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In order to study stochastic viscosity solutions for a class of semilinear stochastic integral-partial differential equations (SIPDEs), a new class of generalized backward doubly stochastic differential equations with general jumps is investigated. The definition of stochastic viscosity solutions of SIPDEs is introduced. A probabilistic representation for stochastic viscosity solutions of semilinear SIPDEs with nonlinear Neumann boundary conditions is given.

In this work, we present the analytical approach to the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the behavior of the harmonic as well the an-harmonic parts of the oscillator. We present the an-harmonic part of the result in the form of the operator function., Comment: 40 pages, revised argument in Appendix F, results unchanged

In this work, we are concerned with the asymptotic properties of solutions for an impulsive neutral stochastic functional integro-differential equation. By applying the theory of resolvent operators, the Banach fixed point principle theorem, and results on stochastic analysis, we study respectively the existence, uniqueness, and global attracting and quasi-invariant sets of mild solutions for the considered equation. We also derive some sufficient conditions of pth moment exponential stability and almost surely exponential stability of the mild solutions. An example is provided in the end to illustrate the applications of the obtained results.

The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, quantum q-Krawtchouk, q-Krawtchouk, affine q-Krawtchouk, q-Charlier, Al-Salam–Carlitz, q-Hahn, little q-Jacobi, little q-Laguerre/Wall, q-Bessel, q-Racah, and dual q-Hahn polynomials.

In this paper, previous considerations concerning differential equations involving a para-Grassmann variable are extended by allowing two independent ordinary variables. For the differential equation of second order, the structure of its solutions is determined and several examples representing analogs of well-known second order differential equations are considered in detail. As a warm-up, the differential equation of first order is treated.

In this article, we study the existence and the boundary behavior of positive blow-up solutions for the semilinear elliptic coupled system. Our arguments are based on the subsupersolution method and the Karamata regular variation theory.

In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on the finite interval x ∈ [0, L] by extending the Fokas unified approach. The solution of this three-component system can be expressed by means of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex spectral k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly obtained using three spectral functions {s(k), S(k), SL(k)} arising from the initial data and Dirichlet-Neumann boundary conditions at x = 0, L, respectively. The global relation is also presented and used to deduce two distinct but equivalent types of representations [i.e., one via the large k of asymptotics of the eigenfunctions and another one in terms of the Gel’fand-Levitan-Marchenko (GLM) approach] for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problem on the finite interval can be extended to the ones on a half-line as the length L of the interval approaches to infinity. Moreover, we also give the linearizable boundary conditions for the GLM representations.

In this manuscript, we initiate a study on a class of stochastic fractional differential equations driven by Levy noise. The existence and uniqueness theorem of solutions to equations of this class is established under global and local Caratheodory conditions. Our analysis makes use of the Caratheodory approximation as well as a stopping time technique. The results obtained here generalize the main results from Pedjeu and Ladde [Chaos, Solitons Fractals 45, 279–293 (2012)], Xu et al. [Appl. Math. Comput. 263, 398–409 (2015)], and Abouagwa et al. [Appl. Math. Comput. 329, 143–153 (2018)]. Finally, an application to the stochastic fractional Burgers differential equations is designed to validate the theory obtained.

In this paper, we consider the mth order Cauchy-Euler’s differential equation involving a para-Grassmann variable of order p. In the Grassmann case (i.e., p = 1), we determine the solution for arbitrary order m. In the case of arbitrary order p, we give a solution for the cases m = 1, 2.

We propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations, which has a non-negative solution. The method applies the maximum entropy principle to approximating the solution numerically. The theoretical analysis and numerical examples show that our method is convergent.We propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations, which has a non-negative solution. The method applies the maximum entropy principle to approximating the solution numerically. The theoretical analysis and numerical examples show that our method is convergent.

The differential equation that defines the Zernike system, originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil, had been shown to separate in three distinct coordinate systems obtained from polar coordinates on a half-sphere. Here we find and examine the separation in the generic elliptic coordinate system on the half-sphere and its projected disk, where the solutions, separated in Jacobi coordinates, contain Heun polynomials.

Journal of mathematical physics 59(6), 062305 (2018). doi:10.1063/1.4986417, We calculate 3-loop master integrals for heavy quark correlators and the 3-loop quantum chromodynamics corrections to the $ρ$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of $_2$F$_1$ Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi’s $ϑ_i$ functions and Dedekind’s $η$-function. The corresponding representations can be traced back to polynomials out of Lambert–Eisenstein series, having representations also as elliptic polylogarithms, a $q$-factorial $1/η^{k}(τ)$, logarithms, and polylogarithms of $q$ and their $q$-integrals. Due to the specific form of the physical variable $x(q)$ for different processes, different representations do usually appear. Numerical results are also presented., Published by American Inst. of Physics, College Park, Md.

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.

We solve the metrisability problem for the six Painleve equations, and more generally for all 2nd order ordinary differential equations with the Painleve property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian metric on a surface.

In this work, the Riccati-Bernoulli sub-ordinary differential equation and modified tanh-coth methods are used to reach soliton solutions of the nonlinear evolution equation. We acquire new types of traveling wave solutions for the governing equation. We show that the equation is nonlinear self-adjoint by obtaining suitable substitution. Therefore, we construct conservation laws for the equation using new conservation theorem. The obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium. The constraint condition for the existence of solitons is stated. Some three dimensional figures for some of the acquired results are illustrated.

Moving frames and Clifford algebras will be used to illustrate an interconnected approach to the study of integrable systems, their surfaces, and methods for producing integrable partial differential equations. After a system of one-forms is defined, moving frame equations can be integrated and the resulting equations for the surface can be obtained. Other differential equations which involve quantities relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case of constant mean curvature, they can be expressed in terms of Jacobi elliptic functions.

Reference EPFL-ARTICLE-234160doi:10.1063/1.5012089View record in Web of Science Record created on 2018-01-15, modified on 2018-01-15

This paper studies the existence, uniqueness, and exponential stability in mean square for the mild solution of neutral second order stochastic partial differential equations with infinite delay and Poisson jumps. By utilizing the Banach fixed point theorem, first the existence and uniqueness of the mild solution of neutral second order stochastic differential equations is established. Then, the mean square exponential stability for the mild solution of the stochastic system with Poisson jumps is obtained with the help of an established integral inequality.

In position dependent mass (PDM) problems, the quantum dynamics of the associated systems have been understood well in the literature for particular orderings. However, no efforts seem to have been made to solve such PDM problems for general orderings to obtain a global picture. In this connection, we here consider the general ordered quantum Hamiltonian of an interesting position dependent mass problem, namely the Mathews-Lakshmanan oscillator, and try to solve the quantum problem for all possible orderings including Hermitian and non-Hermitian ones. The other interesting point in our study is that for all possible orderings, although the Schr\"odinger equation of this Mathews-Lakshmanan oscillator is uniquely reduced to the associated Legendre differential equation, their eigenfunctions cannot be represented in terms of the associated Legendre polynomials with integral degree and order. Rather the eigenfunctions are represented in terms of associated Legendre polynomials with non-integral degree and order. We here explore such polynomials and represent the discrete and continuum states of the system. We also exploit the connection between associated Legendre polynomials with non-integral degree with other orthogonal polynomials such as Jacobi and Gegenbauer polynomials., Comment: Submitted for publication

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I), serves to define a classical and a quantum system which have been found to be superintegrable. We have determined two new orthogonal polynomial solutions (indicated as II and III) that are separable, and which involve Legendre and Gegenbauer polynomials. Here we report on their three interbasis expansion coefficients: between the I--II and I--III bases they are given by $_3F_2(\cdots|1)$ polynomials that are also special su($2$) Clebsch-Gordan coefficients and Hahn polynomials. Between the II--III bases, we find an xpansion expressed by $_4F_3(\cdots|1)$'s and Racah polynomials that are related to the Wigner $6j$ coefficients., Comment: Submitted to Journal of Mathematical Physics (5 Figures)

We take advantage of the simple approach, recently discussed, which associates to (solvable) matrix equations (solvable) dynamical systems interpretable as (interesting) many-body problems, possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles. Starting from a solvable matrix evolution equation, we obtain the corresponding many-body model and note that in one case the auxiliary variables can be altogether eliminated, obtaining thereby an (also Hamiltonian) extension of the “goldfish” model. The solvability of this novel model, and of its isochronous variant, is exhibited. A related, as well solvable, model, is also introduced, as well as its isochronous variant. Finally, the small oscillations of the isochronous models around their equilibrium configurations are investigated, and from their isochronicity certain diophantine relations are evinced.

The motion of a one-dimensional harmonic oscillator caused by recurring pushes in the absence of friction is considered. In particular, two cases are studied: the case when the pushes become more frequent and the other one when the pushes become less frequent. By means of an application of the Hardy–Littlewood–Vinogradov–Van der Corput theorem on the approximation of exponential sums by shorter ones, new asymptotic formulas for the solution of the problem are obtained.

