Searcharxiv⌕ Search

arXiv subjects

A. S. Fokas

Publications and source records attributed to A. S. Fokas.

51 records · Page 3Linked to original sources
arXiv2009

A Novel Method of Solution for the Fluid Loaded Plate

We study the Cauchy problem associated with the equations governing a fluid loaded plate formulated on either the line or the half-line. We show that in both cases the problem can be solved by employing the unified approach to boundary value problems introduced by on of the authors in the late 1990s. The problem on the full line was analysed by Crighton et. al. using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in $L^1_{dt}(R^+)$ and furthermore yields a simpler solution representation which immediately implies the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, but this formula involves two unknown functions. The main difficulty with the half-line problem is the characterisation of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex valued integral equation of the convolution type. This equation can be solved in closed form using the Laplace transform. By prescribing the initial data $η_0$ to be in $H^3(R^+)$, we show that the solution depends continuously on the initial data, and hence, the problem is well-posed.

math.AP↗
arXiv2008

On a novel integrable generalization of the nonlinear Schrödinger equation

We consider an integrable generalization of the nonlinear Schrödinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.

nlin.SI↗
arXiv2006

The generalised Dirichlet to Neumann map for moving initial-boundary value problems

We present an algorithm for characterising the generalised Dirichlet to Neumann map for moving initial-boundary value problems. This algorithm is derived by combining the so-called global relation, which couples the initial and boundary values of the problem, with a new method for inverting certain one-dimensional integrals. This new method is based on the spectral analysis of an associated ODE and on the use of the d-bar formalism. As an illustration, the Neumann boundary value for the linearised Schroedinger equation is determined in terms of the Dirichlet boundary value and of the initial condition.

math-ph↗
arXiv2006

The Generalized Dirichlet to Neumann map for the KdV equation on the half-line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if $q_{t}$ and $q_{xxx}$ have the same sign (KdVI) or two boundary conditions if $q_{t}$ and $q_{xxx}$ have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if $\{q(x,0),q(0,t) \}$ and $\{q(x,0),q(0,t),q_{x}(0,t) \}$ are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values $\{q_{x}(0,t),q_{xx}(0,t) \}$ and $\{q_{xx}(0,t) \}$, respectively. We show that this can be achieved without solving for $q(x,t)$ by analysing a certain ``global relation'' which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function $Φ^{(t)}(t,k)$, where $Φ^{(t)}$ satisifies the $t$-part of the associated Lax pair evaluated at $x=0$. Indeed, by employing a Gelfand--Levitan--Marchenko triangular representation for $Φ^{(t)}$, the global relation can be solved \emph{explicitly} for the unknown boundary values in terms of the given initial and boundary conditions and the function $Φ^{(t)}$. This yields the unknown boundary values in terms of a nonlinear Volterra integral equation.

nlin.SI↗
arXiv2005

The Nonlinear Schrödinger Equation on the Interval

Let $q(x,t)$ satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation on the finite interval, $0 < x < L$, with $q_{0}(x) = q(x,0)$, $g_{0}(t) = q(0,t)$, $f_{0}(t) = q(L,t)$. Let $g_{1}(t)$ and $f_{1}(t)$ denote the {\it unknown} boundary values $q_{x}(0,t)$ and $q_{x}(L,t)$, respectively. We first show that these unknown functions can be expressed in terms of the given initial and boundary conditions through the solution of a system of nonlinear ODEs. Although the question of the global existence of solution of this system remains open, it appears that this is the first time in the literature that such a characterization is explicitely described for a nonlinear evolution PDE defined on the interval; this result is the extension of the analogous result of [4] and [6] from the half-line to the interval. We then show that $q(x,t)$ can be expressed in terms of the solution of a $2\times 2$ matrix Riemann-Hilbert problem formulated in the complex $k$ - plane. This problem has explicit $(x,t)$ dependence in the form $\exp[2ikx + 4ik^2t]$, and it has jumps across the real and imaginary axes. The relevant jump matrices are explicitely given in terms of the spectral data $\{a(k), b(k)\}$, $\{A(k), B(k)\}$, and $\{\A(k), \B(k)\}$, which in turn are defined in terms of $q_{0}(x)$, $\{g_{0}(t), g_{1}(t)\}$, and $\{f_{0}(t), f_{1}(t)\}$, espectively.

nlin.SI↗
arXiv2004

Reconstruction Algorithms for Positron Emission Tomography and Single Photon Emission Computed Tomography and their Numerical Implementation

The modern imaging techniques of Positron Emission Tomography and of Single Photon Emission Computed Tomography are not only two of the most important tools for studying the functional characteristics of the brain, but they now also play a vital role in several areas of clinical medicine, including neurology, oncology and cardiology. The basic mathematical problems associated with these techniques are the construction of the inverse of the Radon transform and of the inverse of the so called attenuated Radon transform respectively. We first show that, by employing mathematical techniques developed in the theory of nonlinear integrable equations, it is possible to obtain analytic formulas for these two inverse transforms. We then present algorithms for the numerical implementation of these analytic formulas, based on approximating the given data in terms of cubic splines. Several numerical tests are presented which suggest that our algorithms are capable of producing accurate reconstruction for realistic phantoms such as the well known Shepp--Logan phantom.

physics.med-ph↗
arXiv2004

The Unique Determination of Neuronal Currents in the Brain via Magnetoencephalography

