on self similar singular solutions of the complex ginzburg landau equation we address the open problem of existence of singularities for the complex ginzburg landau equation using a combination of rigorous results and numerical computations we describe a countable family of self similar singularities our analysis includes the supercritical nonlinear schrodinger equation as a special case we also consider the problem of stability of these singularities 2001 john wiley sons inc

new singular solutions of the nonlinear schrodinger equation we present numerical simulations of a new type of singular solutions of the critical nonlinear schrodinger equation nls that collapse with a quasi self similar ring profile at a square root blowup rate we find and analyze the equation of the ring profile we observe that the self similar ring profile is an attractor for a large class of radially symmetric initial conditions but is unstable under symmetry breaking perturbations the equation for the ring profile admits also multi ring solutions that give rise to collapsing self similar multi ring solutions but these solutions are unstable even in the radially symmetric case and eventually collapse with a single ring profile collapsing ring solutions are also observed in the supercritical nls

dissipation at singularities of the nonlinear schrodinger equation through limits of regularisations abstract the possibility of physically relevant singular solutions of the nonlinear schrodinger equation nlse with sustained dissipation into the singularity is considered through numerical study of a dissipative regularisation and its small dissipation limit a new form of such dissipative solutions is conjectured for certain parameter ranges where this behaviour was previously not expected involving a multi focusing mechanism a possible mechanism is discussed involving a new family of stationary singular solutions of the nlse

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

steady flow of non newtonian fluids monotonicity methods in generalized orlicz spaces abstract our purpose is to show the existence of weak solutions to steady flow of non newtonian incompressible fluids with nonstandard growth conditions of the cauchy stress tensor we are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening since l p framework is not sufficient to capture the described model we describe the growth conditions with the help of a general x dependent convex function and formulate our problem in generalized orlicz spaces

poincare meets korn via maxwell extending korn s first inequality to incompatible tensor fields for a bounded domain r3 r3 with lipschitz boundary and some relatively open lipschitz subset t t of we prove the existence of some c 0c 0 such that equation 0 1 c t l2 r3 3 symt l2 r3 3 curlt l2 r3 3 holds for all tensor fields in h curl h curl i e for all square integrable tensor fields t r3 3t r3 3 with square integrable generalized rotation curlt r3 3 having vanishing restricted tangential trace on t t if t t 0 1 still holds at least for simply connected and for all tensor fields t h curl t h curl which are l2 l2 perpendicular to so 3 so 3 i e to all skew symmetric constant tensors here both operations curl and tangential trace are to be understood row wise for compatible tensor fields t vt v 0 1 reduces to a non standard variant of the well known korn s first inequality in r3r3 namely c v l2 r3 3 sym v l2 r3 3 for all vector fields v h1 r3 v h1 r3 for which vn vn n 1 3n 1 3 are normal at t t on the other hand identifying vector fields v h1 r3 v h1 r3 having the proper boundary conditions with skew symmetric tensor fields t 0 1 turns to poincare s inequality since 2c v l2 r3 c t l2 r3 3 curlt l2 r3 3 2 v l2 r3 therefore 0 1 may be viewed as a natural common generalization of korn s first and poincare s inequality from another point of view 0 1 states that one can omit compatibility of the tensor field t at the expense of measuring the deviation from compatibility through curl t decisive tools for this unexpected estimate are the classical korn s first inequality helmholtz decompositions for mixed boundary conditions and the maxwell estimate

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

sobolev inequalities for the symmetric gradient in arbitrary domains abstract a form of sobolev inequalities for the symmetric gradient of vector valued functions is proposed which allows for arbitrary ground domains in r n in the relevant inequalities boundary regularity of domains is replaced with information on boundary traces of trial functions the inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of sobolev functions in regular domains furthermore they involve constants independent of the geometry of the domain and hence yield novel results yet for smooth domains our approach relies upon a pointwise estimate for the functions in question via a riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace

3d euler equations and ideal mhd mapped to regular systems probing the finite time blowup hypothesis abstract we prove by an explicit construction that solutions to incompressible 3d euler equations defined in the periodic cube 0 l 3 can be mapped bijectively to a new system of equations whose solutions are globally regular we establish that the usual beale kato majda criterion for finite time singularity or blowup of a solution to the 3d euler system is equivalent to a condition on the corresponding regular solution of the new system in the hypothetical case of euler finite time singularity we provide an explicit formula for the blowup time in terms of the regular solution of the new system the new system is amenable to being integrated numerically using similar methods as in euler equations we propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite time singularity problem of euler equations based on the conservation of quantities directly related to energy and circulation the method of mapping to a regular system can be extended to any fluid equation that admits a beale kato majda type of theorem e g 3d navier stokes 2d and 3d magnetohydrodynamics and 1d inviscid burgers we discuss briefly the case of 2d ideal magnetohydrodynamics in order to illustrate the usefulness of the mapping we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1d inviscid burgers equation

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

the focusing singularity of the davey stewartson equations for gravity capillary surface waves abstract we analyze the self focusing of gravity capillary surface waves modeled by the davey stewartson equations this system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two dimensional schrodinger equation with a non local cubic nonlinearity with accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical as in the usual two dimensional cubic schrodinger equation which is the deep water limit of the davey stewartson equations we also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation

optimal sobolev trace embeddings optimal target spaces are exhibited in arbitrary order sobolev type embeddings for traces of n dimensional functions on lower dimensional subspaces sobolev spaces built upon any rearrangement invariant norm are allowed a key step in our approach consists in showing that any trace embedding can be reduced to a one dimensional inequality for a hardy type operator depending only on n and on the dimension of the relevant subspace this can be regarded as an analogue for trace embeddings of a well known symmetrization principle for first order sobolev embeddings for compactly supported functions the stability of the optimal target space under iterations of sobolev trace embeddings is also established and is part of the proof of our reduction principle as a consequence we derive new trace embeddings with improved optimal target spaces for classical sobolev lorentz sobolev and orlicz sobolev spaces

stability of isotropic singularities for the nonlinear schro dinger equation abstract we describe a method by which the fully two and three dimensional cubic schrodinger equations can be accurately integrated numerically up to times very close to the formation of singularities in both cases anisotropic initial data collapse very rapidly towards isotropic singularities in three dimensions the solutions become self similar with a blowup rate t t 1 2 in two dimensions the self similarity is weakly broken and the blowup rate is t t ln ln 1 t t 1 2 the stability of the singular isotropic solutions is very firmly backed by the numerical results

a new approach to counterexamples to l 1 estimates korn s inequality geometric rigidity and regularity for gradients of separately convex functions the derivation of counterexamples to l1 estimates can be reduced to a geometric decomposition procedure along rank one lines in matrix space we illustrate this concept in two concrete applications firstly we recover a celebrated and rather complex counterexample by ornstein proving the failure of korn s inequality and of the corresponding geometrically nonlinear rigidity result in l1 secondly we construct a function f 2 which is separately convex but whose gradient is not in bvloc in the sense that the mixed derivative 2f x1 x2 is not a bounded measure

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

a general formula for soliton form factors in the quantum sine gordon model a general formula for soliton form factors of several local fields in the quantum sine gordon model is presented

korn type inequalities in orlicz spaces abstract we exhibit balance conditions between a young function a and a young function b for a korn type inequality to hold between the l b norm of the gradient of vector valued functions and the l a norm of its symmetric part in particular we extend a standard form of the korn inequality in l p with 1 p and an orlicz version involving a young function a satisfying both the 2 and the 2 condition

local structure of the self focusing singularity of the nonlinear schro dinger equation abstract for the cubic schrodinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d 2 tod 2

semi infinite weil complex and the virasoro algebra we define a semi infinite analogue of the weil algebra associated an infinite dimensional lie algebra it can be used for the definition of semi infinite characteristic classes by analogy with the chern weil construction the second term of a spectral sequence of this weil complex consists of the semi infinite cohomology of the lie algebra with coefficients in its adjoint semi infinite symmetric powers we compute this cohomology for the virasoro algebra this is just the brst cohomology of the bosonic system with central charge 26 we give a complete description of the fock representations of this bosonic system as modules over the virasoro algebra using friedan martinec shenker bosonization we derive a combinatorial identity from this result

