explicit series solutions to nonlinear evolution equations the sine cosine method in this paper a sine cosine method is used to construct many periodic and solitary wave solutions to two nonlinear evolution systems the coupled quadratic nonlinear equations and the coupled klein gordon schrodinger equations under different parameter conditions explicit formulas for some new periodic and solitary wave solutions are successfully obtained the proposed solutions are found to be important for the explanation of some practical physical problems

riemann hilbert problem for camassa holm equation with step like initial data abstract we consider the cauchy problem for the camassa holm equation the initial data is supposed to be a step like function i e it tends to different limits as spatial variable tends to plus or minus infinity we derive a parametric representation of the solution of the initial value problem in terms of the solution of an associated riemann hilbert problem this allows to study effectively the long time asymptotic behavior of the solution by using the nonlinear steepest descent method at least for two sectors of the x t half plane

long time behavior of the solution to the mkdv equation with step like initial data we consider the modified korteweg de vries equation on the line the initial data is the pure step function i e q x 0 cr for x 0 and q x 0 cl for x cr 0 are arbitrary real numbers long time behavior of the solution to the mkdv equation in the case cl cr 0 and t was studied recently in kotlyarov and minakov 2010 j math phys 51 093506 where the structure of the compression wave was obtained in the form of a modulated elliptic wave the goal of this paper is to study the asymptotic behavior of the solution of the initial value problem as t i e we study the long time dynamics of the rarefaction wave using the steepest descent method and the so called g function mechanism we deform the original oscillatory matrix riemann hilbert problem to the explicitly solvable model forms and show that the solution of the initial value problem has a different asymptotic behavior in different regions of the xt plane in the regions x 6c2rt the main term of asymptotics of the solution is equal to cl and cr respectively in the region the asymptotics of the solution tends to an influence of the dispersion is also studied the second term of the asymptotics is obtained in the region x 6c2rt where the background constant cr is perturbed by the self similar vanishing wave

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

riemann hilbert problems and the mkdv equation with step initial data short time behavior of solutions and the nonlinear gibbs type phenomenon we produce a family of matrix riemann hilbert rh problems parametrized by a given scalar function r k the jump matrix of the rh problem depends on r k and external real parameters x and t we prove the unique solvability of such a problem a solution m x t k of this matrix rh problem gives a smooth solution q x t of the modified korteweg de vries mkdv equation qt 6q2qx qxxx 0 for a special choice of the function r k we show that the corresponding initial data q x 0 is a heaviside type function we consider the small time limit of the solution q x t the leading order term is explicitly calculated as an integral of the airy function the first summand is an exact solution to the linear kdv equation pt pxxx 0 with initial data thus this specific solution q x t to the mkdv equation being smooth with respect to and t 0 takes pointwise its limit values and exhibits a gibbs type phenomenon with rapid oscillations near x 0 as t 0

periodic boundary data for an integrable model of stimulated raman scattering long time asymptotic behavior we study the long time asymptotic behavior of the solution to the initial boundary value ibv problem in the quarter plane x 0 t 0 for nonlinear integrable equations of stimulated raman scattering we consider the case of zero initial condition and periodic boundary data p e i t using the steepest descent method for oscillatory matrix riemann hilbert problems we show that the solution of the ibv problem has different asymptotic behavior in different regions the solution takes the form of a plane wave of finite amplitude when 0 2 t for the ibv problem with nonzero initial condition and the same periodic boundary data the solution to this problem is qualitatively similar to that of this study with the only difference that the solitons of finite amplitude can appear in the region x 2 t

compact and noncompact structures for a variant of kdv equation in higher dimensions in this paper we study two mathematical variants of kdv equation we show that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with cusps or infinite slopes a framework is developed for studying these problems in higher dimensions two distinct sets of general formulas for each type of variants are developed to present a fairly complete understanding of the compact and noncompact dispersive structures

a numerical application of the decomposition method for the combined kdv mkdv equation we consider solitary wave solutions of the combined korteweg de vries modified korteweg de vries kdv mkdv equation we prove the convergence of adomian decomposition method adm applied to the combined kdv mkdv equation then we obtain the exact solitary wave solutions and numerical solutions of the combined kdv mkdv equation for the initial conditions the numerical solutions are compared with the known analytical solutions their remarkable accuracy are finally demonstrated for the combined kdv mkdv equation

