non existence of l 2 compact solutions of the kadomtsev petviashvili ii equation we prove that there is no nontrivial solution of the kadomtsev petviashvili ii equation kp ii equation u t u xxx uu x x u yy 0 quad x y in bf r 2 which is l 2 compact i e uniformly localized in l 2 norm and travel to the right in the x variable this result extends the previous work of de bouard and saut 3 stating that there is no traveling wave solution for the kp ii equation the proof uses a monotonicity property of the l 2 mass for solutions of the kp ii equation similar to the one for the kdv equation 12 14 and two virial type relations the result still holds for some natural generalizations of the kp ii equation general nonlinearity higher dispersion and does not rely on the integrability of the equation

on critical behaviour in generalized kadomtsev petviashvili equations an asymptotic description of the formation of dispersive shock waves in solutions to the generalized kadomtsev petviashvili kp equation is conjectured the asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the painleve i hierarchy several examples are discussed numerically to provide strong evidence for the validity of the conjecture the numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the subcritical kp equations while in the supercritical kp equations a blow up occurs after the formation of the dispersive shock waves

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

on equations of kp type we discuss the local cauchy problem for the generalised kadomtsev petviashvili equation namely in the periodic and nonperiodic settings

global well posedness for the kp i equation on the background of a non localized solution we prove that the cauchy problem for the kp i equation is globally well posed for initial data which are localized perturbations of arbitrary size of a non localized i e not decaying in all directions traveling wave solution e g the kdv line solitary wave or the zaitsev solitary waves which are localized in x and y periodic or conversely

exact solutions of the bogoyavlenskii equation using the multiple g g expansion method in this work exact traveling wave solutions of the bogoyavlenskii equation are studied by the g g expansion method and its variants the solutions obtained are general solutions which are in the form of hyperbolic trigonometric and rational functions and a variety of special solutions like kink shaped antikink shaped bell type soliton solutions etc can easily be derived from the general results under certain domain these methods are more effective and simple than other methods and a number of solutions can be obtained at the same time in order to visualize the underlying dynamics of the obtained solutions two and three dimensional plots are drawn

descent for n bundles abstract given a lie group g one constructs a principal g bundle on a manifold x by taking a cover u x specifying a transition cocycle on the cover and descending the trivialized bundle u g along the cover we demonstrate the existence of an analogous construction for local n bundles for general n we establish analogues for simplicial lie groups of moore s results on simplicial groups these imply that bundles for strict lie n groups arise from local n bundles our construction leads to simple finite dimensional models of lie 2 groups such as string n from cocycle data

correction global well posedness for the kp i equation as kindly pointed out by a faminskii the anisotropic sobolev inequality 7 inproposition 3 of 5 is not known to hold note that 7 is only used in the proofof propositions 7 8 in 5 propositions 7 8 of 5 however still hold as will beexplained here the main new point is the following extension of proposition 6of 5

the cauchy problem for the rotation modified kadomtsev petviashvili type equation abstract this paper is devoted to studying the cauchy problem for the rotation modified kadomtsev petviashvili rmkp type equation x u t x 3 u x u 2 y 2 u u 0 in the anisotropic sobolev spaces h s 1 s 2 r 2 when 0 and 0 0 we show that the cauchy problem is locally well posed in h s 1 s 2 r 2 with s 1 1 2 and s 2 0 the main difficulty in establishing bilinear estimates related to nonlinear term of rmkp type equation is that the resonant function 3 1 2 1 2 1 2 2 2 1 2 1 2 1 1 2 2 2 may tend to zero since 0 0 and 0 when 0 and 0 and 0 we also prove that the cauchy problem for rmkp equation is ill posed in h s 1 0 r 2 with s 1 1 2 in the sense that the flow map associated to the rotation modified kadomtsev petviashvili is not c 3

the prolongation structure of the inhomogeneous equation of the reaction diffusion type the inhomogeneous extension of equations of the reaction diffusion type is investigated by means of the covariant prolongation structures theory we construct the sl 2 r r t prolongation structure for an inhomogeneous equation of the reaction diffusion type and give the corresponding akns type

global well posedness of the kp i initial value problem in the energy space we prove that the kp i initial value problem begin cases partial tu partial x 3u partial x 1 partial y 2u partial x u 2 2 0 text on mathbb r 2 x y times mathbb r t u 0 phi end cases is globally well posed in the energy space mathbf e 1 mathbb r 2 big phi mathbb r 2 to mathbb r phi mathbf e 1 mathbb r 2 approx phi l 2 partial x phi l 2 big partial x 1 partial y phi big l 2 infty big

