non commutative polynomials and the transport properties in multichannel multilayer systems using the transfer matrix technique to describe transport properties in multichannel multilayer systems a three term non commutative matrix recurrence relation is deduced and solved the matrix polynomials obtained in this way allow one to write compact expressions for the n cell transmission amplitudes from channel to channel i in the one dimensional one channel limit the non commutative polynomials reduce to the well known chebyshev orthogonal polynomials to illustrate the role of these polynomials in the resonant tunnelling and channel mixing behaviour we discuss one and two channel examples

classification of systems of nonlinear ordinary differential equations with superposition principles a system of n first order nonlinear ordinary differential equations x t f x t is said to admit a superposition principle if its general solution can be written as a function of a finite number m of particular solutions and n constants such a system can be associated with the nonlinear action of a lie group g on a space m we show that indecomposable systems of ode s with supersposition principles are obtained if and only if the lie algebras l0 l corresponding to the isotropy group h of a point and the group g respectively define a transitive primitive filtered lie algebra l l0 using known results from the theory of transitive primitive lie algebras we deduce that l0 must be a maximal subalgebra of l and that g must be an affine group a simple lie group or the direct product of two identical simple lie groups affine groups lead to linear equations the other types to nonlinear equations with polynomial or rational nonlinearities equations corresponding to the classical complex lie algebra

on the synthesis of a stabilizing feedback control via lie algebraic methods let the n dimensional system of differential equations dx dt x x t have p in mathbb r n as a rest solution i e x p 0 even in cases when this rest solution is unstable one can often induce a strong stability asymptotic stability by the inclusion of one or more controls e g via a controlled system a dx dt x x uy x where say u leqq 1 lie theory gives a computable sufficient condition to determine when by use of the control u one can steer a full n dimensional neighborhood of p to p by solutions of a this condition is assumed to hold one prefers a feedback control i e that u u x the main result in this paper is an algorithm which determines a modified stabilizing feedback control specifically for given varepsilon 0 one measures the current state q and the algorithm determines u t q 0 leqq t leqq varepsilon such that the solution x cdot u of a initiating from q and corresponding to this control u satisfies distance x varepsilon u p q p in fact iterates are theoretically shown to converge to p numerical examples computed via a simple fortran program are included these substantiate the strong stability achieved via such a modified feedback control

an approximation algorithm for nonholonomic systems in siam j control optim 37 1997 to appear limiting process of control affine systems with holder continuous inputs submitted we have studied the limiting behavior of trajectories of control affine systems sigma dot x sum k 1 m u k f k x generated by a sequence u j subseteq l 1 0 t rr m where the f k are smooth vector fields on a smooth manifold m we have shown that under very general conditions the trajectories of sigma generated by the u j converge to trajectories of an extended system of sigma of the form sigma ext dot x sum k 1 r v kf k x where f k k 1 ldots m are the same as in sigma and f m 1 ldots f r are lie brackets of f 1 ldots f m in this paper we will apply these convergence results to solve the inverse problem i e given any trajectory gamma of an extended system sigma ext find trajectories of sigma that converge to gamma uniformly this is done by means of a universal construction that only involves the knowledge of the v k k 1 ldots r and the structure of the lie brackets in sigma ext but does not depend on the manifold m and the vector fields f 1 ldots f m these results can be applied to approximately track an arbitrary smooth path in m for controllable systems sigma which in particular gives an alternative approach to the motion planning problem for nonholonomic systems

reduction by invariants and projection of linear representations of lie algebras applied to the construction of nonlinear realizations a procedure for the construction of nonlinear realizations of lie algebras in the context of vessiot guldberg lie algebras of first order systems of ordinary differential equations odes is proposed the method is based on the reduction of invariants and projection of lowest dimensional irreducible representations of lie algebras applications to the description of parameterized first order systems of odes related by contraction of lie algebras are given in particular the kinematical lie algebras in 2 1 and 3 1 dimensions are realized simultaneously as vessiot guldberg lie algebras of parameterized nonlinear systems in r3 and r4 respectively