The problem of determining the neuronal current inside the brain from measurements of the induced magnetic field outside the head is discussed under the assumption that the space occupied by the brain is approximately spherical. By inverting the Geselowitz equation, the part of the current which can be reconstructed from the measurements is precisely determined. This actually consists of only certain moments of one of the two functions specifying the tangential part of the current. The other function specifying the tangential part of the current as well as the radial part of the current are completely arbitrary. However, it is also shown that with the assumption of energy minimization, the current can be reconstructed uniquely. A numerical implementation of this unique reconstruction is also presented.

physics.med-ph↗
arXiv2004

Boundary Value Problems for Linear PDEs with Variable Coefficients

A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs with {\it constant} coefficients. As illustrative examples the following boundary value problems are solved: (a) A Dirichlet and a Neumann problem on the half line for the time-dependent Schrödinger equation with a space dependent potential. (b) A Poincaré problem on the quarter plane for a variable coefficient eneralisation of the Laplace equation.

math.AP↗
arXiv2004

The Davey-Stewartson I Equation on the Quarter Plane with Homogeneous Dirichlet Boundary Conditions

Dromions are exponentially localised coherent structures supported by nonlinear integrable evolution equations in two spatial dimensions.In the study of initial-value problems on the plane, such solutions occur only if one imposes nontrivial boundary conditions at infinity, a situation of dubious physical significance. However it is established here that dromions appear naturally in the study of boundary-value problems. In particular, it is shown that the long time asymptotics of the solution of the Davey-Stewartson I equation in the quarter plane with arbitrary initial conditions and with zero Dirichlet boundary conditions is dominated by dromions. The case of non-zero Dirichlet boundary conditions is also discussed.

nlin.SI↗
arXiv2004

Linearizable Initial-Boundary Value Problems for the sine-Gordon Equation on the Half-Line

A rigorous methodology for the analysis of initial boundary value problems on the half-line, $0 0$, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the solution $q(x,t)$ of the sine-Gordon equation can be obtained in terms of the solution of a $2\times 2$ matrix Riemann-Hilbert problem. This problem is formulated in the complex $k$-plane and is uniquely defined in terms of the so called spectral functions $a(k)$, $b(k)$, and $B(k)/A(k)$. The functions $a(k)$ and $b(k)$ can be constructed in terms of the given initial conditions $q(x,0)$ and $q_t(x,0)$ via the solution of a system of two {\it linear} ODE's, while for \emph{arbitrary} boundary conditions the functions $A(k)$ and $B(k)$ can be constructed in terms of the given boundary condition via the solution of a system of four {\it nonlinear} ODEs. In this paper we analyse two \emph{particular} boundary conditions: the case of constant Dirichlet data, $q(0,t) = χ$, as well as the case that $q_x(0,t)$, $\sin (q(0,t)/2)$, and $\cos(q(0,t)/2)$ are linearly related by two constants $χ_1$ and $χ_2$. We show that for these particular cases, the system of the above nonlinear ODEs can be avoided, and $B(k)/A(k)$ can be computed explicitly in terms of $\{a(k),b(k),χ\}$ and $\{a(k),b(k), χ_1, χ_2\}$ respectively. Thus these ``linearizable'' initial-boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line.

nlin.SI↗
arXiv2004

The Basic Elliptic Equations in an Equilateral Triangle

In his deep and prolific investigations of heat diffusion, Lamé was led to the investigation of the eigenvalues and eigenfunctions of the Laplace operator in an equilateral triangle. In particular he derived explicit results for the Dirichlet and Neumann cases using an ingenious change of variables. The relevant eigenfunctions are complicated infinite series in terms of his variables. Here we first show that boundary value problems with simple boundary conditions, such as the Dirichlet and the Neumann problems, can be solved in an elementary manner. In particular for these problems, the unknown Neumann and Dirichlet boundary values respectively, can be expressed in terms of a Fourier series. Our analysis is based on the so called global relation, which is an algebraic equation coupling the Dirichlet and the Neumann spectral values on the perimeter of the triangle. As Lamé correctly pointed out, infinite series are inadequate for expressing the solution of more complicated problems, such as mixed boundary value problems. Here we show, utilizing further the global relation, that such problems can be solved in terms of {\it generalized Fourier integrals}.

math.AP↗
arXiv2004

The Time Periodic Solution of the Burgers Equation on the Half-Line and an Application to Steady Streaming

The phenomenon of steady streaming, or acoustic streaming, is an important physical phenomenon studied extensively in the literature. Its mathematical formulation involves the Navier-Stokes equations, thus due to the complexity of these equations is usually studied heuristically using formal perturbation expansions. It turns out that the Burgers equation formulated on the half line provides a simple model of the above phenomenon. The physical situation corresponds to the solution of the Dirichlet problem on the half-line, which decays as $x \to \infty$ and which is {\it time periodic}. We show that the Dirichlet problem, where the usual prescription of the initial condition is now replaced by the requirement of the time periodicity, yields a well posed problem. Furthermore, we show that the solution of this problem tends to the ``inner'' and ``outer'' solutions obtained by the perturbation expansions.

nlin.SI↗
arXiv2001

On the construction of evolution equations admitting a master symmetry

A method for constructing evolution equations admitting a master symmetry is proposed. Several examples illustrating the method are presented. It is also noted that for certain evolution equations master symmetries can be useful for obtaining new conservation laws from a given one.

nlin.SI↗
arXiv1998

The Inverse Spectral Theory for the Ward Equation and for the 2+1 Chiral Model

We solve the Cauchy problem of the Ward model in light-cone coordinates using the inverse spectral (scattering) method. In particular we show that the solution can be constructed by solving a $2\times 2$ local matrix Riemann-Hilbert problem which is uniquely defined in terms of the initial data. These results are also directly applicable to the 2+1 Chiral model.

hep-th↗