on similarity solutions of zabolotskaya khokhlov equation abstract the new closed form solutions of the 2 1 dimensional zabolotskaya khokhlov equation are constructed by using the similarity transformations method via lie group theory the zabolotskaya khokhlov equation has been reduced into a new partial differential equation with smaller number of independent variables further using the similarity transformations method the new partial differential equation is reduced into an ordinary differential equation the ordinary differential equation provides semi analytical solutions elastic behaviour of multisolitons and dromion annihilation of solutions are analyzed physically through their evolutional profiles

blow up of solutions to semilinear wave equations with variable coefficients and boundary this paper is devoted to studying initial boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions we will establish blow up results for the initial boundary value problems it is proved that there can be no global solutions no matter how small the initial data are and also we give the life span estimate of solutions for the problems

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

some new periodic solitary wave solutions of 3 1 dimensional generalized shallow water wave equation by lie symmetry approach abstract many important physical situations such as fluid flows marine environment solid state physics and plasma physics have been represented by shallow water wave equations in this article we construct new solitary wave solutions for the 3 1 dimensional generalized shallow water wave gsww equation by using optimal system of lie symmetry vectors the governing equation admits twelve lie dimension space a variety of analytic closed form solutions such as new periodic solitary wave cross kink soliton and doubly periodic breather type solutions have been obtained by using invariance of the concerned 3 1 dimensional gsww equation under one parameter lie group of transformations lie symmetry transformations have applied to generate the different forms of invariant solutions of the 3 1 dimensional gsww equation for different lie algebra lie symmetry method reduces 3 1 dimensional gsww equation into various ordinary differential equations odes while one of the lie algebra it is transformed into the well known 2 1 dimensional blmp equation it is affirmed that the proposed techniques are convenient genuine and powerful tools to find the exact solutions of nonlinear partial differential equations pdes under the suitable choices of arbitrary functions and parameters 2d 3d and contour graphics to the obtained results of gsww equation are also analyzed

a search for bilinear equations passing hirota s three soliton condition ii mkdv type bilinear equations in this paper the results of a search for complex bilinear equations with two soliton solutions are presented the following basic types are discussed a the nonlinear schrodinger equation b dx g f 0 a dx dt f f gg and b the benjamin ono equation p dx f f 0 it is found that the existence of two soliton solutions is not automatic but introduces conditions that are like the usual three and four soliton conditions the search was limited by the degree of a 2 and by degree of p 4 the main results are the following 1 iad3x dxdt idy b g f 0 d2xf f gg 2 d2x ad2y idt b g f 0 dxdy f f gg 3 iad3x d2x idt f f 0 and 4 dxdt i adx bdt f f 0

the glassey conjecture on asymptotically flat manifolds we verify the 3 dimensional glassey conjecture on asymptotically flat manifolds r1 3 g where the metric g is a small space time perturbation or nontrapping time independent perturbation of the minkowski metric and approaches the minkowski metric like 1 x with 0 moreover for radial asymptotically flat manifolds r1 n g with n 3 we verify the glassey conjecture in the radial case high dimensional wave equations with higher regularity are also discussed the main idea is to exploit local energy and kss estimates with variable coefficients together with the weighted sobolev estimates including trace estimates

the extended tanh method for new solitons solutions for many forms of the fifth order kdv equations abstract the extended tanh method is used to derive new solitons solutions for several forms of the fifth order nonlinear kdv equation the forms include the lax sawada kotera sk sawada kotera parker dye skpd kaup kupershmidt kk kaup kupershmidt parker dye kkpd and the ito equations the criteria established in a m wazwaz abundant solitons solutions for several forms of the fifth order kdv equation by using the tanh method appl math comput in press doi 10 1016 j amc 2006 02 047 to build up reliable relations between the parameters of the equation are confirmed by using this new approach entirely new bell shaped solitons are determined

