symmetries and algebras of the integrable dispersive long wave equations in 2 1 dimensional spaces similarly to the kadomtsev petviashvili kp equation a set of generalized symmetries with arbitrary functions of t is given by a simple constructable formula for the integrable dispersive long wave equations in 2 1 space dimensions these symmetries constitute an infinite dimensional lie algebra which is a generalization to the known winfinity algebra

the 2 1 dimensional sine gordon equation integrability and localized solutions in this paper the 2 1 dimensional sine gordon equation 2dsg introduced by konopelchenko and rogers is investigated and is shown to satisfy the painleve property a variable coefficient hirota bilinear form is constructed by judiciously using the painleve analysis with a non conventional choice of the vacuum solutions first the line kinks are constructed then exact localized coherent structures in the 2dsgi equation are generated by the collision of two non parallel ghost solitons which drive the two non trivial boundaries present in the system also the reason for the difficulty in identifying localized solutions in the 2dsgii equation is indicated we also highlight the significance of the asymptotic values of the boundaries of the system

quantum monodromy and nonconcentration near a closed semi hyperbolic orbit for a large class of semiclassical operators p h z which includes schrodinger operators on manifolds with boundary we construct the quantum monodromy operator m z associated to a periodic orbit of the classical flow using estimates relating m z and p h z we prove semiclassical estimates for small complex perturbations of p h z in the case is semi hyperbolic as our main application we give logarithmic lower bounds on the mass of eigenfunctions away from semi hyperbolic orbits of the associated classical flow as a second application of the monodromy operator construction we prove if is an elliptic orbit then p h admits quasimodes which are well localized near

similarity and conditional similarity reductions of a 2 1 dimensional kdv equation via a direct method to get the similarity solutions of a nonlinear physical equation one may use the classical lie group approach nonclassical lie group approach and the clarkson and kruskal ck direct method in this paper the direct method is modified to get the similarity and conditional similarity reductions of a 2 1 dimensional kdv type equation ten types of usual similarity reductions including the 1 1 dimensional shallow water wave equation and the variable kdv equation and six types of conditional similarity reductions of the 2 1 dimensional kdv equation are obtained some special solutions of the conditional similarity reduction equations are found to show the nontriviality of the conditional similarity reduction approach the conditional similarity solutions cannot be obtained by using the nonclassical lie group approach in its present form how to modify the nonclassical lie group approach to obtain the conditional similarity solutions is still open

the analysis of activity networks under generalized precedence relations gprs we present a model for activity networks under generalized precedence relations gprs discuss its temporal analysis and the issues that may arise relative to inconsistency among the specified relations and the activity durations we also give more precise definition to the concept of criticality of an activity and introduce the new concept of flexibility of an activity which is akin to the traditional concept of activity floats in regular cpm with the latter taking on different meaning from its common interpretation in standard cpm issues of optimization are raised when one assumes for each activity a piecewise linear time cost function that permits positive and negative deviations from its least cost duration between specified lower and upper bounds on that duration we seek the optimal activity durations subject to the specified gprs and a given due date we also seek the construction of the complete project duration cost function between the project minimum duration and its least cost duration when the due date is interpreted first as a deadline and second as a target date with rewards for early and penalties for late completion the relations between the problems posed and the uncapacitated minimum cost flow problems are revealed and are utilized in the algorithmic solution of the problems

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

refined and microlocal kakeya nikodym bounds of eigenfunctions in higher dimensions we prove a kakeya nikodym bound on eigenfunctions and quasimodes which sharpens a result of the authors blair and sogge in anal pde 8 747 764 2015 and extends it to higher dimensions as in the prior work the key intermediate step is to prove a microlocal version of these estimates which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic geodesic flow in a companion paper blair and sogge in j differ geom 2015 it will be seen that these sharpened estimates yield improved l q m bounds on eigenfunctions in the presence of nonpositive curvature when 2 q frac 2 d 1 d 1

