freedom in the expansion and obstacles to integrability in multiple soliton solutions of the perturbed kdv equation abstract the construction of solutions of the perturbed kdv equation encounters obstacles to asymptotic integrability beyond the first order when the zero order approximation is not a single soliton wave in the standard analysis the obstacles lead to the loss of integrability of the normal form resulting in a zero order term which does not have the simple structure of the solution of the unperturbed equation exploiting the freedom in the perturbative expansion an algorithm is proposed that shifts the effect of the obstacles from the normal form to the higher order terms the expansion has been carried out through third order through this order the normal form remains integrable and the zero order approximation retains the structure of the unperturbed solution for multiple soliton solutions the resulting obstacles decay exponentially away from the soliton interaction region which is a finite domain around the origin in the x t plane the effect is demonstrated in detail through second order for the two soliton case where the obstacles generate a second order radiative tail that emanates from the origin and decays exponentially away from the origin these results suggest a new meaning to asymptotic integrability the zero order term is determined by an integrable normal form and the higher order terms in the expansion of the full solution tend to those of the integrable case when no obstacles exist asymptotically away from the origin

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 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

four 2 1 dimensional integrable extensions of the kadomtsev petviashvili equation in this work four 2 1 dimensional nonlinear extensions of the kadomtsev petviashvili kp equation are developed the complete integrability of these models are investigated multiple soliton solutions and multiple singular soliton solutions are determined to demonstrate the compatibility of these models the resonance phenomenon does not exist for any of the derived models

a hierarchy of coupled burgers systems possessing a hereditary structure a hierarchy of typical integrable coupled burgers systems is proposed by introducing a special spectral problem involving q dependent variables u0 u1 uq 1 these systems possess a hereditary structure and may reduce to a hierarchy of burgers equations under the reduction ui 0 0 or i or q 2 it is shown that their flows and lax operators commute mutually but each system is not hamiltonian

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

time optimal output deadbeat regulators with internal stability for invertible systems abstract hassan and amin 1987 and amin and hassan 1987 have presented a simple algebraic solution for the time optimal output deadbeat tood regulator problem this solution which consists of a state feedback control law has been obtained for right invertible systems these results are generalized here to accom modate all classes of output controllable left invertible systems s a b c e the design of tood regulators with internal stability for all classes of right or output controllable left invertible systems s a b c e is also considered it is shown that the right wiener hopf factorization at infinity of h z the transfer function matrix of s is the key tool for solving all tood regulator problems moreover the internal stability problem which will arise if any invariant zero of s is unstable is solved by designing a tood regulator with internal stability finally numerical examples are worked out to illustrate the generality of the proposed method

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

feedback block decoupling and infinite structure of linear systems abstract using factorizations at infinity of the transfer matrix two interesting necessary and sufficient conditions arc obtained for the block decoupling problem of linear systems by state feedback and a constant non singular input transformation these conditions are expressed in a very simple way in terms of integers which are feedback invariants for the system under consideration these feedback invariants are related either to the system infinite structure or to the reachability indices of the feedback reduced system

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

minimal time minimal order deadbeat regulator with internal stability this paper is concerned with the output deadbeat finite settling time regulation with internal stability for linear discrete time multivariable systems the two cases are treated separately the one is the regulation by state feedback and the other is the regulation through function observers for both cases the basic solvability conditions the minimal settling time and the characterizations of the minimal time regulators are derived throughout the paper the well posedness condition plays a fundamental role a deterministic version of the so called separation theorem in lqg problem is derived namely the minimal time state feedback regulator combined with the minimal time minimal order function observer yields the overall minimal time minimal order regulator by output feedback

wave interactions and the analysis of the perturbed burgers equation in multiple front solutions of the burgers equation all the fronts except for two are generated through the inelastic interaction of exponential wave solutions of the lax pair associated with the equation the inelastically generated fronts are the source of two interrelated difficulties encountered in the standard normal form expansion of the approximate solution of the perturbed burgers equation when the zero order term is a multiple front solution i the higher order terms in the expansion are not bounded ii the normal form equation obeyed by the zero order approximation is not asymptotically integrable its solutions lose the simple wave structure of the solutions of the unperturbed equation the freedom inherent in the normal form method allows a simple modification of the expansion procedure making it possible to overcome both problems in more than one way the loss of asymptotic integrability is shifted from the normal form to the higher order terms part of which has to be computed numerically in the expansion of the solution the front velocity update is different from the one obtained in the standard analysis