the sine cosine method for obtaining solutions with compact and noncompact structures in this paper we establish compact and noncompact solutions for nonlinear dispersive equations a sine cosine method is used to demonstrate this work the different physical structures of the focusing branch and the defocusing branch are emphasized many models are approached to illustrate the usage of our main results

patterns on liquid surfaces cnoidal waves compactons and scaling abstract localized patterns and nonlinear oscillation formations on the bounded free surface of an ideal incompressible liquid are investigated cnoidal modes solitons and compactons as traveling non axially symmetric shapes are discussed a finite difference differential generalized korteweg de vries kdv equation is shown to describe the three dimensional motion of the fluid surface and in the limit of long and shallow channels one recovers the well known kdv equation a tentative expansion formula for the representation of the general solution of a nonlinear equation for given initial conditions is introduced the model is useful in multilayer fluid dynamics cluster formation and nuclear physics since up to an overall scale these systems display a free liquid surface behavior

modulated elliptic wave and asymptotic solitons in a shock problem to the modified korteweg de vries equation we study the long time asymptotic behavior of the cauchy problem for the modified korteweg de vries equation with an initial function of the step type this function rapidly tends to zero as and to some positive constant c as in 1989 khruslov and kotlyarov have found khruslov and kotlyarov 1989 inverse problems 5 1075 88 that for a large time the solution breaks up into a train of asymptotic solitons located in the domain is a constant the number n of these solitons grows unboundedly as in 2010 kotlyarov and minakov have studied temporary asymptotics of the solution of the cauchy problem on the whole line kotlyarov and minakov 2010 j math phys 51 093506 and have found that in the domain this solution is described by a modulated elliptic wave we consider here the modulated elliptic wave in the domain our main result shows that the modulated elliptic wave also breaks up into solitons which are similar to the asymptotic solitons in khruslov and kotlyarov 1989 inverse problems 5 1075 88 but differ from them in phase it means that the modulated elliptic wave does not represent the asymptotics of the solution in the domain the correct asymptotic behavior of the solution is given by the train of asymptotic solitons given in khruslov and kotlyarov 1989 inverse problems 5 1075 88 however in the asymptotic regime as in the region we can watch precisely a pair of solitons with numbers n one of them is the asymptotic soliton while the other soliton is generated from the elliptic wave their phases become closer to each other for a large n i e these solitons are also close to each other this result gives the answer on a very important question about matching of the asymptotic formulas in the mentioned region where the both formulas are well defined thus we have here a new and previously unknown mechanism 5 35 of matching of the asymptotics of the solution in the adjacent regions

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

on a class of nonlinear dispersive dissipative interactions abstract we study a prototype dissipative dispersive equation ut a um x un xxx uk xx a consts which represents a wide variety of interactions at the critical value k m n 2 which separates dispersive and dissipation dominated phenomena these effects are in a detailed balance and the patterns formed do not depend on the amplitude in particular when m n 2 k 1 the equation can be transformed into a form free of convection and dissipation making it accessible to analysis both bounded and unbounded oscillations as well as solitary waves are found a variety of exact solutions are presented with a notable example being a solitary doublet for n 1 and a 2 3 2 the problem may be mapped into a linear equation leading to rational periodic or aperiodic solutions among others

the effect of the order of nonlinear dispersive equation on the compact and noncompact solutions the effect of the order of nonlinear dispersive partial differential equation on the compact and noncompact solutions is studied in this paper two variants of the kdv equation of odd orders are examined our analysis reveals that the physical nature of these nonlinear kdv type of equations are different two distinct sets of general formulas for each type of variants are performed to present a fairly complete understanding of the compact and noncompact structures

composition of pseudo almost automorphic and asymptotically almost automorphic functions this paper is concerned with pseudo almost automorphic functions which are more general and complicated than pseudo almost periodic functions and asymptotically almost automorphic functions new results concerning the composition of pseudo almost automorphic functions are established

adm pade technique for the nonlinear lattice equations adm pade technique is a combination of adomian decomposition method adm and pade approximants we solve two nonlinear lattice equations using the technique which gives the approximate solution with higher accuracy and faster convergence rate than using adm alone bell shaped solitary solution of belov chaltikian bc lattice and kink shaped solitary solution of the nonlinear self dual network equations sdnes are presented comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the technique