a branch and bound procedure for the resource constrained project scheduling problem with generalized precedence relations abstract we present an optimal solution procedure for the resource constrained project scheduling problem rcpsp with generalized precedence relations rcpsp gpr with the objective of minimizing the project makespan the rcpspgpr extends the rcpsp to arbitrary minimal and maximal time lags between the starting and completion times of activities the proposed procedure is suited for solving a general class of project scheduling problems and allows for arbitrary precedence constraints activity ready times and deadlines multiple renewable resource constraints with time varying resource requirements and availabilities several types of permissible and mandatory activity overlaps and multiple projects it can be extended to other regular and non regular measures of performance essentially the procedure is a depth first branch and bound algorithm in which the nodes in the search tree represent the original project network extended with extra precedence relations to resolve a number of resource conflicts these conflicts are resolved using the concept of minimal delaying modes which is an extension of the notion of minimal delaying alternatives for the rcpsp several bounds and dominance rules are used to fathom large portions of the search tree extensive computational experience is reported

symmetry group and non propagating solitons in the 2 1 dimensional sine gordon equation starting from the compact version of the 2 1 dimensional sine gordon equation the symmetry group and then the lie symmetries and the related algebra can be obtained via a simple combination of a gauge transformation of the spectral function and the transformations of the space time variables thereout applying the group transformation theorem on a two straight line soliton solution one can derive two types of meaningful non propagating dromion lattice excitations

nonlinear schrodinger type equations from multiscale reduction of pdes ii necessary conditions of integrability for real pdes the multiscale method has been applied in j math phys 41 6399 2000 to evolution pdes to obtain various nonlinear schrodinger type equations here these results are exploited to derive necessary conditions of integrability for nonlinear real evolution pdes in 1 1 dimensions these conditions are given by explicit formulas and several examples are detailed

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

on generalized loewner systems novel integrable equations in 2 1 dimensions a reinterpretation and generalization of a class of infinitesimal backlund transformations originally introduced in a gas dynamics context by loewner in 1952 leads to a linear representation for a novel class of 2 1 dimensional nonlinear equations the latter may be parametrized in terms of a triad of eigenfunctions moreover like the well known integrable davey stewartson and nizhnik novikov veselov equations the nonlinear systems typically contain the two spatial variables on an equal footing the dressing method is outlined for these generalized loewner systems it is noted that basic reductions lead to 2 1 dimensional versions of the principal chiral fields model toda lattice system and notably the classical sine gordon equation loewner systems on grassmannian the projective cpn and rpn manifolds are considered in particular the 2 1 dimensional integrable sine gordon system in which x and y occur in a symmetric manner arises naturally in this context a gauge equivalent system likewise eme

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

generalized dromion solutions of the 2 1 dimensional kdv equation the rich dromion structures for a 2 1 dimensional kdv equation are revealed the dromions in a high dimensional integrable model may have a free shape in one or more directions multi dromion solutions can be driven by perpendicular line non perpendicular line and curved line ghost solitons

on no wait and no idle flow shops with makespan criterion the paper deals with the np hard problems of minimizing the makespan in m machine no wait and no idle permutation flow shops we identify networks whose longest path lengths represent the makespans these networks reveal the duality between the two problems and show graphical explanations of the fact that under no wait and no idle conditions the makespan can be a decreasing function of some job processing times moreover they also lead to a natural reduction of the no wait flow shop problem to the traveling salesman problem some lower bounds on the shortest makespan and new efficiently solvable special cases

riemannian manifolds with maximal eigenfunction growth on any compact riemannian manifold m g of dimension n the l sp 2 normalized eigenfunctions phi sb lambda satisfy lvert rvert phi sb lambda lvert rvert sb infty leq c lambda sp n 1 2 where delta phi sb lambda lambda sp 2 phi sb lambda the bound is sharp in the class of all m g since it is obtained by zonal spherical harmonics on the standard n sphere s sp n but of course it is not sharp for many riemannian manifolds for example for flat tori mathbb r sp n gamma we say that s sp n but not mathbb r sp n gamma is a riemannian manifold with maximal eigenfunction growth the problem that motivates this paper is to determine the m g with maximal eigenfunction growth our main result is that such an m g must have a point x where the set mathscr l sb x of geodesic loops at x has positive measure in s sp ast sb xm we show that if m g is real analytic this puts topological restrictions on m for example only m s sp 2 or m mathbb r p sp 2 topologically in dimension 2 can possess a real analytic metric of maximal eigenfunction growth we further show that generic metrics on any m fail to have maximal eigenfunction growth in addition we construct an example of m g for which mathscr l sb x has positive measure for an open set of x but which does not have maximal eigenfunction growth thus it disproves a naive converse to the main result