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

the tanh coth and the sine cosine methods for kinks solitons and periodic solutions for the pochhammer chree equations abstract in this paper the tanh coth method and the sine cosine method are used for a reliable treatment of the pochhammer chree equations which include a variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the two methods demonstrate power reliability and efficiency

n soliton solutions for shallow water waves equations in 1 1 and 2 1 dimensions abstract a variety of shallow water waves equations in 1 1 and 2 1 dimensions are investigated we first show that these models are completely integrable we next determine multiple soliton solutions for each equation the simplified hirota s bilinear method developed by hereman will be employed to achieve this goal a comparison between dispersion relations and the phase shifts will be conducted but possess the same coefficients for the polynomials of exponentials

multiple soliton solutions for the boussinesq equation abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions for the boussinesq equation the one soliton solutions will be handled by the tanh coth method the work highlights the significant features of the employed methods

two integrable lattice hierarchies and their respective darboux transformations two integrable lattice hierarchies associated with two discrete matrix spectral problems are derived and the infinitely many conservation laws of the first integrable model is obtained moreover the darboux transformations based on different darboux matrixes 2 3 4 and 3 2 4 for the above mentioned integrable hierarchies are established with the help of two different gauge transformations of lax pairs in this paper as an application the explicit solutions of the two integrable hierarchies are presented

generalization of exp function and other standard function methods exp function and other standard function methods for solving nonlinear differential equations are generalized in this paper an analytical criterion determining if a solution can be expressed in a form comprising standard functions is derived new computational algorithm for automatic identification of the structure of the solution is constructed the algorithm provides information if the solution can be expressed as a sum of standard functions a ratio of sums of standard functions or even a more complex algebraic form involving standard functions several examples are used to illustrate the proposed concept

on the extended applications of homogenous balance method in this paper we study the travelling wave reductions for certain 2 1 and 3 1 dimensional physically important nonlinear evolutionary equations by using the recently proposed homogenous balance method hbm through this analysis we explore certain new solutions for the equations we have studied

the hirota s direct method and the tanh coth method for multiple soliton solutions of the sawada kotera ito seventh order equation abstract in this work the sawada kotera ito seventh order equation is studied the tanh coth method is applied to obtain soliton solution of this equation the hirota s direct method combined with the simplified hereman s method are applied to determine the n soliton solutions for this equation the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

explicit solutions of the 2 1 dimensional akns shallow water wave equation with variable coefficients abstract using an improved direct reduction method we find the equivalence transformations of 2 1 dimensional akns shallow water wave equation with variable coefficients and obtain the corresponding relationship between explicit solutions of akns equation and those of the corresponding reduced equation in addition we get some new explicit solutions of akns equation by applying lie symmetry method

price s law on nonstationary space times abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate price s law price 1972 54 for linear waves as a corollary we also prove price s law for certain small perturbations of the kerr metric this result was previously established by the second author in tataru 65 on stationary backgrounds the present work was motivated by the problem of nonlinear stability of the kerr schwarzschild solutions for the vacuum einstein equations which seems to require a more robust approach to proving linear decay estimates

multiple front solutions for the burgers kadomtsev petviashvili equation abstract in this work we study a completely integrable dissipative equation the burgers equation is extended by using the sense of the kadomtsev petviashvili kp equation the new established burgers kp equation is studied by using the tanh coth method to obtain kink solutions and periodic solutions we also apply the powerful hirota s bilinear method to establish exact n soliton solutions for the derived integrable equation

group classification for isothermal drift flux model of two phase flows in this paper a full symmetry group classification for isothermal multiphase drift flux model is presented all invariant functions are developed for the lie algebra which play a vital role in construction of optimal systems further with the help of one dimensional optimal classification group invariant solutions are obtained which describe the asymptotic behavior of general solution

exact travelling wave solutions for the dissipative 2 1 dimensional akns equation abstract this paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative 2 1 dimensional akns equation by qualitative analysis global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions furthermore the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated in addition the possible bell profile solitary wave solution kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method error estimates indicate that the approximate solutions are meaningful based on above studies a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative 2 1 dimensional akns equation

global parametrices and dispersive estimates for variable coefficient wave equations in this article we consider variable coefficient time dependent wave equations in mathbb r times mathbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

exact travelling wave solutions for the generalized nonlinear schrodinger gnls equation with a source by extended tanh coth sine cosine and exp function methods the capability of extended tanh coth sine cosine and exp function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed in this study the generalized nonlinear schrodinger gnls equation is solved by three different methods to obtain the single soliton solutions for the equation the extended tanh coth and sine cosine methods are used furthermore for this nonlinear evolution equation the exp function method is applied to derive various travelling wave solution results show that while the first two procedures easily provide a concise solution the exp function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics

combined effects of two nonlinearities in lifespan of small solutions to semi linear wave equations this paper investigates the combined effects of two distinctive power type nonlinear terms with parameters p q 1 in the lifespan of small solutions to semi linear wave equations we determine the full region of p q to admit global existence of small solutions at least for spatial dimensions n 2 3 moreover for many p q when there is no global existence we obtain sharp lower bound of the lifespan which is of the same order as the upper bound of the lifespan

solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov equation by lie symmetry approach abstract in this paper the solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov ezk equation are constructed which appear in the magnetized two ion temperature dusty plasma and quantum physics lie group of transformation method is proposed to investigate the solution of 3 1 dimensional ezk equation via lie symmetry method the optimal system of one dimensional lie subalgebra is constructed by using lie point symmetries the three dimensional ezk equation reduced into number of ordinary differential equations odes by applying similarity reductions consequently solutions so extracted are more general than erstwhile known results we have obtained twenty one solutions in the explicit form some of them are likewise general and some are new for the best study of us eventually single soliton quasi periodic soliton multisoliton lump type soliton traveling wave and solitary wave interaction behavior are illustrated graphically through numerical simulation for physical affirmation of the results please check whether the affiliations are correct

nonlinear effects in focused sound fields focused finite amplitude sound fields are investigated with numerical solutions of the khokhlov zabolotskaya kuznetsov kzk equation the numerical solution is based on the algorithm developed by aanonsen et al j acoust soc am 75 749 768 1984 who used a fourier series expansion of the sound pressure to reduce the kzk equation to a set of coupled parabolic equations the basic algorithm has been modified by introducing a coordinate system that follows the convergent geometry of focused sound fields in this way more efficient numerical evaluation of the detailed field structure within the focal region is achieved arbitrary axisymmetric sources can be modeled here circular sources having linear focusing gains of order 50 will be considered the calculated time waveforms propagation curves and beam patterns illustrate clearly the combined effects of nonlinearity diffraction and absorption on finite amplitude sound that passes through a focal region among the new results are power curves

on abstract strichartz estimates and the strauss conjecture for nontrapping obstacles we establish the strauss conjecture concerning small data global existence for nonlinear wave equations in the setting of exterior domains to compact obstacles for space dimensions n 3 and 4 the obstacle is assumed to be nontrapping and the solution is assumed to satisfy either dirichlet or neumann conditions along the boundary of the obstacle the key step in the proof is establishing certain abstract strichartz estimates for the linear wave equation on exterior domains

on the zakharov and zakharov schulman systems abstract we consider the initial value problem for the zakharov system formula which models the long wave langmuir turbulence in a plasma using the standard iteration scheme in the original system it is shown that the system is locally well posed uniformly on the ionic speed of sound in appropriate sobolev spaces the method provides similar results for the ivp for the zakharov schulman system formula where u r d 0 c r d 0 r and formula with a i j k a k i j real constants which models the interactions of small amplitude high frequency waves with acoustic type waves

some counterexamples to separation of variables abstract among the other things a linear inhomogenous algebraic partial differential equation with integer coefficients is written down that has no differentially algebraic solutions such an equation has no solution that can be generated by a general purpose analog computer i e differential analyzer

multiple front solutions for the burgers equation and the coupled burgers equations abstract in this work multiple front solutions for the burgers equation and the coupled burgers equations are examined the tanh coth method and the cole hopf transformation are used the work highlights the power of the proposed schemes and the structures of the obtained multiple front solutions

on closed form solutions of 2 1 breaking soliton system by similarity transformations method abstract in the present research similarity transformation method via lie group theory is proposed to seek some more exact closed form solutions of the 2 1 dimensional breaking soliton system the system describes the interactions of the riemann wave along y axis and long wave along x axis some explicit solutions of breaking soliton system are attained with appropriate choices of the arbitrary functions and making use of arbitrary constants involved in the infinitesimals in order to obtain physically meaningful solutions numerical simulation is performed on the basis of graphical representation the physical analysis of solutions reveals into multi solitons periodic quadratic asymptotic and stationary profiles

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

concerning the strauss conjecture on asymptotically euclidean manifolds abstract in this paper we verify the strauss conjecture for semilinear wave equations on asymptotically euclidean manifolds when n 3 4 we also give an almost sharp lifespan for the subcritical case 2 p p c when n 3 the main ingredients include a keel smith sogge type estimate with 0 1 2 and weighted strichartz estimates of order two

on some invariant solutions of 2 1 dimensional korteweg de vries equations abstract in the present research similarity transformation method is proposed to obtain some more general invariant solutions of 2 1 dimensional korteweg de vries equations this system of equations describes nonlinear waves propagation on the surface of shallow water the method reduces the number of independent variables by one using invariance property of lie group theory thus korteweg de vries equations are reduced into a system of ordinary differential equations employing twice of similarity transformation method this system of ordinary differential equations is solved under some parametric restrictions and provides invariant solutions the obtained results are supplemented by numerical simulation taking suitable choice of arbitrary constants and functions eventually the elastic behavior of multisoliton compacton negaton positon kink wave solution and dromion annihilation profiles are shown to make this research more admirable

local decay of waves on asymptotically flat stationary space times author s tataru daniel abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate for linear waves this work was motivated by open problems concerning decay rates for linear waves on schwarzschild and kerr backgrounds where such a decay rate has been conjectured by r price our results apply to both of these cases

traveling wave solutions u v to a generalized drinfel d sokolov system which satisfy u a1vm a0 abstract an analysis of the coupled generalized drinfel d sokolov equations u t 1 uu x 1 u xxx v x 0 and v t 2 uv x 2 v xxx 0 is performed in the case of traveling wave solutions u and v satisfying the condition u a 1 v m a 0 we are able to classify all such solutions in terms of the model parameters and we then discuss the planar dynamics of such solutions numerical solutions are obtained and discussed for a variety of parameter regimes such results are one possible generalization of those given by wu et al l wu s chen c pang traveling wave solutions for generalized drinfeld sokolov equations applied mathematical modeling 33 2009 4126 4130

an example where separation of variables fails we present in detail a linear constant coefficient initial boundary value problem for which the classical method of eigenfunction expansions fails we note that the new method recently introduced by a fokas and b pelloni 2005 in 3 can be successfully applied to the same problem

new solitary wave solutions for two nonlinear evolution equations abstract in this paper multiple exp function method is employed to investigate exact multiple wave solutions for 2 1 dimensional potential kadomtsev petviashvili equation and 3 1 dimensional jimbo miwa equation not only already known multiple wave solutions are recovered but also several new or more general multiple wave solutions are obtained

soliton solutions of kd system using similarity transformations method abstract present article deals with some exact solutions of 2 1 dimensional system of coupled konopelchenko dubrovsky equations similarity transformations method is proposed to seek the solution of the system using lie group theory the lie group theory is a very strong tool by which complicated nonlinear partial differential equations under the group transformations remains invariant a brief review of lie symmetries of a system of partial differential equations has been described during the process method reduces the number of independent variables by one hence the system of partial equations reduces into a new system of ordinary differential equations in addition the infinitesimals after first reductions are more general than the previous established results by us kumar et al 2016 consequently solutions so derived are more general than previously known results we have obtained nine solutions in the explicit form some of them are more general and some are new for the best knowledge of us ultimately single solitons multi solitons and shocks behaviour is represented graphically through the numerical simulation for physical validation of the results

the hirota s direct method for multiple soliton solutions for three model equations of shallow water waves abstract multiple soliton solutions for three model equations for shallow water waves are determined the three models are completely integrable the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these equations the tanh coth method is used to obtain single soliton solutions and other solutions for these three models the three models have different linear dispersion relations but possess the same coefficients for the polynomials of exponentials

two systems of two component integrable equations multiple soliton solutions and multiple singular soliton solutions abstract in this work we study two systems of two component integrable equations the cole hopf transformation and hirota s bilinear method are applied to emphasize the integrability of each system multiple soliton solutions and multiple singular soliton solutions are formally derived

invariant solutions and conservation laws to nonconservative fp equation we generate conservation laws for the one dimensional nonconservative fokker planck fp equation also known as the kolmogorov forward equation which describes the time evolution of the probability density function of position and velocity of a particle and associate these where possible with lie symmetry group generators we determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied as the fokker planck equation is evolution type no recourse to a lagrangian formulation is made moreover we obtain invariant solutions for the fp equation via potential symmetries

blow up for small amplitude semilinear wave equations with mixed nonlinearities on asymptotically euclidean manifolds abstract in this work we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small amplitude semilinear wave equations with mixed nonlinearities c 1 u t p c 2 u q posed on asymptotically euclidean manifolds which is related to both the strauss conjecture and the glassey conjecture in some cases we obtain existence results where the lower bound of the lifespan agrees with the upper bound in order in addition our results apply for semilinear damped wave equations when the coefficient of the dissipation term is integrable without sign condition and space independent

multiple soliton solutions for the lax kadomtsev petviashvili lax kp equation a new completely integrable dispersive equation is derived the new equation is obtained by extending the lax fifth order equation using the sense of the kadomtsev petviashvili kp equation in extending the kdv equation the newly derived lax kadomtsev petviashvili lax kp equation is investigated by using the tanh coth method and the hirota bilinear method to derive single soliton solution and n soliton solutions respectively the study highlights the multiple soliton solutions of the derived completely integrable equation

analytical solutions to a generalized drinfel d sokolov equation related to dssh and kdv6 abstract analytical solutions to the generalized drinfel d sokolov gds equations u t 1 uu x 1 u xxx v x 0 and v t 2 uv x 2 v xxx 0 are obtained for various values of the model parameters in particular we provide perturbation solutions to illustrate the strong influence of the parameters 1 and 2 on the behavior of the solutions we then consider a miura type transform which reduces the gds equations into a sixth order nonlinear differential equation under the assumption that 1 under such a transform the gds reduces to the sixth order drinfel d sokolov satsuma hirota dssh equation also known as kdv6 in the very special case 1 2 the method of homotopy analysis is applied in order to obtain analytical solutions to the resulting equation for arbitrary 1 and 2 an error analysis of the obtained approximate analytical solutions is provided

the glassey conjecture for nontrapping obstacles abstract we verify the 3 dimensional glassey conjecture for exterior domain m g where the metric g is asymptotically euclidean provided that certain local energy assumption is satisfied the radial glassey conjecture exterior to a ball is also verified for dimension three or higher the local energy assumption is satisfied for many important cases including exterior domain with nontrapping obstacles and flat metric exterior domain with star shaped obstacle and small asymptotically euclidean metric as well as the nontrapping asymptotically euclidean manifolds r n g

the strauss conjecture on kerr black hole backgrounds we examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the strauss conjecture on the schwarzschild and kerr with small angular momentum black hole backgrounds the key estimates are a class of weighted strichartz estimates which are used near infinity where the metrics can be viewed as small perturbations of the minkowski metric and a localized energy estimate on the black hole background which handles the behavior in the remaining compact set

the tanh method a tool for solving certain classes of nonlinear evolution and wave equations the tanh or hyperbolic tangent method is a powerful technique to search for travelling waves coming out from one dimensional nonlinear wave and evolution equations in particular in those problems where dispersive effects reaction diffusion and or convection play an important role to show the strength of the method an overview is given to find out which kind of problems are solved with this technique and how in some nontrivial cases this method adapted to the problem at hand still can be applied single as well as coupled equations arising from wave phenomena which appear in different scientific domains such as physics chemical kinetics geochemistry and mathematical biology will be treated next attention is focussed towards approximate solutions as a result solitary and shock wave profiles are derived together with the associated width and velocity the same method can be easily extended so that difference differential equations can be similarly solved finally some extensions of the method are discussed

trigonometric and hyperbolic type solutions to a generalized drinfel d sokolov equation abstract a class of trigonometric and hyperbolic type solutions to the generalized drinfel d sokolov gds equations u t 1 uu x 1 u xxx v x 0 and v t 2 uv x 2 v xxx 0 is obtained for the case in which 2 0 for various values of the other model parameters the method of homotopy analysis is then applied to obtain local analytical solutions for nonzero values of the parameter 2 in effect extending the exact solutions we do not assume traveling wave solution forms for the analytical solutions that is we solve the generalized drinfel d sokolov equations as pdes without resorting to transforming the system to odes an error analysis of the obtained approximate local analytical solutions is provided then we outline a general framework by which one many construct solutions in either sine cosine or sinh cosh basis we provide the general perturbation expansion via homotopy analysis and we also discuss a method of selecting the convergence control parameter so as to minimize residual errors travelling solutions with time dependent amplitude are then discussed

generalized differential galois theory a galois theory of differential fields with parameters is developed in a manner that generalizes kolchin s theory it is shown that all connected differential algebraic groups are galois groups of some appropriate differential field extension

rational solutions in grammian form for the 3 1 dimensional generalized shallow water wave equation abstract in this paper the 3 1 dimensional generalized shallow water wave equation is investigated using the hirota bilinear method and kadomtsev petviashvili hierarchy reduction the explicit rational solutions for such an equation have been presented in the grammian form based on the grammian form solution for the equation the one rational two rational and three order rational solutions are obtained when complex parameters p i with nonzero real and imaginary parts are chosen the lump soliton solutions which are localized in all directions for the 3 1 dimensional generalized shallow water wave equation can be derived from the corresponding rational solutions as the figures illustrate the one lump soliton solution with one peak and one trough propagates stably on the x y plane the two lump solitons with different velocities interact with each other and separate with their original shapes and propagation directions different from the case of two lump solitons the propagation directions of the third order lump solitons change after the interaction

similarity solutions of the konopelchenko dubrovsky system using lie group theory this research deals with the similarity solutions of 2 1 dimensional konopelchenko dubrovsky kd system solutions so obtained are derived by using similarity transformations method based on lie group theory the method reduces the number of independent variables by one exploiting lie symmetries and using invariance property thus the kd system can further be reduced to a new system of ordinary differential equations under a suitable choice of functions and the arbitrary constants these new equations yield the explicit solutions of the kd system which are discussed in the similarity solutions section of the article moreover the physical analysis of the solutions is illustrated graphically in the analysis and discussions section based on numerical simulations in order to highlight the importance of the study

