constructing solutions for the prescribed scalar curvature problem via local pohozaev identities abstract this paper deals with the following prescribed scalar curvature problem u q y y u n 2 n 2 u 0 y y y r 2 r n 2 where q y is nonnegative and bounded by combining a finite reduction argument and local pohozaev type of identities we prove that if n 5 and q r y has a stable critical point r 0 y 0 with r 0 0 and q r 0 y 0 0 then the above problem has infinitely many solutions whose energy can be made arbitrarily large here instead of estimating directly the derivatives of the reduced functional we apply some local pohozaev identities to locate the concentration points of the bump solutions moreover the concentration points of the bump solutions include a saddle point of q y

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

coupling integrable field theories to mechanical systems at the boundary we present an integrable hamiltonian which describes the sinh gordon model on the half line coupled to a non linear oscillator at the boundary we explain how we apply sklyanin s formalism to a dynamical reflection matrix to obtain this model this method can be applied to couple other integrable field theories to dynamical systems at the boundary we also show how to find the dynamical solution of the quantum reflection equation corresponding to our particular example

multisymplectic approach to integrable defects in the sine gordon model ideas from the theory of multisymplectic systems introduced recently in integrable systems by the author and kundu to discuss liouville integrability in classical field theories with a defect are applied to the sine gordon model the key ingredient is the introduction of a second poisson bracket in the theory that allows for a hamiltonian description of the model that is completely equivalent to the standard one in the absence of a defect in the presence of a defect described by frozen backlund transformations our approach based on the new bracket unifies the various tools used so far to attack the problem it also gets rid of the known issues related to the evaluation of the poisson brackets of the defect matrix which involve fields at coinciding space point the location of the defect the original lagrangian approach also finds a nice reinterpretation in terms of the canonical transformation representing the defect conditions

bivariate composite vector valued rational interpolation in this paper we point out that bivariate vector valued rational interpolants bvri have much to do with the vector grid to be interpolated when a vector grid is well defined one can directly design an algorithm to compute the bvri however the algorithm no longer works if a vector grid is ill defined taking the policy of divide and conquer we define a kind of bivariate composite vector valued rational interpolant and establish the corresponding algorithm a numerical example shows our algorithm still works even if a vector grid is ill defined

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

the profile of bubbling solutions of a class of fourth order geometric equations on 4 manifolds abstract we study a class of fourth order geometric equations defined on a 4 dimensional compact riemannian manifold which includes the q curvature equation we obtain sharp estimates on the difference near the blow up points between a bubbling sequence of solutions and the standard bubble

a compactness theorem for a fully nonlinear yamabe problem under a lower ricci curvature bound abstract we prove compactness of solutions of a fully nonlinear yamabe problem satisfying a lower ricci curvature bound when the manifold is not conformally diffeomorphic to the standard sphere this allows us to prove the existence of solutions when the associated cone satisfies 1 which includes the k yamabe problem for k not smaller than half of the dimension of the manifold

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

concentration compactness phenomena in the higher order liouville s equation we investigate different concentration compactness and blow up phenomena related to the q curvature in arbitrary even dimension we first treat the case of an open domain in r 2 then that of a closed manifold and finally the particular case of the sphere s 2m in all cases we allow the sign of the q curvature to vary and show that in the case of a closed manifold contrary to the case of open domains in r 2m blow up phenomena can occur only at points of positive q curvature as a consequence on a locally conformally flat manifold of non positive euler characteristic we always have compactness

a pointwise finite dimensional reduction method for einstein lichnerowicz type systems the six dimensional case abstract we construct blowing up solutions for smooth perturbations of a focusing einstein lichnerowicz system on a closed 6 dimensional manifold our construction combines asymptotic analysis techniques with a finite dimensional reduction approach following the scheme of proof developed in premoselli 2016 the blow up examples constructed here highlight in particular the role of the coupling field in dimension 6

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

compactness of solutions to some geometric fourth order equations we prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds the proof is based on a refined bubbling analysis for which the main estimates are given in integral form our result is used in a subsequent paper to find critical points via minimax arguments of some geometric functional which give rise to conformal metrics of constant q curvature as a byproduct of our method we also obtain compactness of such metrics

the q curvature on a 4 dimensional riemannian manifold m g with mqdvg 8π2 abstract we deal with the q curvature problem on a 4 dimensional compact riemannian manifold m g with m q g d v g 8 2 and positive paneitz operator p g let q be a positive smooth function the question we consider is when can we find a metric g which is conformal to g such that q is just the q curvature of g a sufficient condition to this question is given in this paper

