symmetry analysis and exact solutions of the 2 1 dimensional sine gordon system according to the space variable exchange symmetry we change the 2 1 dimensional sine gordon system obtained by konopelchenko and rogers kr to a variant form some types of similarity reductions are obtained by using some lie symmetry analysis from these similarity reductions we find that the soliton structure of the system possesses quite a rich structure the line solitons parallel to the lines x y 0 and x y 0 may have an arbitrary shape in addition to the well known dromion solution which is constructed by two line solitons one may find many other kinds of soliton solutions localized in all directions for instance some kinds of ring type basin like plateau like and bowl like and instanton type soliton solutions can be found directly by selecting the arbitrary functions included in the single soliton solution the dromion solutions may also be constructed by straight line and curved line solitons

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

full symmetry groups painleve integrability and exact solutions of the nonisospectral bkp equation abstract based on the generalized symmetry group method presented by lou and ma lou and ma non lie symmetry groups of 2 1 dimensional nonlinear systems obtained from a simple direct method j phys a math gen 38 2005 l129 firstly both the lie point groups and the full symmetry group of the nonisospectral bkp equation are obtained at the same time a relationship is constructed between the new solutions and the old ones of equation secondly the nonisospectral bkp can be proved to be painleve integrability by combining the standard wtc approach with the kruskal s simplification some solutions are obtained by using the standard truncated painleve expansion finally based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated painleve expansion some interesting solution are constructed

a lax pair for the two dimensional euler equation a lax pair for the 2d euler equation is found a lax pair for the 2d euler equation is found

semiclassical limit of focusing nls for a family of square barrier initial data the small dispersion limit of the focusing nonlinear schrodinger equation nls exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales the non self adjoint scattering problem and ill posed limiting whitham equations associated to focusing nls make rigorous asymptotic results difficult previous studies have focused on special classes of analytic initial data for which the limiting elliptic whitham equations are wellposed in this paper we consider another exactly solvable family of initial data the family of square barriers 0 x q l l for real amplitudes q using riemann hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing nls globally in space and up to an o 1 maximal time in particular we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data to the best of our knowledge this is the first case in which the genus structure of the semiclassical asymptotics for focusing nls have been calculated for nonanalytic initial data 2013 wiley periodicals inc

non lie symmetry groups of 2 1 dimensional nonlinear systems obtained from a simple direct method a new direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems using the direct method for the well known 2 1 dimensional kadomtsev petviashvili equation and the ablowitz kaup newell segur system both the lie point symmetry groups and the non lie symmetry groups are obtained the lie symmetry groups obtained via traditional lie approaches are only special cases furthermore the expressions of the exact finite transformations of the lie groups are much simpler than those obtained via the standard approaches

painleve property soliton like solutions and complexitons for a coupled variable coefficient modified korteweg de vries system in a two layer fluid model abstract as a model derived from a two layer fluid system which describes the atmospheric and oceanic phenomena a coupled variable coefficient modified korteweg de vries system is concerned in this paper with the help of symbolic computation its integrability in the painleve sense is investigated furthermore hirota s bilinear method is employed to construct the bilinear forms through the dependent variable transformations and soliton like solutions and complexitons are derived finally effects of variable coefficients are discussed graphically and it is concluded that the variable coefficients control the propagation trajectories of solitons and complexitons

vortices circumfluence symmetry groups and darboux transformations of the 2 1 dimensional euler equation the euler equation ee is one of the basic equations in many physical fields such as fluids plasmas condensed matter astrophysics and oceanic and atmospheric dynamics a symmetry group theorem of the 2 1 dimensional ee is obtained via a simple direct method which is thus utilized to find exact analytical vortex and circumfluence solutions a weak darboux transformation theorem of the 2 1 dimensional ee can be obtained for an arbitrary spectral parameter from the general symmetry group theorem possible applications of the vortex and circumfluence solutions to tropical cyclones especially hurricane katrina 2005 are demonstrated

kinematic characterization of valvular opening the evaluation of the valvular opening due to a pulse flow as is the case of cardiac valves requires the knowledge of the leaflets material properties and the coupled solution of the fluid and solid equations this approach is not commonly feasible a different approach is introduced here to describe the opening behavior of valvular leaflets by a functional kinematic relationship the asymptotic analysis in the limit of leaflet opening without vortex shedding is presented for a two dimensional rigid leaflet model under the irrotational scheme the approach is then verified by numerical solution of the navier stokes equation in asymptotic and nonasymptotic conditions

