duct acoustics a numerical technique for the higher order mode solution of three dimensional problems with rigid walls and no flow abstract a three dimensional time dependent finite difference technique is developed for the steady state solution of the acoustic wave equation in the absence of flow the acoustic characteristics of a mitred bend are used to illustrate the use of the technique and numerical results are corroborated by experimental data

regularized long wave equation and inverse scattering transform the sech2 solitary wave solution of the regularized long wave equation is reobtained via the inverse scattering transform the wave function of the eigenvalue problem of the relevant schrodinger equation is proved reflectionless with sech2 potential of arbitrary amplitude moreover the nonexistence of n solitary wave solution sech2 form and n 2 is confirmed for the regularized long wave rlw equation and the method provides the possibility of solving some incompletely integrable equations via the inverse scattering transform

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

duct acoustics a time dependent difference approach for steady state solutions abstract a time dependent numerical technique is developed to study the propagation of sound in ducts in the absence of flow the method is shown to be stable and convergent for propagating and decaying modes sample calculations are compared with analytical solutions in straight ducts and with numerical and experimental results in a mitred bend

on exact solutions of the regularized long wave equation a direct approach to partially integrable equations ii periodic solutions in this second of two articles designated i and ii the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long wave rlw equation these solutions are expressed in terms of riemann theta functions and this approach leads to a new and compact expression for the important dispersion relation the periodic solution or cnoidal wave can be represented as an infinite sum of sech2 solitary waves this remarkable property may be interpreted in the context of a nonlinear superposition principle the rlw cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes similar results are presented in brief for the related rlw boussinesq rlwb equation

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

on the periodic solution of the burgers equation a unified approach for the most part the various known expressions for the periodic solution to the burgers equation have been derived by a mixture of ad hoc argument and approximation methods these include the well known solutions due to r d fay and d f parker remarkably these solutions were obtained without recourse to the celebrated hopf cole transformation which linearizes the burgers equation to the heat conduction equation we present here an alternative approach to the periodic problem which uses the fact that the riemann theta functions solve the burgers equation via the hopf cole transformation and thereby provide a unifying framework in which all previous representations of the periodic solution find their natural place yet another expression for the periodic solution in the form of an amplitude modulated sine wave is given and would appear to be new the limiting forms of the periodic solution as t rightarrow 0 embryonic sawtooth profile and t rightarrow infty old age sinusoidal profile are examined in detail the representation due to d f parker which expresses the periodic wave as a superposition of spreading taylor shock profiles is discussed within the context of a nonlinear superposition principle the composite asymptotic expansion due to m b lesser and d g crighton is examined in the light of our results the curious behaviour of this expansion in the embryonic region t ll 1 whereby a first order approximation yields an exact periodic solution the fay solution is explained

application of the time dependent finite difference theory to the study of sound and vibration interactions in ducts abstract the time dependent finite difference theory is extended to the solution of the acoustic wave equation in rectangular ducts when acoustic structural interactions are allowed at a duct wall the treatment of the boundary condition which describes the coupling is examined and the stability of the procedure is studied and found to depend on the nature of this coupling the convergence of solutions is discussed as a function of the discretization of the solution domain particularly at frequencies approaching resonance

non holonomic geodesic flows on lie groups and the integrable suslov problem on so 4 we study geodesic flows on lie groups with the left invariant non holonomic constraint in the case of the existence of an invariant measure we find new integrable non hamiltonian systems on so 4 and other six dimensional lie groups

on exact solutions of the regularized long wave equation a direct approach to partially integrable equations i solitary wave and solitons hitherto the bilinear transformation method has been used to obtain analytic solutions of completely integrable or nonlinear evolution equations in this first of two studies designated i and ii it is shown that the method may apply equally well to those nonlinear wave equations dubbed partially integrable here the regularized long wave rlw and rlw boussinesq rlwb equations are considered as examples in each case the bilinear form is found to have an extra equation which would appear to signal their nonintegrability their solitary wave solutions are reconstructed and the nonexistence of multisoliton solutions is demonstrated the present work provides the basis for the construction of periodic solutions in ii

periodic solutions of the camassa holm equation based on the bilinear form since the famous work of camassa and holm various analytic techniques have been used to find the exact solutions of camassa holm ch equation and many interesting results have been found by the different methods in this paper we first give an extended bilinear form of the ch equation which contains four more free parameters than those in the results proposed in parkers finding for soliton solutions thus we construct the smooth periodic solution of the ch equation in a parametric form based on the previous work of nakumura we also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of is the parameter appeared from the riemann theta function