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

blow up phenomena for scalar flat metrics on manifolds with boundary abstract let m n g be a compact riemannian manifold with boundary m this article is concerned with the set of scalar flat metrics on m which are in the conformal class of g and have m as a constant mean curvature hypersurface we construct examples of metrics on the unit ball b n in dimensions n 25 for which this set is non compact these manifolds have umbilic boundary but they are not conformally equivalent to b n

a non compactness result on the fractional yamabe problem in large dimensions abstract let x n 1 g be an n 1 dimensional asymptotically hyperbolic manifold with conformal infinity m n h the fractional yamabe problem addresses to solve p g h u c u n 2 n 2 u 0 on m where c r and p g h is the fractional conformal laplacian whose principal symbol is the laplace beltrami operator on m in this paper we construct a metric on the half space x r n 1 which is conformally equivalent to the unit ball for which the solution set of the fractional yamabe equation is non compact provided that n 24 for 0 and n 25 for 1 where 0 1 is a certain transition exponent the value of turns out to be approximately 0 940197

q curvature on a class of manifolds with dimension at least 5 for a smooth compact riemannian manifold with positive yamabe invariant positive q curvature and dimension at least 5 we prove the existence of a conformal metric with constant q curvature our approach is based on the study of an extremal problem for a new functional involving the paneitz operator 2015 wiley periodicals inc

stability of solitary waves in finite ablowitz ladik lattices we discuss the effects of different types of boundary conditions b c on the linear stability of a solitary wave in a finite length dynamical lattice described by the ablowitz ladik al model types of b c considered are fixed dirichlet no flux neumann and free as well as periodic b c the behaviour of eigenvalues around a stationary nonlinear wave consistent with several types of b c is studied analytically and numerically the translational eigenvalues are found to move away from the origin it is shown that for b c of the fixed type these eigenvalues bifurcate into a neutrally stable oscillatory kind while the no flux and free b c lead to an exponentially weak instability the rotational eigenmodes those related to the gauge invariance of the lattice strictly remain at the origin as gauge invariance is not broken by the linear homogeneous b c these numerical results are compared to the analytical and semi analytical predictions obtained within the approximation of the images generated by the solitary wave beyond the lattice edges a potential effect of continuous spectrum band edge bifurcations on stability is also evaluated with the conclusion that that these bifurcations cannot destabilize the soliton for periodic b c the translational invariance is preserved and so is the lax pair structure sustaining the integrable nature of the problem the al model was chosen so that the effects of boundary conditions are monitored in the absence of non integrable discreteness finally the possibility of the generation of multipulse configurations due to the interplay of finiteness effects with the exponential tail tail interaction of the pulses is examined

a compactness theorem for scalar flat metrics on 3 manifolds with boundary abstract let m g be a compact riemannian three dimensional manifold with boundary we prove the compactness of the set of scalar flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface this involves a blow up analysis of a yamabe type equation with critical sobolev exponent on the boundary

standard nearest neighbour discretizations of klein gordon models cannot preserve both energy and linear momentum we consider nonlinear klein gordon wave equations and illustrate that standard discretizations thereof involving nearest neighbours may preserve either standardly defined linear momentum or standardly defined total energy but not both this has a variety of intriguing implications for the non potential discretizations that preserve only the linear momentum such as the self accelerating or self decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

aspects of sine gordon solitons defects and gates it was recently noted how the classical sine gordon theory can support discontinuities or defects and yet maintain integrability by preserving sufficiently many conservation laws since soliton number is not preserved by a defect a possible application to the construction of logical gates is suggested

vector valued rational interpolants over triangular grids abstract given in this paper are some results about vector valued rational interpolants over triangular grids by means of thiele type branched continued fractions and the samelson inverse characterization theorem and uniqueness theorem in a certain sense are obtained moreover numerical examples are provided to support other properties of vector valued rational interpolants such as boundary interpolation and duality and it is pointed out that all the results in the vector case can be transplanted to the matrix valued case by the technique of expansion of matrix into vectors

bound states of two dimensional solitons in the discrete nonlinear schrodinger equation the existence and stability of bound states bss of solitary excitations in the two dimensional discrete nonlinear schrodinger lattice are considered stable bss of two or three solitons are categorized according to their vorticity s s 0 corresponds to ordinary pulses while s 1 corresponds to discrete vortex solitons interactions of s 0 solitons are found to have clear particle like characteristics which can be very accurately predicted in the framework of the perturbation theory that we develop triangular bound states of three solitons are also constructed and particle like characteristics in their behaviour are followed in numerical experiments finally stable bound states of two vortex solitons and s 0 1 stable complexes are constructed for the latter state we find a weak dependence of the interaction generated stability eigenvalue on the distance between the two solitons