l p norms nodal sets and quantum ergodicity abstract for small range of p 2 we improve the l p bounds of eigenfunctions of the laplacian on negatively curved manifolds our improvement is by a power of logarithm for a full density sequence of eigenfunctions we also derive improvements on the size of the nodal sets our proof is based on a quantum ergodicity property of independent interest which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

on the complexity of project scheduling to minimize exposed time we consider project scheduling where the project manager s objective is to minimize the time from when an adversary discovers the project until the completion of the project we analyze the complexity of the problem identifying both polynomially solvable and np hard versions of the problem the complexity of the problem is seen to be dependent on the nature of renewable resource constraints precedence constraints and the ability to crash activities in the project

localized solutions with chaotic and fractal behaviours in a 2 1 dimensional dispersive long wave system it is known that in high dimensions there exist abundant localized excitations such as dromions lumps ring soliton solutions and so on in this paper the possible chaotic and fractal localized structures are revealed for the 2 1 dimensional dispersive long wave equation the chaotic and fractal dromion and lump patterns of the model are constructed by some types of lower dimensional chaotic and fractal patterns

formal variable separation approach for nonintegrable models using a formal variable separation approach a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations taking the kdv burgers and modified kdv burgers equations with background interaction as simple examples some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained

extended multilinear variable separation approach and multivalued localized excitations for some 2 1 dimensional integrable systems the multilinear variable separation approach and the related universal formula have been applied to many 2 1 dimensional nonlinear systems starting from the universal formula abundant 2 1 dimensional localized excitations have been found in this paper the universal formula is extended in two different ways one is obtained for the modified nizhnik novikov veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the 2 1 dimensional symmetric sine gordon system the other is for the dispersive long wave equation the broer kaup kupershmidt system the higher order broer kaup kupershmidt system and the burgers system where arbitrary number of variable separated functions can be involved because of the existence of the arbitrary functions in both the original universal formula and its extended forms the multivalued functions can be used to construct a new type of localized excitations folded solitary waves fsws and foldons the fsws

new similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations the author discusses similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations 1972 both of which arise in several physical applications including shallow water waves both equations are thought to non integrable i e not solvable by inverse scattering since numerical studies show that the interaction of solitary waves in inelastic in particular we determine some new similarity reductions of the symmetric regularised long wave equation these new similarity reductions are not obtainable using the classical lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques it is shown that every similarity reduction of both the symmetric regularised long wave and modified benjamin bona mahoney equations obtained using the classical lie group method reduces the associated partial differential equation to an ordinary differential equation of painleve type whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of painleve type it is also shown that neither the symmetric regularised long wave equation nor the modified benjamin bona mahoney equation possesses the painleve property for partial differential equations as defined by weiss et al 1983

on the painleve property of nonlinear field equations in 2 1 dimensions the davey stewartson system with the purpose of clarifying some aspects of the complete integrability of nonlinear field equations a singular point analysis is performed of the davey stewartson system which can be considered as an extension in 2 1 dimensions of the nonlinear schrodinger equation it is found that the system under consideration possesses the painleve property and allows a set of backlund transformations obtained by truncating the series expansions of the solutions about the singularity manifold