some more similarity solutions of the 2 1 dimensional blp system some more similarity solutions of the coupled boiti leon pempinelli system have been obtained by using similarity transformations method via lie group theory the solutions are attained under the restrictions imposed on the functions involved in infinitesimals then solutions are investigated through the graphical representation in different cases the cases depend upon some suitable choice of arbitrary functions involved there i e in the solutions to get physically meaningful structures analytical solutions are supplemented by numerical simulations with appropriate choice of arbitrary functions and constants eventually profiles show single soliton elastic multisolitons quasi periodic parabolic and asymptotic nature of solutions

frobenius integrable decompositions of nonlinear evolution equations with modified term frobenius integrable decompositions are introduced for nonlinear evolution equations with modified term two classes of polynomial equations with modified term are transformed into frobenius integrable ordinary differential equations by two special backlund transformations the resulting solutions are illustrated to describe the solution phenomena shared with the mkdv equation sharma tasso olver equation the modified kawachara equation the fifth order kdv equation and its various forms

almost global existence for semilinear wave equations with mixed nonlinearities in four space dimensions abstract we study the long time existence for the small amplitude semilinear wave equations with mixed nonlinearities of the form c 1 u t p c 2 u 2 with p 2 when the spatial dimension is four by exploiting the local energy estimates and their recent variants we prove the almost global existence up to exp c e 2 which is sharp in general for the case p 2 3 due to technical reason we need to assume the initial data to be radial

multiple soliton solutions of two extended model equations for shallow water waves abstract multiple soliton solutions for two extensions of model equations for shallow water waves presented in m j ablowitz d j kaup a c newell h segur the inverse scattering transform fourier analysis for nonlinear problems stud appl math 53 1974 249 315 r hirota j satsuma n soliton solutions of model equations for shallow water waves j phys soc jpn 40 2 1976 611 612 are obtained the two extensions possess identical dispersion relations but with different structures for the n soliton solutions for n 1 the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these extended equations the tanh coth method is used to obtain single soliton solutions

a direct algorithm of one dimensional optimal system for the group invariant solutions a direct and systematic algorithm is proposed to find one dimensional optimal system for the group invariant solutions which is attributed to the classification of its corresponding one dimensional lie algebra since the method is based on different values of all the invariants the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system with no further proof to leave the algorithm clear we illustrate each stage with a couple of well known examples the korteweg de vries equation and the heat equation finally we apply our method to the novikov equation and use the found optimal system to investigate the corresponding invariant solutions

similarity reduction and similarity solutions of zabolotskay khoklov equation with a dissipative term via symmetry method abstract similarity reductions of the zabolotskay khoklov equation with a dissipative term to one dimensional partial differential equations including burger s equation are investigated by the symmetry method some similarity solutions of the z k equation are obtained via the method of travelling wave technique due to jeffery and xu and the method of separation of variables

laminar flow and heat transfer in the boundary layer of non newtonian fluids over a stretching flat sheet a theoretical analysis of the laminar boundary layer flow and heat transfer of power law non newtonian fluids over a stretching sheet with the sheet velocity distribution of the form u w cx m and the wall temperature distribution of the form t w t ax c is presented where x denotes the distance from the slit from which the surface emerges and c and a are constants m and c denote the sheet velocity exponent and the temperature exponent respectively within the framework of the boundary layer approximations the nonlinear boundary layer momentum equation and the energy equation are reduced to a set of ordinary differential equations it is found that when the velocity exponent m 1 3 or the power law index n 1 the similarity solutions are in existence for both the momentum equation and the energy equation analytical approximations with high accuracy for the reduced velocity and temperature profiles are obtained using a new procedure based on the homotopy analysis method besides the effects of the parameters m n and the prandlt number pr on the flow are investigated

application of lie groups to compressible model of two phase flows this paper presents some exact solutions for the drift flux model of two phase flows using lie group analysis the analysis involves an isentropic no slip conservation of mass for each phase and the conservation of momentum for the mixture the present analysis employs a complete lie algebra of infinitesimal symmetries subsequent to these theoretical analysis a symmetry group is established the symmetry generators are used for constructing similarity variables which reduce the model equations to a system of ordinary differential equations odes in particular a general framework is discussed for solving the model equations analytically as a consequence of this new classes of exact group invariant solutions are developed this provides new insights into the fundamental properties of weak discontinuities and helps one to understand better on existence of solutions