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

on a class of discretizations of hamiltonian nonlinear partial differential equations abstract we present a new class of discretizations of partial differential equations pdes that preserve a momentum like integral of the motion this results in an effective translational invariance for the dynamical problem and the absence of a peierls nabarro barrier that is usually present in discretizations of hamiltonian pdes a general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential difference equations are analyzed in a number of different cases for nonlinear klein gordon as well for nonlinear schrodinger type systems while for the former nonintegrability of the dynamical problem is evident in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems we also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones

internal modes in sine gordon chain we address the issue of internal modes of a stable kink in a discrete sine gordon equation the aim of the present study is to elucidate the effects due to the detachment of the frequency dependence of antisymmetric internal mode from the spectrum we analyze the frequencies of the lowest modes as functions of both the number of sites and the discreteness parameter using a simplified approach we explain the origin of the spectrum peculiarity which arises when the frequency dependence detaches from the quasi continuous spectrum at some value of the intersite coupling

ground states for a system of schrodinger equations with critical exponent abstract we study the following system of nonlinear schrodinger equations u u u p 1 u v x r n v v v 2 2 v u x r n where n 3 2 2 n n 2 1 p 2 1 and are positive parameters satisfying 0 we show that there is some critical value 0 0 1 such that this system has a positive ground state solution if 0 0 in the case 0 there exists 0 such that this system has no ground state solutions if while this system has a positive ground state solution if in particular if p 2 1 the system has no nontrivial solutions some further properties of the ground state solutions are also studied this seems to be the first result for such a critical schrodinger system

vector valued rational interpolants i it is shown explicitly that the samelson inverse may be used to define vector valued thiele type rational interpolants and the equivalence with claessens algorithm is established for the special case of vector valued pade approximants the general form of mccleod s theorem follows as a corollary

from periodic ode s to supercritical pde s abstract let m g be an m dimensional compact riemannian manifold without boundary assume c 2 m is such that g is coercive we prove the existence of solutions to the supercritical problems g u u u p u 0 in m g and g u u e u in m g which concentrate along an m 1 dimensional submanifold of m as p and 0 respectively under suitable symmetry assumptions on the manifold m a connection with the solutions of a class of periodic ode s is established

q curvature on a class of 3 manifolds motivated by the strong maximum principle for the paneitz operator in dimension 5 or higher found in a preprint by gursky and malchiodi and the calculation of the second variation of the green s function pole s value on 3 in our preprint we study the riemannian metric on 3 manifolds with positive scalar and q curvature among other things we show it is always possible to find a constant q curvature metric in the conformal class moreover the green s function is always negative away from the pole and the pole s value vanishes if and only if the riemannian manifold is conformal diffeomorphic to the standard 3 compactness of constant q curvature metrics in a conformal class and the associated sobolev inequality are also discussed 2014 wiley periodicals inc

multiphase patterns in periodically forced oscillatory systems periodic forcing of an oscillatory system produces frequency locking bands within which the system frequency is rationally related to the forcing frequency we study extended oscillatory systems that respond to uniform periodic forcing at one quarter of the forcing frequency the 4 1 resonance these systems possess four coexisting stable states corresponding to uniform oscillations with successive phase shifts of pi 2 using an amplitude equation approach near a hopf bifurcation to uniform oscillations we study front solutions connecting different phase states these solutions divide into two groups pi fronts separating states with a phase shift of pi and pi 2 fronts separating states with a phase shift of pi 2 we find a type of front instability where a stationary pi front open quotes decomposes close quotes into a pair of traveling pi 2 fronts as the forcing strength is decreased the instability is degenerate for an amplitude equation with cubic nonlinearities at the instability point a continuous family of pair solutions exists consisting of pi 2 fronts separated by distances ranging from zero to infinity quintic nonlinearities lift the degeneracy at the instability point but do not change the basic nature of the instability we conjecture the existence of similar instabilities in higher 2n 1 resonances n 3 4 hor ellipsis where stationarymore pi fronts decompose into ital n traveling pi n fronts the instabilities designate transitions from stationary two phase patterns to traveling 2n phase patterns as an example we demonstrate with a numerical solution the collapse of a four phase spiral wave into a stationary two phase pattern as the forcing strength within the 4 1 resonance is increased copyright ital 1999 ital the american physical society less