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

sobolev inequalities in arbitrary domains abstract a theory of sobolev inequalities in arbitrary open sets in r n is established boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order the relevant sobolev inequalities exhibit the same critical exponents as in the classical framework moreover they involve constants independent of the geometry of the domain and hence yield genuinely new results even in the case when just smooth domains are considered our approach relies upon new representation formulas for sobolev functions and on ensuing pointwise estimates which hold in any open set

when is the weakly nonlinear evolution of a localized disturbance governed by the ginzburg landau equation the evolution of weakly nonlinear modulated disturbances in marginally unstable systems has been the subject of intensive study in a wide range of problems it has been proposed that such disturbances are described by the complex ginzburg landau equation or in the case of two dimensional modulations in certain systems by the davey stewartson equations with complex coefficients in this paper we re examine the evolution of an initially linear localized disturbance into the weakly nonlinear regime we show that for one dimensional modulations the weakly nonlinear evolution is governed initially at least by the landau equation rather than the ginzburg landau equation for two dimensional modulations the evolution is governed by a reduced form of the davey stewartson equations we apply our revised theory to three dimensional poiseuille couette flow for both one and two dimensional modulations we find that a localized disturbance can always grow to order one amplitude if its initial magnitude is sufficiently small contrary to earlier predictions

focusing and multi focusing solutions of the nonlinear schro dinger equation abstract a method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic schrodinger equations in two and three dimensions and describe their universal properties the same method is applied to simulate the multi focusing phenomena produced by simple models of saturating nonlinearities

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

efficient and stable numerical methods for the generalized and vector zakharov system we present efficient and stable numerical methods for approximations of the generalized zakharov system gzs and vector zakharov system for multicomponent plasma vzsm with without a linear damping term the key points in the methods are based on i a time splitting discretization of a schrodinger type equation in gzs or vzsm ii discretizing a nonlinear wave type equation by a pseudospectral method for spatial derivatives and iii solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the methods are explicit unconditionally stable and of spectral order accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping term in gzs or vzsm conserve or keep the same decay rate of the wave energy as that in gzs or vzsm without a or with a linear damping term and give exact results for the plane wave solution extensive numerical tests are presented for plane waves and solitary wave collisions in one dimensional gzs and we also give the dynamics of three dimensional vzsm to demonstrate our new efficient and accurate numerical methods furthermore the methods are applied to study the convergence and quadratic convergence rates of vzsm to gzs and of gzs to the nonlinear schrodinger nls equation in the subsonic limit regime 0 varepsilon ll1 where the parameter varepsilon is inversely proportional to the acoustic speed our tests also suggest that the following meshing strategy or varepsilon resolution is admissible in this regime spatial mesh size h o varepsilon and time step k o varepsilon

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

solenoidal lipschitz truncation and applications in fluid mechanics abstract we extend the lipschitz truncation method to the setting of solenoidal functions in particular we approximate a solenoidal sobolev function by a solenoidal lipschitz function which differs from the original function only on a small set our main application is the existence of weak solutions to the two dimensional prandtl eyring fluid model which has almost linear growth in this situation a correction via bogovski operators does not work furthermore we extend the concept of almost a harmonicity to the fluid context in the pressure free formulation

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

orlicz sobolev boundary trace embeddings we prove a sharp trace embedding theorem for orlicz sobolev spaces into orlicz spaces on the boundary an optimal trace embedding into more general rearrangement invariant spaces on the boundary is also established

wave collapse and instability of solitary waves of a generalized kadomtsev petviashvili equation abstract the solutions of a family of generalized kadomtsev petviasvili equation in two diemnsions u t u m u x u xxx x 2 u yy are studied with appropriate boundary conditions and constraints on the initial data for m in a dense set m 0 and for 2 1 if m 4 then the solution can blow up in finite time we show that if 1 m 4 a solitary wave so exists which is unstable if m 4 3 in this and related problems th e instability of solitary waves is associated with onset of wave collapse we give numerical evidence to support this

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

on the exact solution of 2 1 dimensional cubic nonlinear schrodinger nls equation in this paper symmetry reductions for a cubic nonlinear schrodinger nls equation to complex ordinary differential equations are presented these are obtained by means of lie s method of infinitesimal transformation groups it is shown that ten types of subgroups of the symmetry group lead via symmetry reduction to ordinary differential equations these equations are solved and the similarity solutions are obtained

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star