3 c this model is motivated by the two dimensional water wave equation which has a somewhat similar structure in the eulerian formulation in the case of irrotational flows we show that one cannot expect linear scattering even in this simplified model more precisely we identify a suitable nonlinear logarithmic correction and prove global existence and modified scattering of solutions

fractional nonlinear schrodinger equation of order α 0 1 abstract we consider the cauchy problem for the fractional nonlinear schrodinger equation i t u 1 x u u 2 u t 0 x r u 0 x u 0 x x r where r the order of the fractional derivative 0 1 we obtain the large time asymptoitic behavior of solutions which has a logarithmic phase modifications for large time comparing with the linear problem

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

a second zero of a function a sufficient condition is given to assert that a c m n 1 mapping between r m and r n which has a zero also has a second zero the condition can still be useful for continuous mappings when m n the constructive proof of the result is based upon continuation methods and supplies the existence of a path linking the zero points

global analysis of a model for capillary water waves in two dimensions in this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 the capillary waves system the model we consider here retains most of the difficulties of the full capillary water waves system including the delicate time resonance structure and modified scattering it is slightly simpler however at the technical level and our goal here is to present our method in this simplified situation the analysis of the full system is presented in a preprint by the authors 2014 2016 wiley periodicals inc

on fractional schrodinger equation in rn with critical growth in this paper we study the following nonlinear fractional schrodinger equation with critical exponent h2 u v x u u 2 2u u q 2u x rn where h is a small positive parameter 0 2 0 is a parameter the potential v rn r is a positive continuous function satisfying some natural assumptions by using variational methods we obtain the existence of solutions in the following case if 2 0 such that for all 0 we show that it has one nontrivial solution and there exist at least cat nontrivial solutions if max 2 4 n 2 0

mappings sharing a value on finite dimensional spaces sufficient conditions are given to assert that differentiable mappings and differentiable proper mappings between r n and r m share a value the proof of the result is constructive and is based upon continuation methods

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

global minimum point of a convex function in this paper we prove the existence of an unique global minimum point of a convex function under some smoothness conditions our proof permits us to calculate numerically such a minimum point utilizing a constructive homotopy method

eigenvalue problems for fractional differential equations with right and left fractional derivatives this paper studies the eigenvalue problem of a class of fractional differential equations with right and left fractional derivatives with the aid of the spectral theory of compact self adjoint operators in hilbert spaces we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system furthermore the lower bound of the eigenvalues is obtained

holder regularity for certain strongly coupled parabolic systems abstract we present a shorter proof to show holder continuity of bounded solutions to a general class of quasilinear parabolic equations the proof will be extended to obtain regularity results for bounded solutions to certain strongly coupled or cross diffusion quasilinear parabolic systems

critical fractional p laplacian problems with possibly vanishing potentials abstract we obtain nontrivial solutions of a critical fractional p laplacian equation in the whole space and with possibly vanishing potentials in addition to the usual difficulty of the lack of compactness associated with problems involving critical sobolev exponents the problem is further complicated by the absence of a direct sum decomposition suitable for applying classical linking arguments we overcome this difficulty using a generalized linking construction based on the z 2 cohomological index

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

a fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrodinger equations abstract in this paper a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional schrodinger equations we prove that the scheme preserves both the mass and energy which are defined by virtue of some recursion relationships using the sobolev inequalities and then employing the mathematical induction the discrete scheme is proved to be unconditionally convergent in the sense of l 2 norm and h 2 norm which means that there are no any constraints on the grid ratios then the prior bound of the discrete solution in l 2 norm and l norm are also obtained moreover we propose an iterative algorithm by which the coefficient matrix is independent of the time level and thus it leads to toeplitz like linear systems that can be efficiently solved by krylov subspace solvers with circulant preconditioners this method can reduce the memory requirement of the proposed linearized finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m log m in each iterative step where m is the number of grid nodes finally numerical results are carried out to verify the correction of the theoretical analysis simulate the collision of two solitary waves and show the utility of the fast numerical solution techniques

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

asymptotics for large time of solutions to the nonlinear schrodinger and hartree equations we study the asymptotic behavior in time of solutions to the cauchy problems for the nonlinear schrodinger equation with a critical power nonlinearity and the hartree equation we prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the cauchy problem with small initial data this estimate is very important for the proof of the existence of modified scattering states to the nonlinear schrodinger equations with a critical nonlinearity and the hartree equation in order to derive the desired estimates we introduce a certain phase function since the previous methods based solely on a priori estimates of the operator x it acting on the solution without specifying any phase function do not work for the critical case under consideration the well known nonexistence of the usual l 2 scattering states shows that our result is sharp

long range scattering for non linear schrodinger and hartree equations in space dimension n 2 we consider the scattering problem for the non linear schrodinger nls equation with a power interaction with critical powerp 1 2 n in space dimensionsn 2 and 3 and for the hartree equation with potential x 1 in space dimensionn 2 we prove the existence of modified wave operators in thel2 sense on a dense set of small and sufficiently regular asymptotic states

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

a compactness condition sufficient conditions are given to assert that a differentiable mapping between k m and k n where k r or k c has a zero the constructive proof of the result is based upon continuation methods and supplies the existence of a path leading to the zero point

the global solution for a class of systems of fractional nonlinear schrodinger equations with periodic boundary condition in this paper we consider a class of systems of fractional nonlinear schrodinger equations we prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the faedo galerkin method

existence and multiplicity of solutions for superlinear fractional schrodinger equations in ℝn in this paper we are concerned with superlinear fractional schrodinger equation su v x u f x u x n where f is continuous when v and f are asymptotically periodic in x we show the existence of ground states when v and f are periodic in x we obtain infinitely many geometrically distinct solutions the method used here is based on the method of nehari manifold and lusternik schnirelmann category theory

stability of standing waves for the fractional schrodinger hartree equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger hartree equation by using the profile decomposition theory and variational methods we firstly obtain the best constant of a generalized gagliardo nirenberg inequality and then prove that the standing waves are orbitally stable

a mesh free pseudospectral approach to estimating the fractional laplacian via radial basis functions abstract this paper investigates the use of radial basis function rbf interpolants to estimate a function s fractional laplacian of a given order through a mesh free pseudospectral method the mesh free approach yields an algorithm that can be implemented in high dimensional settings without adjustment moreover the fractional laplacian is defined in terms of the fourier transform and the symmetry of rbfs can be exploited to simplify the estimation problem convergence rates are established for rbfs when the function whose fractional laplacian to be estimated is compactly supported further results demonstrate convergence when a function is in the native space for a wendland rbf i e a sobolev space and satisfies a certain l 1 condition numerical experiments demonstrate the developed method by estimating the fractional laplacian of several functions and by solving a fractional poisson equation with extended dirichlet condition in one and two dimensions

an initial boundary value problem for a certain density dependent diffusion system on considere le systeme parabolique quasilineaire u t x c 1 d 1 v v e 1 a 1 v b 1 u u v t x c 2 d 2 u v e 2 a 2 u b 2 o v dans o t u x s 0 v x s sur o t u x o w 1 x v x o w 2 x sur ou r n est un domaine borne a frontiere lisse

sharp asymptotic behavior of solutions for cubic nonlinear schrodinger equations with a potential we consider the cubic nonlinear schrodinger equation with a potential we obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions to the cauchy problem

fractional dynamics of systems with long range space interaction and temporal memory abstract field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power law memory function and long range interaction in systems the method is applied to obtain a fractional generalization of the ginzburg landau and nonlinear schrodinger equations as another example dynamical equations for particle chains with power law interaction and memory are considered in the continuous limit the obtained fractional equations can be applied to complex media with without random parameters or processes

nonlinear schrodinger equations with exceptional potentials abstract we consider the cubic nonlinear schrodinger equation with an exceptional potential we obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions we prove the existence of modified scattering for this model that is linear scattering modulated by a phase our approach is based on the spectral theorem for the perturbed linear schrodinger operator and a factorization technique that allows us to control the resonant nonlinear term we make some parity assumptions in order to control the small energy behavior of the scattering coefficients and of the wave functions

extremum points of a convex function abstract we prove the existence of a global minimum point of a convex function under some smoothness conditions the proof is indeed constructive numerical methods can be used to follow a solution curve with a computer

a compact fourth order in space energy preserving method for riesz space fractional nonlinear wave equations in this work we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the riesz type the model under consideration generalizes various nonlinear wave equations including the sine gordon and the nonlinear klein gordon models the system considered in this work is conservative when homogeneous dirichlet boundary conditions are imposed motivated by this fact we propose a finite difference method based on fractional centered differences that is capable of preserving the discrete energy of the system the method under consideration is a nonlinear implicit scheme which has various numerical properties among the most interesting numerical features we show that the methodology is consistent of second order in time and fourth order in space moreover we show that the technique is stable and convergent some numerical simulations show that the method is capable of preserving the energy of the discrete system this characteristic of the technique is in obvious agreement with the properties of its continuous counterpart