symmetry analysis of the infeld rowlands equation a fourth order nonlinear evolution equation in 2 1 dimensions arising in the study of soliton stability is analyzed its symmetry group is shown to be infinite dimensional and is used to obtain particular solutions the equation is shown not to have the painlev e property

generalized riccati equation expansion method and its application to the 3 1 dimensional jumbo miwa equation based on the computerized symbolic system maple and a riccati equation a generalized riccati equation expansion method for constructing soliton like solutions of non linear evolution equations nees is presented by a new general ansatz the proposed method is more powerful than most of the existing tanh methods the extended tanh function method the modified extended tanh function method and generalized hyperbolic function method by use of the method we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nees making use of the method we study the the 3 1 dimensional jumbo miwa equation and obtain rich new families of the exact solutions including the non travelling wave and coefficient functions soliton like solutions singular soliton like solutions periodic form solutions

on a computationally simple form of the generalized campbell baker hausdorff dynkin formula abstract in this paper a computationally simple form of the generalized campbell baker hausdorff dynkin formula gcbhd is given simplifications arise from both combinatorial explicit as well as algorithmic implicit arguments on generating the ph hall basis in a special form and introducing a specific structure representation of the vector field vector fields appearing in gcbhd are automatically transformed into the reduced ph hall basis the formula can be exploited in nonholonomic motion planning what is illustrated with examples

new explicit exact solutions for the generalized coupled hirota satsuma kdv system in this paper we study the generalized coupled hirota satsuma kdv system by using the two new improved projective riccati equations method as a result many explicit exact solutions which contain new solitary wave solutions periodic wave solutions and combined formal solitary wave solutions and combined formal periodic wave solutions are obtained

transfer matrices for one dimensional potentials the one dimensional schrodinger equation can be written as a first order multicomponent equation by considering and d dx or combinations thereof as independent variables a potential barrier is then represented by a matrix belonging to one of the homomorphic groups su 1 1 so 2 1 sp 2 r or sl 2 r the relationship between these groups is clarified in various applications one of them may turn out more convenient than others in particular so 2 1 which is obtained by using as a basis some bilinear combinations of and d dx leads to remarkable results the schrodinger wavefunction is represented by a trajectory on a unit hyperboloid a periodic potential corresponds to a pseudorotation around a fixed axis a random potential gives a random walk on the hyperboloid this method can also be used to calculate bound states in potential wells and may have many other interesting applications

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

nonlinear superposition formulae based on the lie group so n 1 n systems of nonlinear ordinary differential equations are constructed for which the general solution is algebraically expressed in terms of a finite number of particular solutions expressions of that type are called nonlinear superposition formulae these systems are connected with local lie group transformations on their homogeneous spaces in the work presented here the nonlinear superposition formulae are constructed for the case of the so 3 2 group and some aspects of the general case of so n 1 n are studied

normal form solutions of dynamical systems in the basin of attraction of their fixed points abstract the normal form theory of poincare siegel and arnol d is applied to an analytically solvable lotka volterra system in the plane and a periodically forced dissipative duffing s equation with chaotic orbits in its 3 dimensional phase space for the planar model we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by the presence of singularities of the solutions in the complex t plane despite such limitations however we show in the case of a periodically driven system that normal forms can be used to obtain useful estimates of the basin of attraction of a stable fixed point of the poincare map whose boundary is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

superposition principles for matrix riccati equations a superposition rule is obtained for the matrix riccati equation mre w a wb cw wdw where w t a t b t c t and d t are real n n matrices expressing the general solution in terms of five known solutions for all n 2 the symplectic mre w wt a at d dt c bt is treated separately and a superposition rule is derived involving only four known solutions for the unitary and gl n r subcases with d a and c bt or d a and c bt respectively superposition rules are obtained involving only two solutions the derivation of these results is based upon an interpretation of the mre in terms of the action of the groups sl 2n r sp 2n r u n and gl n r on the grassman manifold gn r2n

