Fokas Method Research Articles

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions a,b,A,B. The functions a(k) and b(k) are defined via a nonlinear Fourier transform of the initial data, whereas A(k) and B(k) are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.

A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method

Abstract In this paper we propose a hybrid method for solving inhomogeneous elliptic PDEs based on the unified transform. This approach relies on the derivation of the global relation, containing certain integral transforms of the given boundary data as well as of the unknown boundary values. Herewith, the approximate global relation for the Poisson equation is solved numerically using a collocation method on the complex λ-plane, based on Legendre expansions. The corresponding numerical results are presented using closed-form expressions and numerical approximations for different types of boundary and source data, indicating the applicability of the considered approach. Additionally, the full solution is computed in a recursive manner by splitting the domain into smaller concentric polygons, and by using a spatial-stepping scheme followed by an interpolation step. Furthermore, numerical results are also given for the solution of the Poisson and the inhomogeneous Helmholtz equations on several convex polygons. Additional results are provided for the case of nonconvex polygons as well as for the case of a problem with discontinuities across an interface. The proposed approach provides a framework for solving inhomogeneous elliptic PDEs using the unified transform.

Water waves with moving boundaries

The unified transform, also known as the Fokas method, provides a powerful methodology for studying boundary value problems. Employing this methodology, we analyse inviscid, irrotational, two-dimensional water waves in a bounded domain, and in particular we study the generation of waves by a moving piecewise horizontal bottom, as it occurs in tsunamis. We show that this problem is characterised by two equations which involve only first-order derivatives. It is argued that under the assumptions of ‘small amplitude waves’ but not of ‘long waves’, the above two equations can be treated numerically via a recently introduced numerical technique for elliptic partial differential equations in a polygonal domain. In the particular case that the moving bottom is horizontal and under the assumption of ‘small amplitude waves’, but not of ‘long waves’, these equations yield a non-local generalisation of the Boussinesq system. Furthermore, under the additional assumption of ‘long waves’ the above system yields a Boussinesq-type system, which however includes the effect of the moving boundary.

Initial–boundary value problems associated with the Ablowitz–Ladik system

We employ the Ablowitz–Ladik system as an illustrative example in order to demonstrate how to analyze initial–boundary value problems for integrable nonlinear differential–difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schrödinger and integrable discrete modified Korteweg–de Vries equations in terms of the solutions of appropriate matrix Riemann–Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations and the process of eliminating of the unknown boundary values.

Fokas’s Unified Transform Method for linear systems

We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, allowing the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. The case of wave equations is treated separately in an appendix.

Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method

In this paper, we implement the Fokas method in order to study initial-boundary value problems of the coupled modified Korteweg–de Vries equation formulated on the half-line, with Lax pairs involving matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution can be written in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions and , which are respectively determined by the initial values and boundary values at . Finally, the associated Dirichlet to Neumann map of the equation is analyzed in detail. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.

Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

Integral representations for the solution of the Laplace, modified Helmholtz, and Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well‐posed boundary value problem (BVP) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs), usually referred to as the unified transform or the Fokas method, was introduced in the late 1990s. For linear elliptic PDEs in two dimensions, this method first, by employing two algebraic equations formulated in the Fourier plane, provides an elegant approach for determining the Dirichlet to Neumann map, i.e., for constructing the unknown boundary values in terms of the given boundary data. Second, this method constructs novel integral representations of the solution in terms of integrals formulated in the complex Fourier plane. In the present paper, we extend this novel approach to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three‐dimensional cylindrical domain with a polygonal cross‐section.

The elliptic sinh-Gordon equation in a semi-strip

Abstract We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method.

In this paper, we implement the Fokas method to study initial-boundary value problems of the mixed coupled nonlinear Schrödinger equation formulated on the half-line with Lax pairs involving 3×3 matrices. The solution can be written in terms of the solution to a 3×3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions s(k) and S(k), which are determined by the initial values and boundary values at x=0, respectively. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.