on the camassa holm equation and a direct method of solution i bilinear form and solitary waves over the last decade a variety of techniques has been used to find exact solutions both analytic and other of the camassaholm equation the different approaches have met with varying measures of

the time dependent finite difference procedure for propagation of sound in a non uniform lined duct a comparison with experiments abstract a time dependent finite difference procedure is developed for the calculation of acoustic fields which incorporate finite impedance and rigid wall boundaries a numerical mapping technique is evoked to transform the solution domain into a rectangular geometry and the procedure is used to solve for frequencies where cross modes can propagate in the straight sections of duct numerical results are corroborated with experiments

nonholonomic left and right flows on lie groups nonholonomic systems on lie groups with a left invariant lagrangian and a right invariant nonintegrable distribution are studied a method for construction of the first integrals is given it is applied to three dimensional lie groups and to the four dimensional nonholonomic rigid body motion new integrable problems with an invariant measure are obtained

a comparison of measured and computed sound pressure levels in a non uniform acoustically lined duct a comparison of measured and numerically calculated acoustical fields is presented for a non uniform lined duct in the absence of appreciable mean flow the frequency range investigated includes the cut on frequencies of several transverse modes in certain portions of the duct measured pressure fields are compared to those predicted by one and two dimensional numerical models the validity of the one dimensional model is confirmed for frequencies below cut on of the first transverse mode for higher frequencies the one dimensional model is clearly unsatisfactory as might be expected the two dimensional model gives reasonable results for frequencies below and above cut on of the transverse modes although it indicates a very strong sensitivity of acoustical fields to small variations in local wall impedance data

cnoidal waves as exact sums of repeated solitary waves new series for elliptic functions via poisson summation of the corresponding fourier series it is shown that the cnoidal wave of the korteweg de vries equation can be written as an infinite sum of periodically repeated solitary waves a result first proved by toda through a different method similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn sn and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves as a by product new series are also obtained for the complete elliptic integral k and modulus k

exact periodic solutions of the sixth order generalized boussinesq equation this paper examines a class of nonlinear sixth order generalized boussinesq like equations sgbe utt uxx 3 u2 xx uxxxx uxxxxxx r depending on the positive parameter hirota s bilinear transformation method is applied to the above class of non integrable equations and exact periodic solutions have been obtained the results confirmed the well known nonlinear superposition principle

generalizations of classical integrable nonholonomic rigid body systems nonholonomic systems with an invariant measure the suslov chaplygin and veselov veselova problem are considered new families of integrable potential perturbations of these non hamiltonian systems are constructed we also obtain a similar result for a classical hamiltonian system of the motion of a rigid body fixed at a point

solitons from sine waves analytical and numerical methods for non integrable solitary and cnoidal waves abstract the fkdv equation u t uu x u xxxxx 0 is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of non integrable differential equations that is to say to equations which cannot be solved by the inverse scattering transform the basic tools are i pade approximants formed from power series in the amplitude ii a newton kantorovich pseudospectral fourier continuation numerical method iii singular perturbation theory for two interacting solitons of almost identical phase speed iv bifurcation and branch switching methods v the imbricate soliton series a number of new results for the fkdv equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak cnoidal wave and two peaks bicnoidal wave per period an analytical expression for the double peaked soliton bion calculation of both the limit and bifurcation points for the bicnoidal wave and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes more important all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform work now in progress will apply these methods to non integrable equations in two space dimensions

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

periodic solutions of the dabo equation as a sum of repeated solitons it is shown that the periodic solution of the non linear dabo equation can be written as an infinite sum of an equally spaced row of identical lorentzian solitons this may be considered as a generalisation of a similar result which has been found for the kdv equation and is related to the clean interaction properties of colliding solitons under non linear coupling

a general mapping procedure for variable area duct acoustics a general mapping procedure is described and applied to the study of noise propagation in variable area ducts the mapping provides a boundary fitted co ordinate system which is ideal for the finite difference solution of acoustic fields with irregular boundaries without the burden of large matrices required by finite element methods the procedure is first described in general and then applied to a particular two dimensional geometry under current experimental investigation this method should be ideally suited to the study of high frequency noise propagation in variable area ducts and in cases where the far field is included in the calculation procedure moreover the current approach can be directly extended to three dimensions resulting in numerical calculation over a rectangular parallelepiped in the transformed plane