We study the discrete part of spectrum of a singular non-self-adjoint second-order differential equation on a semiaxis with an operator coefficient. Its boundedness is proved. The result is applied to the Schrodinger boundary value problem −Δu+q(x)u=λ2u, u|∂D=0, with a complex potential q(x) in an angular domain.

We study an inverse problem for the Zakharov–Shabat system which is motivated by an application to the design of co-directional couplers with prescribed response properties.

By using two different procedures we show that on the space of solutions of a certain class of second-order ordinary differential equations, u″=Λ(s,u,u′), a two-dimensional definite or indefinite metric, gab, can be constructed such that the two-dimensional Hamilton–Jacobi equation, gabu,au,b=1 holds. Furthermore, we show that this structure is invariant under a certain subset of contact transformations (canonical transformations). Two examples are given.

This article is about the focusing nonlinear Schrodinger equation on the half-line. The initial function vanishes at infinity while boundary data are local perturbations of periodic or quasi-periodic (finite-gap) functions. We study the corresponding scattering problem for the Zakharov–Shabat compatible differential equations, the representation of the solution of the nonlinear Schrodinger equation in the quarter of the (x,t)-plane through functions, which satisfy Marchenko integral equations. We use this formalism to investigate the asymptotic behavior of the solution for large time. We prove that under certain conditions a periodic (quasi-periodic) behavior at infinity of boundary data generates an unbounded train of asymptotic solitons running away from the boundary. The asymptotics of the solution shows that boundary data with periodic behavior as time tends to infinity generates a train of such asymptotic solitons even in the case when the initial function is identically zero.

The nonlinear equations that describe solitonic behavior in physical systems have, to-date, typically been derived by approximation or expansion methods. Here, by contrast, hidden integrable structure is revealed in diverse areas of nonlinear continuum mechanics through natural geometric constraints.

The homogeneous Bloch equation reduces to the Riccati equation. By linearizing the Riccati equation, a set of three solutions of the homogeneous Bloch equation is found. The fundamental matrix becomes singular. We clarify the utility and limitation of our approach to solve the homogeneous Bloch equation.

We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansatz defines a new class of generalised Hermite polynomials which are explicit functions of the coupling parameter and tend to ordinary Hermite polynomials in the limit of vanishing coupling constant. The polynomials become orthogonal as parts of the eigenvectors of a Hermitian matrix and, consequently, the exponential part of the solution can not be excluded. We have conjectured the general structure of the solution, both with respect to the quantum number and the order of the expansion. An explicit proof is given for the three leading orders of the asymptotical solution and we sketch a proof for the asymptotical convergence of eigenvectors with respect to norm. From a more practical point of view, we can estimate the required effort for improving the known solution and the accuracy of the eigenvectors. The applied method can be generalised in order to accommodate several variables., Comment: 18 pages, ReVTeX, the final version with rather general expressions

Recently, we developed a unified formalism which shows that an exact solution of an integrable nonlinear evolution equation (belonging to a certain class) can be associated with three distinct moving space curves, which represent the geometric characterizations of the equation concerned. Here, we apply our formalism to the Lamb equation, iqu+qs+q∫s|q|2 ds′=0, and obtain its geometric characterizations by exploiting its mapping to the exactly solvable Belavin–Polyakov equation, mu=m×ms; m2=1. We show that the Lamb equation supports “envelope solitons” and novel “envelope instantons” as exact solutions. As illustrative examples, the surfaces swept out by the three moving curves that correspond to each of these solutions are derived and displayed pictorially. Possible applications are suggested.

In a recent article, certain underdetermined linear systems of partial differential equations connected with Lie–Poisson structures have been studied. They were constructed via power series solutions of the evolution equation for a given Hamiltonian. We extend the results to arbitrary Poisson manifolds, correct an error in the case of degenerate Poisson structures, and show that these linear systems simply characterize Hamiltonian vector fields. Our basic tool is the formal theory of differential equations with its central concept of an involutive system.

A new integrable class of systems of nonlinear partial differential equations (NPDEs) in 2+1 dimensions is derived from the matrix Nizhnik–Novikov–Veselov (NVV) equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, by applying the reduction technique also to the Lax pair of the starting matrix equation and thereby obtaining the Lax pair for the new class of systems of equations. A reduction to a system of two interacting complex fields is briefly described.

We present a class of time-dependent potentials in Rn that can be integrated by separation of variables: by embedding them into so-called cofactor pair systems of higher dimension, we are led to a ...