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

initial value problem of the whitham equations for the camassa holm equation we study the whitham equations for the camassa holm equation the equations are neither strictly hyperbolic nor genuinely nonlinear we are interested in the initial value problem of the whitham equations when the initial values are given by a step function the whitham solution is self similar when the initial values are given by a smooth function the whitham solution exists within a cusp in the x t plane on the boundary of the cusp the whitham solution matches the burgers solution which exists outside the cusp

compact structures for variants of the generalized kdv and the generalized kp equations in this paper we discuss mathematical variants in higher dimensions of the generalized kdv and the generalized kp equations it is shown that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with infinite slopes or cusps the compact and the noncompact dispersive structures are examined in a general way

compactons and solitary patterns structures for variants of the kdv and the kp equations in this paper we discuss mathematical variants in higher dimensions of the kdv and the kp equations it is shown that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with infinite slopes or cusps the study presents a fairly complete understanding of the compact and noncompact dispersive structures

numerical solution of the small dispersion limit of the camassa holm and whitham equations and multiscale expansions the small dispersion limit of solutions to the camassa holm ch equation is characterized by the appearance of a zone of rapid modulated oscillations an asymptotic description of these oscillations is given for short times by the one phase solution to the ch equation where the branch points of the corresponding elliptic curve depend on the physical coordinates via the whitham equations we present a conjecture for the phase of the asymptotic solution a numerical study of this limit for smooth hump like initial data provides strong evidence for the validity of this conjecture we present a quantitative numerical comparison between the solution to the ch equation and the asymptotic solution the dependence on the small dispersion parameter epsilon is studied in the interior and at the boundaries of the whitham zone in the interior of the zone the difference between the solution to the ch equation and the asymptotic solution is of the order epsilon at the trailing edge of the order sqrt eps

a computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

new compact and noncompact solutions for two variants of a modified camassa holm equation in this paper exact solutions of two variants of a modified camassa holm equation are determined by using direct ansatze as a result compactons solutions solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns solutions having infinite slopes or cusps and plane periodic solutions are established it is also found that the qualitative change in the physical structure of solutions depends mainly on whether the exponent of the wave function u x t and the expression 2k c are positive or negative

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

the sine cosine and the tanh methods reliable tools for analytic treatment of nonlinear dispersive equations the sine cosine method and the tanh method are used for analytic treatment of nonlinear dispersive equations nonlinear variants of boussinesq equation are used as vehicles to show the strength of these methods solutions of distinct physical structures solitons compactons solitary patterns solutions and periodic solutions are formally derived the results show that the change in physical structures of the obtained solutions depends mainly on exponents and on the coefficients of the derivatives involved

nonlinear dispersive special type of the zakharov kuznetsov equation zk n n with compact and noncompact structures in this paper we study nonlinear dispersive special type of the zakharov kuznetsov equation with positive and negative exponents the approach depends mainly on the sine cosine algorithm compactons solitary patterns solitons and periodic solutions are formally derived

almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional applications to heat and wave equations with nonlinearities in this work we study the existence of bounded and almost automorphic solutions for evolution equations in banach spaces we suppose that the linear part is the infinitesimal generator of a compact c0 semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument we give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on r we use the subvariant functional method to show that every k minimizing mild solution is compact almost automorphic applications are provided for both heat and wave equations with nonlinearities in several functional spaces

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

compact and noncompact solutions for nonlinear dispersive variants of the generalized kdv equation in this paper we examine three distinct nonlinear dispersive variants of the generalized kdv equation we study the compact and the noncompact structures for these variants we also emphasize the fact that purely nonlinear dispersion can cause a deep qualitative change in the genuinely nonlinear phenomenon

general solutions with solitary patterns for the defocusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this work the defocusing branch of the genuinely nonlinear dispersive k n n equation that exhibits solutions with solitary patterns is investigated the equation with constant coefficients to the derivatives is examined in one two and three dimensional spaces general formulas for solutions with solitary patterns are developed for all positive integers n n 1 two sets of entirely new formulas one set produces solutions for even integers n whereas the second set produces solutions in opposite directions for odd integers n n 1 are established

exact solutions of compact and noncompact structures for the kp bbm equation the tanh and the sine cosine methods are used for a reliable treatment of the 2 1 dimensional kp bbm equation generalized forms of the kp bbm equation are investigated as well a variety of exact travelling wave solutions of distinct physical structures compactons solitons solitary patterns and periodic solutions are formally derived