solitary wave solutions of nonlinear evolution and wave equations using a direct method and macsyma the direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized the class of solitary wave solutions is extended to analytic rather than rational functions of the real exponential solutions of the linearized equation expanding the solutions in an infinite series in these real exponentials an exact solution of the nonlinear pde is obtained whenever the series can be summed methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed the algorithm is now suited to solving nonlinear equations by any symbolic manipulation program this direct method is illustrated by constructing exact solutions of a generalized kdv equation the kuramoto sivashinski equation and a generalized fisher equation

the structures and interactions of solitary waves in a 2 1 dimensional coupled nonlinear extension of the reaction diffusion equation the complete integrability of a 2 1 dimensional coupled nonlinear extension of the reaction diffusion cnlerd equation using the technique of painlev p analysis is investigated using the formalism of weiss et al 1983 j math phys 24 522 the arbitrariness of the expansion coefficients are proved besides following the hirota s formalism hirota r 1980 direct methods in soliton theory soliton berlin springer combined to weiss et al s methodology weiss et al 1984 j math phys 25 13 the consistency of the truncation is shown thus without the use of kruskal s simplification the bcklund transformation bt of the equations is obtained via the truncation procedure taking into account the arbitrariness of the expansion coefficients in the foregoing truncation a typical spectrum of localized coherent structures may be unearthed the scattering behavior of such structures is also investigated

local regularity and decay estimates of solitary waves for the rotation modified kadomtsev petviashvili equation this paper is mainly concerned with the local low regularity of solutions and decay estimates of solitary waves to the rotation modified kadomtsevpetviashvili rmkp equation it is shown that with negative dispersion the rmkp equation is locally well posed for data in hs1 s2 r2 for s1 13 and s2 0 and hence globally well posed in the space l2 moreover an improved result on the decay property of the solitary waves is established which shows that all solitary waves of the rmkp equation decay exponentially at infinity

global well posedness for the kp i equation we prove the global well posedness of the kp i equation in bbb r 2 for data of arbitrary size in a suitable sobolev class we use a compactness method to get local solutions we extend the solutions thanks to some conservation laws an almost conserved quantity and strichartz estimates

remarks on solitary waves of the generalized two dimensional benjamin ono equation abstract in this paper we study the generalized 2d bo equation in two dimensions u t h u xx u p u x x u yy 0 x y r 2 t 0 we classify the existence and non existence of solitary waves depending on the sign of and on the nonlinearity we also prove nonlinear stability and some regularity properties of such waves

a smoothing effect and polynomial growth of the sobolev norms for the kp ii equation abstract we prove that the eventual growth in time of the sobolev norms of the solutions of the kp ii equation is at most polynomial

low regularity for the fifth order kadomtsev petviashvili i type equation abstract the cauchy problem of the fifth order kadomtsev petviashvili i equation fifth kp i 0 1 t u x 5 u x 3 u x 1 y y u x u 2 0 x y t r 3 is considered it follows that the fifth order kadomtsev petviashvili i equation 0 1 is locally well posed in h s 0 with s 3 4 where the norm h s r is defined by f h x y s r s r f l 2

the tanh method and adomian decomposition method for solving the foam drainage equation foaming occurs in many distillation and absorption processes the drainage of liquid foams involves the interplay of gravity surface tension and viscous forces in this paper we use a semi analytic method the adomian decomposition method and an analytic method the tanh method to handle the foam drainage equation the powerful tanh method gives the solution in a closed form however adomian decomposition method computes the solution in a rapidly convergent infinite series the comparison between the two approaches is conducted to illustrate the performance of each method

the extended g g expansion method and its applications to the whitham broer kaup like equations and coupled hirota satsuma kdv equations in this paper based on a new general ansatze the extended g g expansion method is proposed to seek exact solutions of nonlinear evolution equations being concise and straightforward this method is applied to construct travelling wave solutions of whitham broer kaup like equations and coupled hirota satsuma kdv equations by using this method new exact solutions involving parameters expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions are obtained when the parameters are taken as special values some solitary wave solutions are derived from the hyperbolic function solutions

exact travelling wave solutions of some nonlinear equations by g g expansion method abstract here we find exact solutions of some nonlinear evolution equations within the framework of the g g expansion method exact solutions of five nonlinear equations of physical importance viz the coupled schrodinger kdv equation the coupled nonlinear reaction diffusion equation the foam drainage equation the phi four equation and the dodd bullough mikhailov equation are obtained these general solutions can be reduced in some standard results derived by some other methods two and three dimensional plots of some of the results are also presented