systems of ordinary differential equations with nonlinear superposition principles abstract this work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations x x t x r n t r in terms of a finite number of particular solutions the point of departure is lie s criterion according to which such a rule exists if and only if the vector fields x t generate a finite dimensional lie algebra we provide three different constructive methods for deriving superposition rules and apply them to systems of coupled riccati equations of the projective and conformal types based respectively on the lie algebra sl n 1 r and o p 1 n p 1

the pinney equation and its discretization the pinney equation is part of the original ermakov system which has been the subject of intensive study recently here we show that it may be related to a two dimensional conformal riccati equation leading to a new method for its linearization a discrete analogue of the pinney equation is constructed using the above connection with the conformal group an alternative discretization is obtained by using a discrete schwarz derivative both of these nonlinear difference equations are linearizable

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

on the integrability of systems of nonlinear ordinary differential equations with superposition principles a new class of solvable nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations ode s describing each member of this class possess nonlinear superposition principles these systems of ode s are generally not derived from a hamiltonian and are classified by associated pairs of lie algebras of vector fields in this paper all such systems of n 3 ode s are integrated in a unified way by finding explicit integrals for them and relating them all to a pivotal member of their class the projective riccati equations moreover by perturbing two parametrically driven projective riccati equations thus making them nonsolvable in the above sense evidence is discovered of chaotic behavior on the poincare surface of section in the form of sensitive dependence on initial conditions near a boundary separating bounded from unbounded motion

averaging theorems for highly oscillatory differential equations and iterated lie brackets using averaging techniques and developing proper algebraic formalisms we study the limiting process of ordinary differential equations with highly oscillatory right hand sides we give sufficient conditions generalizing earlier work by kurzweil and jarnik for a sequence u j u 1 j ldots u m j subseteq l 1 0 t bbb r m to be such that for every choice of smooth vector fields f k k 1 ldots m on a smooth manifold the trajectories of dot x sum k 1 mu k j t f k x converge to the trajectories of an extended system dot x sum k 1 r v k t f k x where the new directions f m 1 ldots f r are lie brackets of f 1 ldots f m

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

new exact solutions for the 2 1 dimensional generalized broer kaup system abstract the 1 1 dimensional broer kaup system which describes the propagation of shallow water waves is extended to a generalized 2 1 dimensional model with painleve property in this paper based on the general variable separation approach and two extended riccati equations we first find several new families of exact soliton like solutions and periodic like wave solutions with arbitrary functions for the 2 1 dimensional simplified generalized broer kaup gbk system b 0 abundant new localized excitations can be found by selecting appropriate functions after that we consider the conditions of b 0 to the gbk and several new results are obtained

pre control form of the generalized campbell baker hausdorff dynkin formula for affine nonholonomic systems abstract in this paper a pre control form of the generalized campbell baker hausdorff dynkin gcbhd formula is presented this form is dedicated to nonholonomic affine systems often encountered in robotics the simplest possible expression for the pre control gcbhd is obtained as control dependent functions pre multiply elements of the ph hall basis algorithmic aspects of deriving the formula are highlighted an application of the pre control form of the gcbhd formula in motion planning of nonholonomic systems is also reported

abnormal minimizers this paper constructs the first example of a singular abnormal minimizer for the lagrange problem with linear velocity constraints and quadratic definite lagrangian or equivalently for an optimal control system of linear controls with k controls n states and a running cost function that is quadratic positive definite in the controls in the example k 2 n 3 and the system is completely controllable the example is stable if both the control law and cost are perturbed the singular minimizer persists its importance is due in part to the fact that it is a counterexample to a theorem that has appeared several times in the differential geometry literature there the problem is called the problem of finding minimizing sub riemannian geodesics and it has been claimed that all minimizers are normal pontryagin extremals the mathematical theory of optimal processes wiley interscience new york 1962 if the number of states equals the number of controls then the problem is that of finding riemannian geodesics the main difficulty is proving minimality to do this the length cost of the abnormal is compared with all competing normal extremals a detailed asymptotic analysis of the differential equations governing the normals shows that they are all longer

symmetries parametrization and group structure of transfer matrices in quantum scattering theory useful and new representations of the transfer matrices are explicitly obtained for systems with and without time reversal or spin rotation symmetries in the symplectic orthogonal and unitary cases the whole group of transfer matrices contains a compact and a noncompact subgroup and the generalized bargman s polar parametrization is obtained in a straightforward way it is shown that the transfer matrices used by luttinger to prove the saxon hutner conjecture on the forbidden energy levels belong to the noncompact sp 2 c subgroup

exp function method for the exact solutions of fifth order kdv equation and modified burgers equation abstract in this paper we implemented the exp function method for the exact solutions of the fifth order kdv equation and modified burgers equation by using this scheme we found some exact solutions of the above mentioned equations

applications of the combined tanh function method with symmetry method to the nonlinear evolution equations based on the symbolic computation we combined the tanh function method with the symmetry method to construct new type of solutions of the nonlinear evolution equations for the first time with the combined method some new types of solutions of the coupled 2 1 dimensional nonlinear system of schrodinger equations were obtained and their properties were studied by analyzing their figures

note on exact solutions for a class of nonlinear diffusion equations an effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms further a new auxiliary equation ansatz is derived consequently new exact traveling wave trigonometric function solitary like and weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized riccati equation expansion method

exact travelling wave solutions of nonlinear evolution equations in 1 1 and 2 1 dimensions abstract based upon the symbolic computation system maple the projective riccati equation method is improved to seek for travelling wave solutions in 1 1 and 2 1 dimensional nonlinear evolution equations by introducing two new elementary bell shaped and kink shaped functions the validity and reliability of the method is tested by applying it to the 2 1 dimensional generalized of the shallow water wave equation potential kdv equation and burgers kdv equation

algebraic solution of the lindblad equation for a collection of multilevel systems coupled to independent environments we consider the lindblad equation for a collection of multilevel systems coupled to independent environments the equation is symmetric under the exchange of the labels associated with each system and thus the open system dynamics takes place in the permutation symmetric subspace of the operator space the dimension of this space grows polynomially with the number of systems we construct a basis of this space and a set of superoperators whose action on this basis is easily specified for a given number of levels m these superoperators are written in terms of a bosonic realization of the generators of the lie algebra in some cases these results enable finding an analytic solution of the master equation using known lie algebraic methods to demonstrate this we obtain an analytic expression for the state operator of a collection of three level atoms coupled to independent radiation baths when analytic solutions are difficult to find the basis and the superoperators can be used to considerably reduce the computational resources required for simulations

towards a unified method for exact solutions of evolution equations an application to reaction diffusion equations with finite memory transport we present a brief report on the different methods for finding exact solutions of nonlinear evolution equations explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones it is shown that most of methods that exist in the literature are equivalent to the generalized mapping method that unifies them by using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained attention is focused to the finite memory transport fisher and nagumo equations

lax pair backlund transformation and n soliton like solution for a variable coefficient gardner equation from nonlinear lattice plasma physics and ocean dynamics with symbolic computation abstract in this paper a generalized variable coefficient gardner equation arising in nonlinear lattice plasma physics and ocean dynamics is investigated with symbolic computation the lax pair and backlund transformation are explicitly obtained when the coefficient functions obey the painleve integrable conditions meanwhile under the constraint conditions two transformations from such an equation either to the constant coefficient gardner or modified korteweg de vries mkdv equation are proposed via the two transformations the investigations on the variable coefficient gardner equation can be based on the constant coefficient ones the n soliton like solution is presented and discussed through the figures for some sample solutions it is shown in the discussions that the variable coefficient gardner equation possesses the right and left travelling soliton like waves which involve abundant temporally inhomogeneous features