solitary wave solutions of the generalized shallow water wave gsww equation by hirota s method tanh coth method and exp function method abstract the generalized short wave equation gsww is studied by three distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for the completely integrable forms of the gsww equation the tanh coth method is used to obtain single soliton solutions for this equation the exp function method is also applied to derive a variety of travelling wave solutions with distinct physical structures for this non linear evolution equation

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

a galois theoretic proof of the differential transcendence of the incomplete gamma function abstract we give simple necessary and sufficient conditions for the t transcendence of the solutions to a parameterized second order linear differential equation of the form 2 y x 2 p y x 0 where p f x is a rational function in x with coefficients in a t field f our criteria imply in particular the t transcendence of the incomplete gamma function t x generalizing a result of johnson reinhart and rubel this result is also an important part of an efficient algorithm to compute the parameterized picard vessiot group of an arbitrary parameterized second order linear differential equation over f x

multiple soliton solutions for the kp equation by hirota s bilinear method and by the tanh coth method abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions n 1 2 3 4 5 for the kadomtsev petviashvili kp equation the one soliton solutions will be handled by the tanh coth method the work emphasizes the power of the two methods that will be used

the cole hopf transformation and multiple soliton solutions for the integrable sixth order drinfeld sokolov satsuma hirota equation abstract the completely integrable sixth order nonlinear drinfeld sokolov satsuma hirota equation is studied three distinct methods namely the cole hopf transformation method the tanh coth method and the exp function method are used for a reliable treatment of this equation solitons multiple soliton solutions multiple singular soliton solutions and plane periodic solutions are obtained for this equations the study highlights the power of each method

soliton solutions of blmp equation by lie symmetry approach abstract shallow water wave equations are usually described by korteweg de vries kdv type equations in this paper we have used lie transformation group theory to solve 2 1 dimensional boiti leon manna pempinelli blmp equation we have obtained some exact solutions of blmp equation in the explicit form through similarity reduction all the reported results are expressed in closed form and analysed physically through their evolution profiles the physical analysis reveals that the nature of solutions is parabolic quasi periodic multisoliton and asymptotic

hirota bilinear equations with linear subspaces of solutions we explore when hirota bilinear equations possess linear subspaces of solutions first we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to hirota bilinear equations second we show that multivariate polynomials whose zeros form a vector space can generate the desired hirota bilinear equations with given linear subspaces of solutions and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero third applying an algorithm using weights we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired hirota bilinear 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

diversity of interaction solutions to the 2 1 dimensional ito equation abstract we aim to show the diversity of interaction solutions to the 2 1 dimensional ito equation based on its hirota bilinear form the proof is given through maple symbolic computations an interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function special cases lead to lump solutions lump soliton solutions and lump kink solutions two illustrative examples of the resulting solutions are displayed by three dimensional plots and contour plots

dynamic behaviors of interaction solutions of 3 1 dimensional shallow water wave equation abstract rational solutions and interaction solutions of the 3 1 dimensional shallow water wave equation are obtained depending on n soliton solution by the hirota bilinear method and long wave limit method and breather lump solutions and line rogue waves are obtained from two soliton solution as well as their interaction solutions are derived from four soliton solution after the collision breathers and lumps remain unchanged but the line rogue wave has a difference furthermore three soliton solution is discussed the interactions between some hybrid solutions such as hybrid solution between the stripe soliton and breather hybrid solution between stripe soliton and lump solution and so on are demonstrated in detail more interestingly stripe soliton remain unchanged but the amplitudes of the periodic line waves lump solutions and rogue waves have a great change after the collision