nonlinear schrodinger type equations from multiscale reduction of pdes i systematic derivation in this article we begin a systematic investigation via multiscale expansions of nonlinear evolution pdes partial differential equations in this first article we restrict consideration to a single autonomous but otherwise generic pde in 1 1 variables space time of first order in time whose linear part is dispersive and to solutions dominated by a single plane wave satisfying the linear part of the pde the expansion parameter is an assumedly small coefficient multiplying this plane wave the main indeed asymptotically exact effect of the weak nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics which is generally described in appropriately defined coarse grained time and space variables by evolution equations of nonlinear schrodinger type a systematic analysis of such equations is presented corresponding to various assumptions on the resonances occurring for the first few harmonics

linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces abstract we obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces as applications we improve the known estimates for oscillatory integrals

localized excitations of the 2 1 dimensional sine gordon system the 2 1 dimensional sine gordon system is investigated via a generalized bilinear operator representation plateau basin bowl and saddle type ring solitons are thereby constructed it is indicated how iterated moutard transformations may be employed to extend the range of the method

a path enumeration approach for the analysis of critical activities in fuzzy networks this paper addresses the problem of determining the degree of possible and necessary criticality of activities as well as determining paths in networks that have fuzzy activity durations in such networks activities and paths are reported in a fuzzy representation as being critical with certain degrees of possibility and necessity instead of being declared critical or not in a binary way although the problem of computing the possibility and necessity degrees of criticality for paths have been investigated in the literature those problems for activities have not yet been addressed herein an efficient algorithm that relies on a path enumeration technique is proposed to compute the possibility degrees of criticality of activities additionally the proposed algorithm computes paths with maximum and minimum degrees for the necessity of criticality which correspond to activities real world project networks were used to evaluate the performance of the algorithm

a time oriented branch and bound algorithm for resource constrained project scheduling with generalised precedence constraints resource constrained project scheduling with generalised precedence constraints is a very general scheduling model with applications in areas such as make to order production planning we describe a time oriented branch and bound algorithm that uses constraint propagation techniques which actively exploit the temporal and resource constraints of the problem in order to reduce the search space extensive computational experiments with systematically generated test problems show that the algorithm solves more problems to optimality than other exact solution procedures which have recently been proposed and that the truncated version of the algorithm is also a very good heuristic

group theoretical analysis of dispersive long wave equations in two space dimensions abstract the symmetry algebra of an integrable dispersive long wave equation in two space dimensions is shown to be infinite dimensional and to have a kac moody virasoro structure the corresponding symmetry group is used to generate a large number of new solutions in particular solitons kinks and periodic waves these waves have wave crests of quite general shapes the results are compared to those for a nonintegrable two space dimensional dispersive long wave equation for which the symmetry group is finite dimensional and wave crests are always straight lines

new similarity solutions for the modified boussinesq equation in this paper the author presents some new similarity solutions of the modified boussinesq equation which is a completely integrable soliton equation these new similarity solutions include reductions to the second and fourth painleve equations which are not obtainable using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques

a new integrable davey stewartson type equation a new integrable nonlinear partial differential equation pde in 2 1 dimensions is derived starting from the konopelchenko dubrovsky equation we use an asymptotically exact reduction method based on fourier expansion and spatio temporal rescaling and obtain a new integrable davey stewartson type equation in order to demonstrate the integrability of the new equation by the inverse scattering method we apply the reduction technique to the lax pair of the konopelchenko dubrovsky equation and find the corresponding lax pair of the new equation the new equation reduces to the davey stewartson or the nonlinear schrodinger equation by appropriate limits

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

infinitely many symmetries of the davey stewartson equation using a formal series symmetry method for the davey stewartson equation dse we find that there exist two sets of infinitely many formal series symmetries of the dse however different from the kadomtsev petviashvili equation kpe we failed to get infinitely many truncated symmetries only six truncated symmetries can be obtained simply from the formal series symmetries

logarithmic improvements in l p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature we consider the problem of proving l p bounds for eigenfunctions of the laplacian in the high frequency limit in the presence of nonpositive curvature and more generally manifolds without conjugate points in particular we prove estimates at the critical exponent p c frac 2 d 1 d 1 where a spectrum of scenarios for phase space concentration must be ruled out our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical l p bounds of the second author