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

design of precompensators via infinite zero structure for dynamic decoupling of linear invertible systems the problem of designing precompensators necessary for decoupling invertible systems having weak inherent coupling is considered a simple algorithm which is directly applicable to systems characterized by square proper transfer function matrices t s is proposed for such design in the proposed algorithm the concept row reducedness at infinity of t s is utilized as an alternative necessary and sufficient condition for the decouplability of t s in addition to the simple design procedure of the compensator the algorithm yields the infinite zero structure of both t s and the resulting compensated system

time optimal output regulators for linear multivariate discrete time systems part 2 all classes of right invertible systems in part 1 of this paper hassan and amin 1987 a simple algebraic solution for the time optimal output regulator problem was presented this solution which consists of a state feedback control law has been obtained for all classes of right invertible decouplable systems s a b c e the results of part 1 are here extended to all classes of right invertible systems s a b c e a set of optimal output deadbeat indices called the optimal set is defined and related to the observability indices of the optimal closed loop system the time optimal output regulator problem for a right invertible non decouplable system s a b c e is resolved by transforming s into a decouplable system sc a b cc ec having the optimal output deadbeat index of s first an algorithm is presented to construct iteratively in a well defined optimal sense a unimodular left compensator l z and a compensated decouplable system sc a b cc ec from the state space parameters of s then a family of optimal state fee

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

2 1 dimensional burgers equations be m n 1 using the recursion operator in this work we develop a variety of 2 1 dimensional burgers equations be n of distinct orders we use the recursion operator of the burgers equation to derive these equations the obtained equations possess multiple kink solutions the same as the multiple kink solutions of the burgers hierarchy but differ only in the dispersion relations we finally establish a generalization for the dispersion relations

travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations in this work exact travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations are obtained the analysis rests mainly on the standard tanh method the work emphasizes the need for a transformation formula for the case where the parameter m is non integer the approach can be used in a variety of many types of nonlinearity

multiple kink solutions for m component burgers equations in 1 1 dimensions and 2 1 dimensions abstract m component burgers equations in 1 1 dimensions and 2 1 dimensions are examined for complete integrability the cole hopf transformation method and the simplified form of hereman s method are used to achieve this goal multiple kink solutions and multiple singular kink solutions are formally derived for each vector equation

n soliton solutions for the combined kdv cdg equation and the kdv lax equation abstract the hirota ito approach presented in r hirota m ito resonance of solitons in one dimension j phys soc jpn 52 3 1983 744 748 for extending fifth order integrable equations with a nonvanishing boundary conditions to combined equations is used in this work the generalized fifth order caudrey dodd gibbon cdg and lax equations are extended to combined integrable equations the hirota s bilinear method is used to derive multiple soliton solutions for the extended kdv cdg and kdv lax equations

four 2 1 dimensional integrable extensions of the kdv equation multiple soliton and multiple singular soliton solutions in this work four 2 1 dimensional nonlinear completely integrable equations generated by extending the kdv equation are developed the necessary condition for the complete integrability of these equation are formally derived multiple soliton solutions and multiple singular soliton solutions are determined to emphasize the compatability of these models the dispersion relations of these models are characterized by distinct physical structures the resonance phenomenon for these equations does not exist for any model

classification of proper bases of rational vector spaces minimal macmillian degree bases abstract the algebraic structure of the get of all proper rational vectors contained in a given rational vector space x1d4af s is shown to be that of a noothorian pr s motiulo m ropf pr a is the ring of proper rational functions the proper submodules mt of m form an ascending chain of submodules partially ordered by an invariant of mt defined as the valuation at s ofmt tho various bases of mt are examined and classified according to their property of column reduceness at s the concept of a prime column reduced at basis of mt is introduced it is shown that tho prime bases of mt can be further classified by their macmillan degrees and the existence of minimal macmillan degree bases for mt is established a prime and minimal macmillan degree basis ofmt extends forney s concept of a minimal polynomial basis of s for the rpr mt module the macmillan degrees of the columns of such bases form a set of invariants mt for which are defined as the x1d4af s generalized invariant dynamicol indices o

time optimal output regulators for linear multivariable discrete time systems part 1 right invertible decouplable systems corsetti and houpis 1985 presented a simple algebraic solution for the timeoptimal output regulator problem this solution has been obtained for a special class of right invertible decouplable systems s a b c satisfying the condition that cb is of full rank in this paper a general method is presented to extend the results of corsetti and houpis to all classes of right invertible decouplable systems s a b c e it is shown that a simple state feedback control law drives each output of s a b c e in a minimum number of steps equal to the order of the associated infinite zero this state feedback represents the optimal solution for the considered time optimal output regulator problem a class of optimal solutions parametrized by a free parameter matrix k is obtained for non square systems the properties of the resulting optimal closed loop system is given and a numerical example is worked out to illustrate the generality and feasibility of the proposed method

qualitative robustness of tests abstract a test is qualitatively robust by definition if its sequence of n 1 log transformed p values n being a measure of the sample size is continuous as a point function of the observations and weakly equicontinuous as a function of discrete probability measures this definition is applitable both to unconditional and to conditional tests under weak regularity conditions an unconditional test is qualitatively robust if and only if its test statistic is continuous a counterexample shows that conditional tests do not share this property the sample mean student s t and ybar xbar permutation tests are not qualitatively robust the sign wilcoxon huber censored likelihood and normal scores tests are qualitatively robust