classical behavior of deformed sine gordon models abstract in this work we deform the 4 model with distinct deformation functions to propose a diversity of sine gordon like models we investigate the proposed models and we obtain all the topological solutions that they engender in particular we introduce non polynomial potentials which support some exotic two kink solutions

the effect of linear perturbations on the yamabe problem in conformal geometry the compactness conjecture asserts that the set of yamabe metrics on a smooth compact aspherical riemannian manifold left m g right is compact established in the locally conformally flat case by schoen lecture notes in mathematics vol 1365 pp 120 154 springer berlin 1989 surveys pure application and mathematics 52 longman science technology pp 311 320 harlow 1991 and for n le 24 by khuri marques schoen j differ geom 81 1 143 196 2009 it has revealed to be generally false for n ge 25 as shown by brendle j am math soc 21 4 951 979 2008 and brendle marques j differ geom 81 2 225 250 2009 a stronger version of it the compactness under perturbations of the yamabe equation is addressed here with respect to the linear geometric potential frac n 2 4 n 1 mathrm scal g mathrm scal g being the scalar curvature of left m g right we show that a priori l infty bounds fail for linear perturbations on all manifolds with n ge 4 as well as a priori gradient l 2 bounds fail for non locally conformally flat manifolds with n ge 6 and for locally conformally flat manifolds with n ge 7 in several situations the results are optimal our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g

compactness of conformal metrics with constant q curvature i abstract we study compactness for nonnegative solutions of the fourth order constant q curvature equations on smooth compact riemannian manifolds of dimension 5 if the q curvature equals 1 we prove that all solutions are universally bounded if the q curvature is 1 assuming that paneitz operator s kernel is trivial and its green function is positive we establish universal energy bounds on manifolds which are either locally conformally flat lcf or of dimension 9 moreover assuming in addition that a positive mass type theorem holds for the paneitz operator we prove compactness in c 4 positive mass type theorems have been verified recently on lcf manifolds or manifolds of dimension 7 when the yamabe invariant is positive we also prove that for dimension 8 the weyl tensor has to vanish at possible blow up points of a sequence of blowing up solutions this implies the compactness result in dimension 8 when the weyl tensor does not vanish anywhere to overcome difficulties stemming from fourth order elliptic equations we develop a blow up analysis procedure via integral equations

reflection transmission quantum yang baxter equations we explore the reflection transmission quantum yang baxter equations arising in factorized scattering theory of integrable models with impurities the physical origin of these equations is clarified and three general families of solutions are described in detail explicit representatives of each family are also displayed these results allow us to establish for the first time a direct relationship with the different previous works on the subject and make evident the advantages of the reflection transmission algebra as a universal approach to integrable systems with impurities

composite schemes for multivariate blending rational interpolation it is demonstrated that newton s interpolation polynomials and thiele s interpolating continued fractions can be incorporated to generate various interpolation schemes based on rectangular grids among them are two kinds of bivariate blending rational interpolants however blending rational interpolants strongly depend on the existence of so called blending differences which means that for some grids of data one may fail to find out the corresponding rational interpolants as a whole in this paper we offer a solution scheme by adopting composite interpolation over triangular sub grids characterization theorem is given error estimation is worked out and vector valued case as well as matrix valued case is discussed

defect structures in sine gordon like models abstract we investigate several models described by real scalar fields searching for topological defects some models are described by a single field and support one or two topological sectors and others are two field models which support several topological sectors almost all the defect structures that we find are stable and finite energy solutions of first order differential equations that solve the corresponding equations of motion in particular for the double sine gordon model we show how to find small and large bps solutions as deformations of the bps solution of the 4 model and also for most of the two field models we find the corresponding integrating factors which lead to the complete set of bps solutions nicely unveiling how they bifurcate among the several topological sectors

compactness for a class of yamabe type problems on manifolds with boundary abstract in this paper we are interested in studying the set of positive solutions to the following yamabe type problem g u a x u 0 in m u g b x u n 2 u n n 2 on m where m n g is a compact riemannian n manifold n 3 with boundary g is the laplace beltrami operator for g g is the pointing unit outward normal to m a c m and b c m assuming suitable hypotheses on the functions a n 2 4 n 1 r g and b n 2 2 h g where r g denotes the scalar curvature of the metric g and h g is the mean curvature of m we prove some results of compactness for the set of positive solutions to the problem