The fundamental elements of the variable separation theory are revisited, including the Eisenhart and Robertson theorems, Kalnins–Miller theory, and the intrinsic characterization of the separation of the Hamilton–Jacobi equation, in a unitary and geometrical perspective. The general notion of complete integrability of first-order normal systems of PDEs leads in a natural way to completeness conditions for separated solutions of the Schrodinger equation and to the Robertson condition. Two general types of multiplicative separation for the Schrodinger equation are defined and analyzed: they are called “free” and “reduced” separation, respectively. In the free separation the coordinates are necessarily orthogonal, while the reduced separation may occur in nonorthogonal coordinates, but only in the presence of symmetries (Killing vectors).

We recall the construction of the common eigenvectors of Gaudin Hamiltonians based on the Bethe ansatz. In the case of an arbitrary Lie algebra, this construction can be done either recursively or explicitly and we prove the equivalence of the two methods. We also prove that Bethe vectors are singular only if the Bethe equations are satisfied. In each eigenspace of the spin operator we construct additional common eigenvectors, having the same eigenvalue as the vacuum vector and which can not be obtained by the Bethe ansatz. These eigenvectors are not singular. We also recall the connection between Bethe vectors and integral solutions of the KZ equation. In an analogous way, the additional vectors lead to solutions of KZ equation which are not singular vectors and do not have an integral representation.

The (2+1)-dimensional (M+N)-component AKNS system that is derived from the inner parameter dependent symmetry constraint of the KP equation is studied in detail. First, the Painleve integrability of the model is proved by using the standard WTC and Kruskal approach. Using the formal series symmetry approach, the generalized KMV symmetry algebra and the related symmetry group are found. The two-dimensional similarity partial differential equation reductions and the ordinary differential equation reductions are obtained from the generalized KMV symmetry algebra and the direct method. Abundant localized coherent structures are revealed by the variable separation approach. Some special types of the localized excitations like the multiple solitoffs, dromions, lumps, ring solitons, breathers and instantons are plotted also.

Focusing of waves in one dimension is analyzed for the plasma-wave equation and the wave equation with variable speed. The existence of focusing causal solutions to these equations is established, and such wave solutions are constructed explicitly by deriving an orthogonality relation for the time-independent Schrodinger equation. The connection between wave focusing and inverse scattering is studied. The potential at any point is recovered from the incident wave that leads to focusing to that point. It is shown that focusing waves satisfy certain temporal-antisymmetry and support properties. Discontinuities in the spatial and temporal derivatives of the focusing waves are examined and related to the discontinuities in the potential of the Schrodinger equation. The theory is illustrated with some explicit examples.

The problem of determining the metric for a nonstatic shear-free spherically symmetric fluid (either charged or neutral) reduces to the problem of determining a one-parameter family of solutions to a second-order ordinary differential equation (ODE) containing two arbitrary functions f and g. Choices for f and g are determined such that this ODE admits a one-parameter family of solutions that have poles as their only movable singularities. This property is strictly weaker than the Painleve property and it is used to identify classes of solvable models. It is shown that this procedure systematically generates many exact solutions including the Vaidya metric, which does not arise from the standard Painleve analysis of the second-order ODE. Interior solutions are matched to exterior Reissner–Nordstrom metrics. Some solutions given in terms of second Painleve transcendents are described.

A recently proposed technique to solve a class of second order functional difference equations arising in electromagnetic diffraction theory is further investigated by applying it to a case of intermediate complexity. The proposed approach is conceptually simple and relies on first obtaining well-defined branched solutions to a pair of associated first order difference equations. The construction of these branched expressions leads to an equation system whose solution requires relationships akin to Riemann’s bilinear relations for differentials of the first and third kinds; their derivation necessitates the application of Cauchy’s theorem on Riemann surfaces of, in this particular instance, genera one and three. Branch-free solutions of the second order difference equation are then obtained by taking appropriate linear combinations of the branched solutions of the first order equations. Analysis and computation demonstrate that the resulting expressions have the desired analytical properties and recover known solutions in the appropriate limit.

Considered is a linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters. Asymptotic solutions have been constructed. As has been shown, an important characteristic determining the qualitative portrait of the system is the real part of Berry’s complex geometric phase, which we call the geometric amplitude. For systems with purely imaginary eigenvalues the equivalence has been proven in the adiabatic approximation of system sets without geometric amplitude, Hamiltonian systems, quantizable systems, and strongly stable systems. Classification of the systems without geometric amplitude is given with respect to the kind of matrix of the initial system.