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

a symbolic computational method for constructing exact solutions to difference differential equations in this paper we extended tanh method to solve difference differential equations and pure difference equations with the projective riccati equation as an example we applied this method to a 2 1 dimensional toda lattice equation as a result many exact solutions are obtained with the help of symbolic system maple including soliton solutions presented by hyperbolic functions sinh and cosh periodic solutions presented by sin and cos and rational solutions this method can also be used to other nonlinear difference differential equations

wave function confinement via transfer matrix methods the exact transfer matrix approach used in studying sectionally constant potentials in one dimension is generalized to cylindrical and spherical geometries where the potential depends only on radius in each geometry two transfer matrices suffice to completely describe the wave function one for handling a discontinuity in potential and one for handling a delta function potential barrier this method is then applied to the problem of confining a wave function in a cylindrical configuration using only a series of carefully placed delta function potential barriers it is found that confinement can be made to increase nearly exponentially with the number of barriers if placed correctly but that this arrangement has an exponentially sharp dependence on both barrier position and energy

a functional realization of 3 ℝ providing minimal vessiot guldberg lie algebras of nonlinear second order ordinary differential equations as proper subalgebras a functional realization of the lie algebra sl3 r as a vessiot guldberg lie algebra of second order differential equation sode lie systems is proposed it is shown that a minimal vessiot guldberg lie algebra lvg is obtained from proper subalgebras of sl3 r for each of the sode lie systems of this type by particularization of one functional and two scalar parameters of the sl3 r realization the relation between the various vessiot guldberg lie algebras by means of a limiting process in the scalar parameters further allows to define a notion of contraction of sode lie systems

travelling wave solutions to the two dimensional kdv burgers equation travelling wave solutions for the two dimensional kdv burgers equation are obtained by using mathematica they are shown to be sums of a shock wave and solitary wave

new exact travelling wave solutions for the shallow long wave approximate equations based upon a generally projective riccati equation method which is a direct and unified algebraic method for constructing more general form travelling wave solutions of nonlinear partial differential equations and implemented in a computer algebraic system we consider the shallow long wave approximate equations new and more general form solutions are obtained including kink shaped solitons bell shaped solitons singular solitons and periodic solutions the properties of the new formal solitary wave solutions are shown by some figures

a combinatorial approach to the generalized baker campbell hausdorff dynkin formula abstract the aim of this paper is to obtain an explicit expression instead of using a recursive method for the nth term coefficient of the generalized baker campbell hausdorff dynkin gbchd formula the gbchd formula has been applied to control theory specially to nonholonomic motion planning

symbolic computation and a uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations on the basis of the computer symbolic system maple and the tanh method the riccati equation method as well as all kinds of improved versions of these methods we present a further uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations compared with the existing methods the presented method can be used to construct more new general solutions and we give some examples to illustrate the key step of our method

a new riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations abstract in this paper based on a new system of riccati equations we present a triple riccati equation expansion method the nonlinear klein gordon equation and the combined kdv mkdv equation are chosen to illustrate the method as a result many new travelling solutions which include bell profile solitary wave solution singular solitary solutions periodic solutions are obtained this method can be used to the other nonlinear evolution equations in mathematics physics

superposition formulas for pseudounitary matrix riccati equations the purpose of this article is to derive a superposition formula for the pseudounitary matrix riccati equation of dimension n 2 the superposition formula will be written in closed form in terms of five particular solutions satisfying certain well specified conditions defining a fundamental set examples will be studied in order to show how the superposition formula works and how it can be used in numerical calculations

exact solutions of a hierarchy of mixing speeds models this paper presents several new aspects of discrete kinetic theory dkt first a hierarchy of d dimensional d 1 2 3 models is proposed with 2d 3 velocities and three moduli speeds 0 2 and a third one that can be arbitrary it is assumed that the particles at rest have an internal energy which for microscopic collisions supplies for the loss of the kinetic energy in a more general way than usual collisions are allowed that mix particles with different speeds second for the 1 1 dimensional restriction of the systems of pde for these models which have two independent quadratic collision terms we construct different exact solutions the usual types of exact solutions are studied periodic solutions and shock wave solutions obtained from the standard linearization of the scalar riccati equations called riccatian shock waves then other types of solutions of the coupled riccati equations are found called non riccatian shock waves and they are compared with the previous ones the main new result