an application of the modified adomian decomposition method for n 1 dimensional sine gordon field abstract based upon the modified adomian decomposition method a scheme is developed to obtain approximate jacobi elliptic function solutions of the n 1 dimensional sine gordon field with initial conditions as a result we derive the approximate solution and exact jacobi elliptic function solution of the n 1 dimensional sine gordon field with initial conditions moreover the approximate solution is compared with the exact solution with n 2 and with different modulus k 0 25 0 5 and 1

numerical study of the regularizing effect of the 3d weakly transverse bbm equations for long times from a spectral method combined with a predictor corrector scheme we numerically study the behavior in time of solutions of the three dimensional generalized kadomtsev petviashvili equations regularized using the bbm trick the solution no longer blows up and the solitonic behavior is observed

compactons structures for fifth order kdv like equations in higher dimensions in this paper we study a variant of the fifth order kdv equation fkdv that exhibits compactons solitons with finite wave lengths the work formally shows how to construct compact dispersive structures in higher dimensions two sets of general formulas for compactons solutions that are of substantial interest are developed for this variant fk n n for all positive integers n n 1

a reliable treatment of the physical structure for the nonlinear equation k m n in this work the nonlinear equation k m n is studied for all possible values of m and n we show that this equation may exhibit compactons solitons or periodic solutions the analysis reveals the change of the physical structure of the solutions as a result of the change of m and n

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

a new fractional order soliton equation hierarchy and its integrable coupling system a fractional order soliton equation hierarchy is presented which is a new liouville integrable hierarchy and possesses bi hamiltonian structure finally we obtain a integrable coupling system of this fractional order soliton equation hierarchy by applying a upper triangular matrix of lie algebras

an analytic study of compacton solutions for variants of kuramoto sivashinsky equation in this paper two mathematical variants of kuramoto sivashinsky equation are examined a framework is developed for studying these variants in one and higher dimensions two distinct sets of general formulae for each variant are developed compactons solitons with compact support and noncompact dispersive structures are formally established

solutions of compact and noncompact structures for nonlinear klein gordon type equation the focusing and the defocusing branches of nonlinear klein gordon type equation kg n n are considered a framework is implemented to show that the first model exhibits compactons solitons that do not have exponential tails whereas the second model demonstrates solitary patterns solutions the two variants of kg n n equation will be examined in one and higher dimensions general formulas will be developed to present a fairly complete understanding of the solutions of compact and noncompact structures

asymptotics of step like solutions for the camassa holm equation abstract we study the long time asymptotics of solution of the cauchy problem for the camassa holm equation with a step like initial datum by using the nonlinear steepest descent method and the so called g function approach we show that the camassa holm equation exhibits a rich structure of sharply separated regions in the x t half plane with qualitatively different asymptotics which can be described in terms of a modulated finite gap elliptic function and a finite number of solitons

existence uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in banach spaces abstract we give sufficient conditions ensuring the existence uniqueness and global attractiveness of a pseudo compact almost automorphic solution of the following differential equation x t f t x t in a banach space e where f r e e is a pseudo almost automorphic function with respect to the first argument we essentially assume that the function f is dissipative then we apply the main results to the following equation x t q t x t x t f t 0

complete linearization of a mixed problem to the maxwell bloch equations by matrix riemann hilbert problems considered in this paper the maxwell bloch mb equations became known after lamb 1967 phys lett a 25 181 2 1971 rev mod phys 43 99 144 1973 phys rev lett 31 196 199 1974 phys rev a 9 422 30 ablowitz kaup and newell 1974 j math phys 15 1852 58 proposed the inverse scattering transform ist to the mb equations for studying a physical phenomenon known as the self induced transparency a description of general solutions to the mb equations and their classification was done by gabitov zakharov and mikhailov 1985 teor mat fiz 63 11 31 in particular they gave an approximate solution of the mixed problem to the mb equations in the domain x t 0 l 0 and on this basis a description of the phenomenon of superfluorescence it was emphasized in gabitov et al 1985 teor mat fiz 63 11 31 that the ist method is non adopted for the mixed problem authors of the mentioned papers have developed the ist method in the form of the marchenko integral equations we propose another approach for solving the mixed problem to the mb equations in the quarter plane we use a simultaneous spectral analysis of both the lax operators and matrix riemann hilbert rh problems firstly we introduce appropriate compatible solutions of the corresponding ablowitz kaup newell segur equations and then we suggest such a matrix rh problem which corresponds to the mixed problem for mb equations secondly we generalize this matrix rh problem prove a unique solvability of the new rh problem and show that the rh problem after a specialization of jump matrix generates the mb equations as a result we obtain solutions defined on the whole line and studied in ablowits 1974 j math phys 15 1852 58 and gabitov et al 1985 teor mat fiz 63 11 31 solutions to the mixed problem studied below in this paper and solutions with periodic finite gap boundary conditions the type of solution is defined by specialization of the conjugation contour and jump matrix suggested matrix rh problems will be useful for studying the long time long distance asymptotic behavior of solutions to the mb equations using the deift zhou method of steepest decent