on the asymptotic stability of n soliton solutions of the defocusing nonlinear schrodinger equation we consider the cauchy problem for the defocusing nonlinear schrodinger nls equation for finite density type initial data using the overline partial generalization of the nonlinear steepest descent method of deift and zhou we derive the leading order approximation to the solution of nls for large times in the solitonic region of space time x 2 t and we provide bounds for the error which decay as t rightarrow infty for a general class of initial data whose difference from the non vanishing background possesses a fixed number of finite moments and derivatives using properties of the scattering map of nls we derive as a corollary an asymptotic stability result for initial data that are sufficiently close to the n dark soliton solutions of nls

asymptotic stability of solitons of the gkdv equations with general nonlinearity we consider the generalized korteweg de vries equation gkdv partial t u partial x partial x 2 u f u 0 quad t x in 0 t times mathbb r with general c3 nonlinearity f under an explicit condition on f and c 0 there exists a solution in the energy space h1 of the type u t x qc x x0 ct called soliton in this paper under general assumptions on f and qc we prove that the family of solitons around qc is asymptotically stable in some local sense in h1 i e if u t is close to qc for all t 0 then u t locally converges in the energy space to some qc as t note in particular that we do not assume the stability of qc this result is based on a rigidity property of the gkdv equation around qc in the energy space whose proof relies on the introduction of a dual problem these results extend the main results in martel siam j math anal 38 759 781 2006 martel and merle j math pures appl 79 339 425 2000 arch ration mech anal 157 219 254 2001 nonlinearity 1 55 80 devoted to the pure power case

on some local cohomology modules we give an explicit description of a certain high order local cohomology module with support in an ideal in terms of some combinatorial information about the minimal primes of the ideal as a consequence we give a combinatorial necessary and sufficient condition for an upper bound on the cohomological dimension of the ring with support in the ideal

symmetry full symmetry groups and some exact solutions to a generalized davey stewartson system the lie symmetry algebra of a generalized davey stewartson gds system is obtained the general element of this algebra depends on eight arbitrary functions of time which has a kac moody virasoro loop algebra structure and is isomorphic to that of the standard integrable davey stewartson equations under certain conditions imposed on parameters and arbitrary functions then based on the symmetry group direct method proposed by lou and ma j phys a 38 l129 2005 the full symmetry groups of the gds system are obtained from the full symmetry groups both the lie symmetry group and a group of discrete transformations can be obtained finally some exact solutions involving sech sech2 sech2 and tanh tanh2 tanh2 type solitary wave solutions are presented by a generalized subequation expansion method

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

universality of the break up profile for the kdv equation in the small dispersion limit using the riemann hilbert approach we obtain an asymptotic expansion for the solution of the cauchy problem for the korteweg de vries kdv equation u t 6uu x epsilon 2 u xxx 0 u x t 0 epsilon u 0 x for epsilon small near the point of gradient catastrophe x c t c for the solution of the dispersionless equation u t 6uu x 0 the sub leading term in this expansion is described by the smooth solution of a fourth order ode which is a higher order analogue to the painleve i equation this is in accordance with a conjecture of dubrovin suggesting that this is a universal phenomenon for any hamiltonian perturbation of a hyperbolic equation using the deift zhou steepest descent method applied on the riemann hilbert problem for the kdv equation we are able to prove the asymptotic expansion rigorously in a double scaling limit

scaling structure of the velocity statistics in atmospheric boundary layers the statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model they contain contributions from various two and three dimensional aspects and from the superposition of inhomogeneous and anisotropic contributions we employ the recently introduced decomposition of statistical tensor objects into irreducible representations of the so 3 symmetry group characterized by j and m indices where j5 0 2 j m j to disentangle some of these contributions separating the universal and the asymptotic from the specific aspects of the flow the different j contributions transform differently under rotations and so form a complete basis in which to represent the tensor objects under study the experimental data are recorded with hot wire probes placed at various heights in the atmospheric surface layer time series data from single probes and from pairs of probes are analyzed to compute the amplitudes and exponents of different contributions to the second order statistical objects characterized by j 50 1 and 2 the analysis shows the need to make a careful distinction between long lived quasi twodimensional turbulent motions close to the ground and relatively short lived three dimensional motions we demonstrate that the leading scaling exponents in the three leading sectors j50 1 and 2 appear to be different but universal independent of the positions of the probe the tensorial component considered and the large scale properties the measured values of the scaling exponent are z 2 j50 50 6860 01 z 2 j51 51 0 60 15 and z 2 j52 51 3860 10 we present theoretical arguments for the values of these exponents using the clebsch representation of the euler equations neglecting anomalous corrections the values obtained are 2 3 1 and 4 3 respectively some enigmas and questions for the future are sketched