STOCHASTIC BLACK-SCHOLES EQUATION WITH TIME-FRACTIONAL DERIVATIVE ON THE HALF-LINE

We investigate the pricing of options using a modified Black-Scholes equation with a time-fractional derivative and additive white noise on the half-line. We construct the Green function for the initial-boundary value problem adapting the main ideas of the Fokas method and we prove existence and uniqueness of solutions. AMS Subject Classification: 35C15, 35A22, 60G22

Initial–boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method

Boundary value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order to study initial–boundary value problems of the general coupled nonlinear Schrödinger equation formulated on the finite interval with 3×3 Lax pairs. The solution can be written in terms of the solution of a 3×3 Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions s(k), S(k), and SL(k). The associated general Dirichlet to Neumann map is also analyzed via the global relation. It is interesting that the relevant formulas can be reduced to the analogous formulas derived for boundary value problems formulated on the half-line in the limit when the length of the interval tends to infinity. It is shown that the formulas characterizing the Dirichlet to Neumann map coincide with the analogous formulas obtained via a Gelfand–Levitan–Marchenko representation.

Evolution PDEs and augmented eigenfunctions. Finite interval

The so-called unified or Fokas method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVPs, which will be referred to as problems of type~I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated IBVPs, which will be referred to as problems of type~II, there does not exist a classical transform pair and the solution cannot be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type~II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalized eigenfunctions described in the sixties by Gel'fand and his co-authors.

A Boundary Value Problem for Conjugate Conductivity Equations

We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half‐plane with conductivity , . The representations are obtained via the so‐called unified transform method or Fokas method, involving a Riemann–Hilbert problem on the complex plane when p is even and on a two‐sheeted Riemann surface when p is odd. They are given in terms of the Dirichlet and Neumann data on the boundary of the domain. For even exponent p, we also show how to make the conversion from one type of conditions to the other by using the global relation that follows from the closedness of some differential form. The method used to derive our integral representations could be applied in any bounded simply connected domain of the right half‐plane with a smooth boundary.

The Elliptic Sinh-Gordon Equation in the Quarter Plane

We study the elliptic sinh-Gordon equation formulated in the quarter plane by using the so-called Fokas method, which is a significant extension of the inverse scattering transform for the boundary value problems. The method is based on the simultaneous spectral analysis for both parts of the Lax pair and the global algebraic relation that involves all boundary values. In this paper, we address the existence theorem for the elliptic sinh-Gordon equation posed in the quarter plane under the assumption that the boundary values satisfy the global relation. We also present the formal representation of the solution in terms of the unique solution of the matrix Riemann- Hilbert problem defined by the spectral functions.

Heat equation on a network using the Fokas method

The problem of heat conduction on networks of multiply connected rods is solved by providing an explicit solution of the one-dimensional heat equation in each domain. The size and connectivity of the rods is known, but neither temperature nor heat flux are prescribed at the interface. Instead, the physical assumptions of continuity at the interfaces are the only conditions imposed. This work generalizes that of Deconinck et al (Proc. R. Soc. A 470 22) for heat conduction on a series of one-dimensional rods connected end-to-end to the case of general configurations.

On the Dirichlet to Neumann problem for the 1-dimensional cubic NLS equation on the half-line44Research funded by ARISTEIA II grant no. 3964 from

Initial-boundary value problems for one-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 suffices 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 sufficiently decaying and that, hence, the Fokas method can be applied.

Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(−2iβx) (or asymptotically periodic, u(x, 0) =α exp(−2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: ux(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.

Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.

Stochastic evolution equation with Riesz-fractional derivative and white noise on the half-line

In this work, we consider an initial boundary-value problem for a stochastic evolution equation with Riesz-fractional spatial derivative and white noise on the half-line,{ut(x,t)=Dxαu(x,t)+Nu(x,t)+B˙(x,t),x>0,t∈[0,T],u(x,0)=u0(x),x>0,ux(0,t)=g1(t),t∈[0,T], where Dxα is the Riesz-fractional derivative, α∈(2,3), N is a Lipschitzian operator and B˙(x,t) is the white noise. To construct the integral representation of solutions we use the Fokas method and Picard scheme to prove existence and uniqueness. Moreover, Monte Carlo methods are implemented to approximate solutions.

On the Inverse Scattering Method for Integrable PDEs on a Star Graph

We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schrödinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is introduced. For such conditions, the method is shown to be as efficient as the ISM on the full-line.