a new riemann hilbert problem in a model of stimulated raman scattering the riemann hilbert problem proposed in 2 for the integrable stimulated raman scattering srs model was shown to be solvable under an additional condition the boundary data have to be chosen in such a way that a corresponding spectral problem has no spectral singularities in the general case it can be shown that a spectral singularity occurs at k 0 on the other hand the initial boundary value ibv problem for the srs equations is known to be well posed using pde techniques this has been established in 3 therefore it seems natural to try to find a new rh problem that is solvable in the presence of arbitrary spectral singularities the formulation of such a rh problem is the main aim of the paper then the solution of the nonlinear initial boundary value problem for the srs equations is expressed in terms of the solution of a linear problem which is the riemann hilbert problem for a sectionally analytic matrix function

variants of the generalized kdv equation with compact and noncompact structures abstract in this paper nonlinear dispersive variants of the generalized kdv equation are investigated the compact and the noncompact physical structures are studied two general formulas for the focusing branch and the defocusing branch that are of substantial interest are formally derived

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

numerical simulation of kadomtsev petviashvili benjamin bona mahony equations using finite difference method in this paper the finite difference method is employed to solve kadomtsev petviashvili benjamin bona mahony ii kp bbm ii partial differential equations the time and space variable are discretized by the crank nicholson method and the central difference scheme respectively the consistence and stability are also proved some examples are investigated to verify the efficiency of the present method

a new modification of adomian decomposition method for solving a kind of evolution equation in this paper we propose a new modification of adomian decomposition method for solving a kind of evolution equation the modification will improve the convergence of the series solution the validity of the modified technique is verified through illustrative examples

pseudo almost automorphic solutions for dissipative differential equations in banach spaces this work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations we prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations

long time asymptotics of the nonlinear schrodinger equation shock problem the long time asymptotics of two colliding plane waves governed by the focusing nonlinear schrodinger equation are analyzed via the inverse scattering method we find three asymptotic regions in space time a region with the original wave modified by a phase perturbation a residual region with a one phase wave and an intermediate transition region with a modulated two phase wave the leading order terms for the three regions are computed with error estimates using the steepest descent method for riemann hilbert problems the nondecaying initial data requires a new adaptation of this method a new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions also the effect of the collision is felt in the plane wave state well beyond the shock front at large times 2007 wiley periodicals inc

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

universality of a double scaling limit near singular edge points in random matrix models we consider unitary random matrix ensembles z n s t 1 e n tr v s t m dm on the space of hermitian n n matrices m where the confining potential vs t is such that the limiting mean density of eigenvalues as n and s t 0 vanishes like a power 5 2 at a singular endpoint of its support the main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit the limiting kernel is built out of functions associated with a special solution of the pi2 equation which is a fourth order analogue of the painleve i equation in order to prove our result we use the well known connection between the eigenvalue correlation kernel and the riemann hilbert rh problem for orthogonal polynomials together with the deift zhou steepest descent method to analyze the rh problem asymptotically the key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint for which we use the model rh problem for the special solution of the pi2 equation

inverse scattering transform for the defocusing nonlinear schrodinger equation with fully asymmetric non zero boundary conditions abstract we formulate the inverse scattering transform ist for the defocusing nonlinear schrodinger nls equation with fully asymmetric non zero boundary conditions i e when the limiting values of the solution at space infinities have different non zero moduli the theory is formulated without making use of riemann surfaces and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem for the direct problem we give explicit single valued definitions of the jost eigenfunctions and scattering coefficients over the whole complex plane and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues we pose the inverse problem as a riemann hilbert problem on an open contour and we reduce the problem to a standard set of linear integral equations finally for comparison purposes we present the single sheet branch cut formulation of the inverse scattering transform for the initial value problem with symmetric equimodular non zero boundary conditions as well as for the initial value problem with one sided non zero boundary conditions and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