We study the invariance of stochastic differential equations under random diffeomorphisms, and establish the determining equations for random Lie-point symmetries of stochastic differential equations, both in Ito and in Stratonovich form. We also discuss relations with previous results in the literature., In new version (November 2017) we have added an important ERRATUM. Due to a trivial mistake in a formula, several examples should be revised; more relevant, the qualitatove results of Section VIII turn out to be wrong, as discussed in the erratum. See also arXiv:1711.01999

We present a unifying solution framework for the linearized compressible equations for two-dimensional linearly sheared unbounded flows using the Lie symmetry analysis. The full set of symmetries that are admitted by the underlying system of equations is employed to systematically derive the one- and two-dimensional optimal systems of subalgebras, whose connected group reductions lead to three distinct invariant ansatz functions for the governing sets of partial differential equations (PDEs). The purpose of this analysis is threefold and explicitly we show that (i) there are three invariant solutions that stem from the optimal system. These include a general ansatz function with two free parameters, as well as the ansatz functions of the Kelvin mode and the modal approach. Specifically, the first approach unifies these well-known ansatz functions. By considering two limiting cases of the free parameters and related algebraic transformations, the general ansatz function is reduced to either of them. This f...

The Hamilton–Jacobi theory is a formulation of classical mechanics equivalent to other formulations as Newtonian, Lagrangian, or Hamiltonian mechanics. The primordial observation of a geometric Hamilton–Jacobi theory is that if a Hamiltonian vector field XH can be projected into the configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field XHdWcan be transformed into integral curves of XH provided that W is a solution of the Hamilton–Jacobi equation. Our aim is to derive a geometric Hamilton–Jacobi theory for physical systems that are compatible with a Nambu–Poisson structure. For it, we study Lagrangian submanifolds of a Nambu–Poisson manifold and obtain explicitly an expression for a Hamilton–Jacobi equation on such a manifold. We apply our results to two interesting examples in the physics literature: the third-order Kummer–Schwarz equations and a system of n copies of a first-order differential Riccati equation. From the first example, we retrieve the origi...

We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is quasi-periodic solutions having the same frequency vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the frequency vector. In this paper we show that the results extend to the case of arbitrary frequency vectors., 18 pages

In order to determine the region where the Borel sum of a Wentzel–Kramers–Brillouin (WKB) solution ψ of an mth order (m⩾3) ordinary differential equation Pψ=0 with polynomial coefficients is well defined, we apply the steepest descent method to the Laplace integral of a WKB solution ψ of Pψ=0, with P denoting the Laplace transform of P. As a counterpart of the connection formula for ψ, we introduce a new notion of the exact steepest descent path, which is the union of bifurcated steepest descent paths. Both theoretical and computer-assisted experimental studies are given to show the importance of this notion.

A new integrable Davey–Stewartson type class of systems of partial differential equations in 2+1 dimensions is derived from a previously known integrable equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio–temporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, because the reduction technique is applied to the Lax pair of the starting equation and the corresponding Lax pair of the class of systems of equations is found. The particular characteristics of the reduction method imply that the new systems are likely to be of applicative relevance.

In this paper, Lax pairs are constructed for two fifth-order nonlinear evolution equations of “Boussinesq”-type which govern wave propagation in two opposite directions. One of the equations is related to the well-known Sawada–Kotera (SK) equation and, through its bilinear form, is identified with the Ramani equation. The second equation—about which very little seems to be known—may be considered a bidirectional version of the Kaup–Kupershmidt (KK) equation and is the main focus of this study. The “anomalous” solitary wave of this latter equation is derived and is found to possess the remarkable property that its profile depends on the direction of propagation. This type of directional dependence would appear to be quite unusual and, to our knowledge, has not been reported in the literature before now. By taking an appropriate undirectional (long wave) limit, it is shown that neither the Ramani, nor the bidirectional Kaup–Kupershmidt (bKK) equation can be classified as truly “Boussinesq” in character (a distinction that is made precise in the study). Recursion formulas are given for generating an infinity of conserved densities for both equations. These are used to obtain the first few conservation laws of the bKK and Ramani equations explicitly; not surprisingly, they exhibit the same lacunary behavior as their unidirectional counterparts. In conclusion, a canonical interpretation of the N-soliton solution of the bKK equation is proposed which provides a basis for constructing these anomalous solitons in a future work.

A constructive algorithm is proposed for the investigation of symmetries of partial differential equations. The algorithm is used to present classical Lie symmetries of systems of two nonlinear reaction diffusion equations.

Many evolution partial differential equations (PDEs) can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs [P. J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1986)]. In this paper we consider symmetries and Noether’s theorem for semidiscrete Hamiltonian equations which are obtained by space discretization of Hamiltonian PDEs. Using symmetries, one can find conservation laws of these equations. Several applications including a transfer equation and the Korteweg–de Vries equation are presented.