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

lax pairs galore a formula that yields an apparently but only apparently nontrivial lax pair for any nonlinear evolution pde in 1 1 dimensions possessing a local conservation law is presented several examples are exhibited

the sine gordon equation in the finite line a rather complete numerical study of the perturbed and nonperturbed sine gordon equation in one dimensional cartesian coordinates and finite domains is presented the numerical study is based on five three point linearly implicit finite difference methods of different spatial accuracy for the unperturbed sge the accuracy of these five methods is found to be very sensitive to the implicitness parameter less sensitive to the spatial order of accuracy and almost no sensitive to the time step for second order accurate temporal discretizations the largest errors of the five methods were found to occur at the front of the soliton solution of the unperturbed sge but the techniques were very accurate for long time computations of the unperturbed sge in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries despite the fact that they do not preserve a discrete energy it is shown that the largest values of the kinetic and total energies and the smallest values of the strain and potential energies occur when solitons and antisolitons strike on the boundaries subject to homogeneous neumann conditions and the four energies remain constant between collisions with the boundaries for the malomed s perturbation one of the fronts of the soliton soliton doublet and soliton antisoliton doublet is trapped at one boundary and the other one undergoes a change in amplitude and its front describes a curved trajectory the soliton antisoliton breather oscillates in a damped manner and slowly drifts towards one boundary the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet of the unperturbed sge are robust under kivshar and malomed s perturbations whereas the soliton antisoliton breather loses its integrity becomes a kink antikink pair and then yields a chaotic behaviour in both space and time small amplifications proportional to the first order temporal derivative of the amplitude do not alter substantially the initial dynamics of the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet but their amplitudes exhibit oscillatory behaviour of increasing amplitude at large times however small amplifications cause first a transition from the soliton antisoliton breather to the kink antikink solution a linearized stability analysis of the spatially homogeneous solutions of the unperturbed sge is performed and analytical solutions are obtained by means of the method of separation of variables for both homogeneous dirichlet and neumann boundary conditions and the results show that the solution corresponding to a zero amplitude is oscillatory in both space and time whereas that corresponding to an amplitude equal to p may exhibit exponential or linear growth depending on the length of the domain if either of these growths is present it is shown that the evolution of small initial perturbations may result in homoclinic crossings and chaotic behaviour in both space and time in the absence of perturbations in the sge and the existence of an attractor when perturbations of the malomed s type are present in either case the solution may exhibit certain spatial symmetries or antisymmetries at large times finally the results presented in this paper indicate that the accurate computation of homoclinic orbits of the sge is not only controlled by the spatial accuracy of the numerical method but also by its energy preserving characteristics and perhaps by the number of grid points used in the numerical technique

a class of integrable davey stewartson type systems a new integrable davey stewartson type class of systems of partial differential equations in 2 1 dimensions is derived from a previously known integrable equation by means of an asymptotically exact nonlinear reduction method based on fourier expansion and spatio temporal rescaling the integrability by the inverse scattering method is explicitly demonstrated because the reduction technique is applied to the lax pair of the starting equation and the corresponding lax pair of the class of systems of equations is found the particular characteristics of the reduction method imply that the new systems are likely to be of applicative relevance

dromion like structures in a 3 1 dimensional kdv type equation dromion structures have been found in many 2 1 dimensional models the similar structure in 3 1 dimensions is studied in this paper for a kdv type equation starting from the bilinear form of the model we found that there are five types of multi dromion solutions for its potentials say the first type of multi dromion solution is driven by multi camber solitons with one of them being non parallel to the x axis the second third and fourth types of multi dromion solutions are driven by two camber solitons and one two and three sets of parallel plane solitons respectively the fifth type of multi dromion solution is driven by one set of parallel plane solitons one set of camber solitons which are parallel to the x axis and one camber soliton which is non parallel to the x axis phase shifts may be involved in the interactions among the multi dromions for the first and fifth types of solutions a single dromion solution of the model may possess quite a free shape for instance the point like dromions ring type dromions extended and sharp dromions and oscillatory dromions can be obtained by selecting some arbitrary functions appropriately