a kam theorem for hamiltonian partial differential equations in higher dimensional spaces in this paper we give a kam theorem for a class of infinite dimensional nearly integrable hamiltonian systems the theorem can be applied to some hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi periodic solutions and its local birkhoff normal form the applications to the higher dimensional beam equations and the higher dimensional schrodinger equations with nonlocal smooth nonlinearity are also given in this paper

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

quasi periodic solutions for 1d wave equation with the nonlinearity u2p 1 abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 2 p 1 0 p n on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that there are many 2 dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation the proof is based on infinite dimensional kam theory and partial birkhoff normal form

an extended simplest equation method and its application to several forms of the fifth order kdv equation abstract in this paper an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations as applications many new exact travelling wave solutions for several forms of the fifth order kdv equation are obtained by using our method the forms include the lax sawada kotera sawada kotera parker dye caudrey dodd gibbon kaup kupershmidt kaup kupershmidt parker dye and the ito forms

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

application of the extended simplest equation method to the coupled schrodinger boussinesq equation in this paper we successfully construct the new exact traveling wave solutions of the coupled schrodinger boussinesq equation by using the extended simplest equation method the exact traveling wave solutions with double arbitrary parameters are expressed by the hyperbolic functions the trigonometric functions and the rational functions respectively when the parameters are taken as special values some solitary wave solutions are derived from the hyperbolic function solutions

quasi periodic solutions in a nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 4 u 0 with the periodic boundary condition is considered it is proved that for each given constant potential m and each prescribed integer n 1 the equation admits a whitney smooth family of small amplitude time quasi periodic solutions with n diophantine frequencies the proof is based on a partial birkhoff normal form reduction and an improved kam method

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

quasi periodic solutions of nonlinear schrodinger equations of higher dimension abstract it is shown that there are plenty of quasi periodic solutions of nonlinear schrodinger equations of higher spatial dimension where the dimension of the frequency vectors of the quasi periodic solutions are equal to that of the space

fuzzy regression models using the least squares method based on the concept of distance fuzzy regression models are developed to construct the relationship between explanatory variables and responses in a fuzzy environment in order to increase the explanatory performance of the model the least squares method is applied to determine the numeric coefficients based on the concept of distance unlike most existing approaches the numeric coefficients in the proposed model can have negative values the proposed model minimizes total estimation error in terms of the sum of the average squared distance between the observed and estimated responses based on a few alpha cuts the proposed approach is not limited to triangular fuzzy numbers it can be used to carry out a large number of fuzzy observations efficiently because the model is based on traditional statistical methods comparisons with existing methods show that based on the total estimation error using the mean squared error and kim and bishu s criterion the explanatory performance of the proposed model is satisfactory

solitons and other solutions for higher order nls equation and quantum zk equation using the extended simplest equation method abstract in this article we apply the extended simplest equation method for constructing the solitons and other solutions of two nonlinear partial differential equations pdes namely the higher order nonlinear schrodinger nls equation with derivative non kerr nonlinear terms and the nonlinear quantum zakharov kuznetsov qzk equation which play an important role in mathematical physics the first equation describes pulse of the propagation beyond ultrashort range in optical communication systems while the second equation arises in quantum magneto plasma comparison of our new results in this article with the well known results is given

invariant tori for 1d quintic nonlinear wave equation abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 5 0 on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that for any integer b 2 there are many b dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation this is an extension of the previous work 6 by the same authors where b 2 the proof is based on infinite dimensional kam theory and partial birkhoff normal form

new exact traveling wave solutions of the 4 1 dimensional fokas equation abstract in this paper two distinct methods are applied to look for exact traveling wave solutions of the 4 1 dimensional nonlinear fokas equation namely the modified simple equation method msem and the extended simplest equation method esem some new exact traveling wave solutions involving some parameters are obtained the solitary wave solutions can be extracted by assigning special values of these parameters the obtained solutions show the simplicity and efficiency of the used approaches that can be applied for nonlinear equations as well as linear ones

extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear schrodinger type equations in this paper we extended the auxiliary equation method proposed by sirendaoreji and kudryashov to construct new types of jacobi elliptic function solutions of nonlinear partial differential equations pdes in mathematical physics the effectiveness of the extended method is demonstrated by applications to three nonlinear pdes namely the 2 1 dimensional nonlinear cubic quintic ginzburg landau equation the 1 1 dimensional resonant nonlinear schrodinger s equation with dual power law nonlinearity and the generalized zakharov system of equations the solitary wave solutions or trigonometric functions solutions are obtained from the jacobi elliptic function solutions when the modulus of the jacobi elliptic functions approaches to one or zero respectively comparison between our new results and the well known results is given

cantor families of periodic solutions for wave equations via a variational principle abstract we prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a cantor set of asymptotically full measure via a variational principle a lyapunov schmidt decomposition reduces the problem to a finite dimensional bifurcation equation variational in nature defined on a cantor set of non resonant parameters the cantor gaps are due to small divisors phenomena to solve the bifurcation equation we develop a suitable variational method in particular we do not require the typical arnold non degeneracy condition of the known theory on the nonlinear terms as a consequence our existence results hold for new generic sets of nonlinearities

a note on the modified simple equation method applied to sharma tasso olver equation abstract jawad et al have applied the modified simple equation method to find the exact solutions of the nonlinear fitzhugh naguma equation and the nonlinear sharma tasso olver equation the analysis of the sharma tasso olver equation obtained by jawad et al is based on variant of the modified simple equation method in this paper we provide its direct application and obtain new 1 soliton solutions

on solving the 3 1 dimensional nleqzk equation and the 3 1 dimensional nlmzk equation using the extended simplest equation method abstract the extended simplest equation method is employed in this article to construct many solitons and other solutions for two nonlinear partial differential equations pdes namely the 3 1 dimensional nonlinear extended quantum zakharov kuznetsov nleqzk equation and the 3 1 dimensional nonlinear modified zakharov kuznetsov nlmzk equation which play an important role in mathematical physics the first equation is performed in quantum electron positron ion magnetoplasmas while the second equation governs the behavior of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field comparison of our new results obtained in this article for the two considered models with the other well known results is also provided

periodic solutions of nonlinear wave equations with general nonlinearities we present a variational principle for small amplitude periodic solutions with fixed frequency of a completely resonant nonlinear wave equation existence and multiplicity results follow by min max variational arguments

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

modified simple equation method for nonlinear evolution equations abstract this paper reflects the implementation of a reliable technique which is called modified simple equation method msem for solving evolution equations the proposed algorithm has been successfully tested on two very important evolution equations namely fitzhugh nagumo equation and sharma tasso olver equation numerical results are very encouraging

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

quasi periodic solutions for 1d schrodinger equation with the nonlinearity u 2pu abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x u 2 p u 0 p n with periodic boundary conditions is considered it is proved that the above equation admits small amplitude quasi periodic solutions corresponding to 2 dimensional invariant tori of an associated infinite dimensional dynamical system the proof is based on infinite dimensional kam theory partial normal form and scaling skills

quasi periodic solutions for 1d wave equation with higher order nonlinearity abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 5 0 on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that there are many 2 dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation the proof is based on infinite dimensional kam theory and partial birkhoff normal form

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

conservation of resonant periodic solutions for the one dimensional nonlinear schrodinger equation we consider the one dimensional nonlinear schrodinger equation with dirichlet boundary conditions in the fully resonant case absence of the mass term we investigate conservation of small amplitude periodic solutions for a large measure set of frequencies in particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes provided the corresponding frequencies are large enough say greater than a certain threshold value depending on the number of resonant modes if the frequencies of the latter are close enough to such a threshold then they can not be too distant from each other hence we can interpret such solutions as perturbations of wave packets with large wave number

the extended tanh method and its applications for solving nonlinear physical models the extended tanh method with a computerized symbolic computation is used for constructing the travelling wave solutions of coupled nonlinear equations arising in physics the obtained solutions include solitons kinks and plane periodic solutions the applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics

quasi periodic solutions of completely resonant nonlinear wave equations it is shown that there are many elliptic invariant tori and thus quasi periodic solutions for the completely resonant nonlinear wave equation subject to periodic boundary conditions via kam theory