A Novel Approach to Elastodynamics: II. The Three-Dimensional Case

A new approach was recently introduced by the authors for constructing analytic solutions of the linear PDEs describing elastodynamics. Here, this approach is applied to the case of a homogeneous isotropic half-space body satisfying arbitrary initial conditions and Lamb's boundary conditions. A particular case of this problem, namely the case of homogeneous initial conditions and normal point load boundary conditions, was first solved by Lamb using the Fourier-Laplace transform. The general problem solved here can also be analysed via the Fourier transform, but in this case, the solution representation involves transforms of \textit{unknown} boundary values; this necessitates the formulation and solution of a cumbersome auxiliary problem, which expresses the unknown boundary values in terms of the Laplace transform of the given boundary data. The new approach, which is applicable to arbitrary initial and boundary conditions, bypasses the above auxiliary problem and expresses the solutions directly in terms of the given initial and boundary conditions

On a Novel Class of Integrable ODEs Related to the Painlev\'e Equations

One of the authors has recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation $h=H(p,q,t),$ where $H$ is a given Hamiltonian containing $t$ explicitly, yields the function $t=T(p,q,h)$, which defines a new Hamiltonian system with Hamiltonian $T$ and independent variable $h.$ By employing this construction and by using the fact that the classical Painlev\'e equations are Hamiltonian systems, it is straightforward to associate with each Painlev\'e equation two new integrable ODEs. Here, we investigate the conjugate Painlev\'e II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlev\'e I and Painlev\'e IV equations.Comment: This paper is dedicated to Professor T. Bountis on the occasion of his 60th birthday with appreciation of his important contributions to "Nonlinear Science

The Unified Method: I Non-Linearizable Problems on the Half-Line

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions.Comment: 39 page

Proof of some asymptotic results for a model equation for low Reynolds number flow

A two-point boundary value problem in the interval [ε, ∞], ε > 0 is studied. The problem contains additional parameters α ≥ 0, β ≥ 0, 0 ≤ U 0; for α = 0 an explicit construction shows that no solution exists unless k > 1. A special method is used to show uniqueness. For ε ↓ 0, k ≥ 1, various results had previously been obtained by the method of matched asymptotic expansions. Examples of these results are verified rigorously using the integral representation. For k < 1, the problem is shown not to be a layer-type problem, a fact previously demonstrated explicitly for k = 0. If k is an integer ≥ 0 the intuitive understanding of the problem is aided by regarding it as spherically symmetric in k + 1 dimensions. In the present study, however, k may be any real number, even negative

An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons.Comment: 4 pages, no figure

Dromion perturbation for the Davey-Stewartson-1 equations

The perturbation of the dromion of the Davey-Stewartson-1 equation is studied over the large time

On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS Equation on the half-line

This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0951-7715/28/9/3073Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data su ces to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also su ciently decaying and that, hence, the Fokas method can be applied

On the location of poles for the Ablowitz-Segur family of solutions to the second Painlev\'e equation

Using a simple operator-norm estimate we show that the solution to the second Painlev\'e equation within the Ablowitz-Segur family is pole-free in a well defined region of the complex plane of the independent variable. The result is illustrated with several numerical examples.Comment: 8 pages, to appear in Nonlinearit