generalized boussinesq type of equations with compactons solitons and periodic solutions generalized boussinesq type of equations with positive and negative exponents are examined the analysis depends mainly on the sine cosine ansatz it is formally shown that these nonlinear models give rise to compactons solitary patterns solitons and periodic solutions depending on the exponents and the coefficients of the derivatives of u x t the presented scheme reveals quite a number of remarkable features that will be helpful for identical problems

numerical solutions of coupled burgers equations with time and space fractional derivatives in this paper by introducing the fractional derivative in the sense of caputo the adomian decomposition method is directly extended to study the coupled burgers equations with time and space fractional derivatives as a result the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components the figures show the effectiveness and good accuracy of the proposed method

numerical doubly periodic solution of the kdv equation with the initial condition via the decomposition method in this paper a scheme is extended to study numerical doubly periodic solutions of the kortweg de vries kdv equation with initial conditions using the adomian decomposition method as a result we obtain the numerical doubly periodic solutions of the kdv equation with initial conditions the numerical doubly periodic solution is compared with the exact doubly periodic solution moreover we analyze the absolute error and relative error between them and also give the contour and density plots of the numerical doubly periodic solutions with two different moduli m 15 78

a composition theorem for weighted pseudo almost automorphic functions and applications abstract in this paper we establish a composition theorem for weighted pseudo almost automorphic functions under a weaker lipschitz condition our composition theorem generalizes some known results moreover the existence and uniqueness of pseudo almost automorphic solutions for abstract semilinear evolution equations are studied

exponentially small asymptotics of solutions to the defocusing nonlinear schrodinger equation ii the riemann hilbert problem approach is used to derive a special set of leading order exponentially small asymptotics as t such that x t 0 of solutions to the cauchy problem for the defocusing nonlinear schrodinger equation i t u x2 u 2 u 2 1 u 0 with finite density initial data u x 0 x exp i 11 1 o 1 0 2

a construction of compact and noncompact solutions for nonlinear dispersive equations of even order a construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered two model equations of even orders k 4 are adopted to carry out this study the analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions

solutions of the kp equation associated to rank three commuting differential operators over a singular elliptic curve abstract we give a method for finding explicit solutions of the kadomtsev petviashvili kp equation which are associated to rank three commuting ordinary differential operators explicit formulae for the coefficients of rank three commuting differential operators are used to reduce the kp equation to a nonlinear partial differential equation in only two variables this equation which is obtained in a similar way to the rank two krichever novikov kn equation is easier to solve than kp and some solutions are given

generation of asymptotic solitons of the nonlinear schrodinger equation by boundary data this article is about the focusing nonlinear schrodinger equation on the half line the initial function vanishes at infinity while boundary data are local perturbations of periodic or quasi periodic finite gap functions we study the corresponding scattering problem for the zakharov shabat compatible differential equations the representation of the solution of the nonlinear schrodinger equation in the quarter of the x t plane through functions which satisfy marchenko integral equations we use this formalism to investigate the asymptotic behavior of the solution for large time we prove that under certain conditions a periodic quasi periodic behavior at infinity of boundary data generates an unbounded train of asymptotic solitons running away from the boundary the asymptotics of the solution shows that boundary data with periodic behavior as time tends to infinity generates a train of such asymptotic solitons even in the case when the initial function is identically zero

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

a study on nonlinear dispersive partial differential equations of compact and noncompact solutions in this paper we study two genuinely nonlinear dispersive partial differential equations in one and higher dimensional spaces we show that the focusing branch and the defocusing branch of these equations each leads to a different physical structure in addition we develop general solutions of compact and noncompact support which are not obtainable through classic techniques

the inverse scattering transform for the focusing nonlinear schrodinger equation with asymmetric boundary conditions the inverse scattering transform ist as a tool to solve the initial value problem for the focusing nonlinear schrodinger nls equation with non zero boundary values ql r t al re 2ial r2t i l r as x is presented in the fully asymmetric case for both asymptotic amplitudes and phases i e with al ar and l r the direct problem is shown to be well defined for nls solutions q x t such that q x t ql r t l1 1 r with respect to x for all t 0 and the corresponding analyticity properties of eigenfunctions and scattering data are established the inverse scattering problem is formulated both via left and right marchenko integral equations and as a riemann hilbert problem on a single sheet of the scattering variables l r k2 al r2 where k is the usual complex scattering parameter in the ist the time evolution of the scattering coefficients is then derived showing that unlike the case of solutions with equal amplitudes as x here both reflection and transmission coefficients have