on the asymptotic behavior of solutions to the generalized kadomtsev petviashvili burgers equations we show global existence and sharp results on the asymptotic decay of solutions to the generalized kp equations with a power nonlinearityupuxand with a burgers type dissipation in the main direction of propagation appended furthermore we prove that forp n the decay rate of thel2 norm of the solution of these equations is that exhibited by the solution of these same equations linearized around the zero solution and that forp 2 this also holds for thel norm this limitation seems due to the presence of a dispersive effect in the transverse direction

new traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized big frac g prime g big expansion method in this paper we construct new traveling wave solutions of some nonlinear evolution equations in mathematical physics via the 3 1 dimensional potential ytsf equation the 3 1 dimensional modified kdv zakharov kuznetsev equation the 3 1 dimensional kadomtsev petviashvili equation and the 1 1 dimensional kdv equation by using a generalized expansion method where g g satisfies the jacobi elliptic equation g 2 p g here we assume that p g is a polynomial of fourth order many new exact solutions in terms of the jacobi elliptic functions are obtained

local and global well posedness for the ostrovsky equation abstract we consider the initial value problem for 0 1 t u x 3 u x 1 u uu x 0 x t r where u is a real valued function and are real numbers such that 0 and x 1 f i 1 f this equation differs from korteweg de vries equation in a nonlocal term nevertheless we obtained local well posedness in x s f h s r x 1 f l 2 r s 3 4 using techniques developed in c e kenig g ponce l vega well posedness of the initial value problem for the korteweg de vries equation j amer math soc 4 1991 323 347 for the case 0 we also obtain a global result in x 1 using appropriate conservation laws

on finite energy solutions of the kp i equation we prove that the flow map of the kadomtsev petviashvili i kp i equation is not uniformly continuous on bounded sets of the natural energy space

applications of extended tanh method to special types of nonlinear equations in this paper we explore more applications of extended tanh method to some special nonlinear equations such equations are not of differential polynomial form so that cannot be directly dealt with by tanh method or extended tanh method as examples we apply a recently proposed extended tanh method to build some new explicit solutions for 2 1 dimensional sine gordon equation dodd bullough mikhailov equation and coupled schrodinger kdv equation

on the low regularity of the fifth order kadomtsev petviashvili i equation we consider the fifth order kadomtsev petviashvili i kp i equation as tu 3 x u 5 x u 1 x 2 y u uux 0 while r we introduce an interpolated energy space es to consider the well posedeness of the initial value problem ivp of the fifth order kp i equation we obtain the local well posedness of ivp of the fifth order kp i equation in es for 0 s 1 to obtain the local well posedness we present a bilinear estimate in the bourgain space in the framework of the interpolated energy space it crucially depends on the dyadic decomposed strichartz estimate the fifth order dispersive smoothing effect and maximal estimate

the g g expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics i the present paper we construct the traveling wave solutions involving parameters of the combined korteweg de vries modified korteweg de vries equation the reaction diffusion equation the compound kdv burgers equation and the generalized shallow water wave equation by using a new approach namely the g g expansion method where g g satisfies a second order linear ordinary differential equation when the parameters take special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

prolongation structure of the 2 1 dimensional integrable heisenberg ferromagnet model we investigate the integrable 2 1 dimensional modified heisenberg ferromagnet hf model using the prolongation structure theory for the integrable 2 1 dimensional modified hf models the corresponding geometrical equivalent counterparts such as the 2 1 dimensional nonlinear schrodinger equation and the coupled 2 1 dimensional integrable equations are also presented through the motion of minkowski space curves endowed with an additional spatial variable

nondegeneracy morse index and orbital stability of the kp i lump solution using backlund transformation we analyze the spectral property of the kp i lump solution it is proved that the lump is nondegenerate and its morse index equals one as a consequence we show that it is orbitally stable

on periodic kp i type equations we analyze bilinear estimates related to periodic kp i type equations we prove that the cauchy problem for the fifth order kp i equation posed on is globally well posed in the energy space despite the loss of a smoothing relation which is essential to recuperate the derivative loss used in the context of the kdv and kp ii type equations we are able to gain a moothing thanks to some specific properties of the set of fourier modes where the smoothing relation fails counterexamples in the context of kp i type equations posed on are given these examples indicate that contrary to the kp ii case it would be difficult to obtain the local well posedness for the periodic kp i type equations by an iteration scheme using bourgain spaces

exact traveling solutions of some nonlinear evolution equation by g g expansion method in this paper we establish exact solutions for some nonlinear evolution equations the g g expansion method was used to construct traveling wave solutions of ramani s equations in this method we take the advantages of the general solutions of the second order lode to solve effectively many nonlinear evolution equations the g g expansion method presents a wider applicability for handling nonlinear wave equations

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations