Integrable Nonlinear PDEs Research Articles

Arnold--Liouville Theorem for Integrable PDEs: A Case Study of the Focusing NLS Equation

We prove an infinite dimensional version of the Arnold-Liouville theorem for integrable non-linear PDEs: In a case study we consider the {\em focusing} NLS equation with periodic boundary conditions.

On quantum and relativistic mechanical analogues in mean-field spin models

Conceptual analogies among statistical mechanics and classical or quantum mechanics have often appeared in the literature. For classical two-body mean-field models, such an analogy is based on the identification between the free energy of Curie–Weiss-type magnetic models and the Hamilton–Jacobi action for a one-dimensional mechanical system. Similarly, the partition function plays the role of the wave function in quantum mechanics and satisfies the heat equation that plays, in this context, the role of the Schrödinger equation. We show that this identification can be remarkably extended to include a wider family of magnetic models that are classified by normal forms of suitable real algebraic dispersion curves . In all these cases, the model turns out to be completely solvable as the free energy as well as the order parameter are obtained as solutions of an integrable nonlinear PDE of Hamilton–Jacobi type. We observe that the mechanical analogue of these models can be viewed as the relativistic analogue of the Curie–Weiss model and this helps to clarify the connection between generalized self-averaging in statistical thermodynamics and the semiclassical dynamics of viscous conservation laws.

Synthesis, as Opposed to Separation, of Variables

Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one-parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the “synthesis” as opposed to the “separation” of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the development of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open. This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearized KdV equation formulated on the half-line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to the radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method. Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE.

Efficient numerical solution of the generalized Dirichlet–Neumann map for linear elliptic PDEs in regular polygon domains

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet–Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation’s coefficient matrix properties. We prove that, for the Laplace’s equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.

Boundary-value problems for the stationary axisymmetric Einstein equations: a rotating disc

The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analysing boundary-value problems (BVPs) for integrable PDEs, and this method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix Riemann–Hilbert (RH) problem formulated in the complex k-plane, where k denotes the spectral parameter of the associated Lax pair. This representation involves, in general, some unknown boundary values, thus the solution formula is not yet effective. (b) Characterize the unknown boundary values by analysing a certain equation called the global relation. This analysis involves, in general, the solution of a nonlinear problem; however, for certain BVPs called linearizable, it is possible to determine the unknown boundary values using only linear operations. Here, we employ the above methodology for the investigation of certain BVPs for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurrence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular BVP corresponding to the physically significant case of a rotating disc is a linearizable BVP. In this way the remarkable results of Neugebauer and Meinel are recovered.

Riemann–Hilbert Approach to the Elastodynamic Equation: Part I

We develop the Riemann–Hilbert (RH) approach to scattering problems in elastic media. The approach is based on the RH method introduced in the 1990s by Fokas (A unified approach to boundary value problems, CBMS-SIAM, 2008) for studying boundary problems for linear and integrable nonlinear PDEs. A suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations derived from this Lax pair are applied to Rayleigh wave propagation in an elastic half space and quarter space. The latter problem is reduced to the analysis of a certain underdetermined RH problem. We show that the problem can be re-formulated as a well-determined vector Riemann–Hilbert problem with a shift posed on a torus.

The Kadomtsev–Petviashvili II equation on the half-plane

The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics, where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey–Stewartson equation (Fokas (2009) [13]), in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d -bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial value problems in 2+1 is that the d -bar formalism now involves a function in the complex plane which is discontinuous across the real axis.

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.

The Klein–Gordon Equation in a Domain with Time‐Dependent Boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.

K‐symmetric AKS systems and flat immersions into spheres

We define a large class of integrable nonlinear PDEs, k‐symmetric AKS systems, with solutions that evolve on finite‐dimensional subalgebras of loop algebras and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the solution, and show that the problem of isometrically immersing n dimensional Euclidean space into a sphere of dimension 2n – 1 can be addressed via this scheme, producing infinitely many real analytic solutions.

The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation

A new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in Fokas [A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 53 (1997) 1411–1443]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. This is based on the analysis of the so-called global relation, an equation which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. A collocation-type numerical method for solving the global relation for the Laplace equation in an arbitrary bounded convex polygon was introduced in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. Here, by choosing a different set of the “ collocation points” (values for k), we present a significant improvement of the results in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. The new collocation points lead to well-conditioned collocation methods. Their combination with sine basis functions leads to a collocation matrix whose diagonal blocks are point diagonal matrices yielding efficient implementation of iterative methods; numerical experimentation suggests quadratic convergence. The choice of Chebyshev basis functions leads to higher order convergence, which for regular polygons appear to be exponential.

A novel analytic technique for the disintegration of coherent structure solutions of non-integrable and integrable nonlinear PDEs under forcing

We consider a technique for deriving exact analytic coherent structure (pulse/front/domain wall) solutions of general NLPDEs via the use of truncated invariant Painlevé expansions. Coupling this to Melnikov theory, we then consider the breakdown to chaos of such analytic coherent structure solutions of the Zakharov–Kuznetsov and reaction-diffusion equations under forcing. We also demonstrate that similar treatments are possible for integrable systems (using the well-studied forced KdV equation as an example) where the soliton/kink solutions represent the homoclinic/heteroclinic structures of the reduced ODEs. A method of treating the dynamics of the system prior to the onset of chaos by the use of intrinsic harmonic balance, multiscale or direct soliton perturbation theory is briefly discussed. It is conceivable that resummation of such perturbation series via the use of Fade approximants or other techniques may enable one to analytically follow the homoclinic or heteroclinic tangling beyond the first transversal intersection of the stable and unstable manifolds and into the chaotic regime.

A new class of linearizable equations

Using the symmetry approach, we find a class of integrable nonlinear PDEs with dispersion law . All these equations turn out to be linearizable by means of a differential parametrization.

Homoclinic orbits for soliton equations solvable via the quadratic bundle

A method is proposed to generate homoclinic solutions for an integrable nonlinear PDE with periodic boundaries. This approach resembles the dressing method known in the theory of solitons. The pole positions in the dressing factor are given by the complex double points of the Floquet spectrum associated with unstable modes of a seed solution of the nonlinear equation. As an example, we obtain the homoclinic orbit for the modified nonlinear Schrödinger equation solvable by the Wadati–Konno–Ichikawa spectral problem.

Long‐time asymptotics of moving boundary problems using an Ehrenpreis‐type representation and its Riemann‐Hilbert nonlinearization

AbstractA new method for solving boundary value problems for linear and integrable nonlinear PDEs has recently been introduced. For a linear evolution equation with dispersion relation ω(k), in a time‐dependent concave domain, 𝓁(t) < x < ∞, 𝓁″(t) < 0, this method yields the solution q(x, t) as a line integral in the complex k‐plane, whose integrand has explicit (x, t)‐dependence of the form exp[ikx − iω(k)t]. For a nonlinear integrable evolution equation the situation is similar, but the integrand also involves the solution M(x, t, k) to a Riemann‐Hilbert problem whose jump function also has explicit (x, t)‐dependence of exponential form. These representations for linear and for nonlinear evolution equations are convenient for the study of the long‐time asymptotics, using the stationary phase and the Deift‐Zhou methods, respectively. Here we study the long‐time asymptotics in a time‐dependent concave domain for a linear evolution equation with a spatial derivative of arbitrary order, and for the defocusing nonlinear Schrödinger equation. For completeness we also study the long‐time asymptotics for a linear evolution equation with a spatial derivative of arbitrary order in the fixed domain 0 < x < ∞. © 2003 Wiley Periodicals, Inc.

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

Lax pairs and a new spectral method for linear and integrable nonlinear PDEs

A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem. The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is linearized.

A unified transform method for solving linear and certain nonlinear PDEs

A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann–Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine–Gordon equation in light–cone coordinates, the nonlinear Schrodinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrodinger equation is derived.

A New Green's Function Method for Solving Linear PDE's in Two Variables

A new spectral method for solving linear and integrable nonlinear PDE's in two variables has recently been developed. This method is based on a Lax pair formulation, and generates an elegant integral representation of the solution to a given initial-boundary value problem that is particularly useful for determining long-time asymptotics. In this paper, it is shown that the integral representation obtained using Lax pairs can also be obtained using Green's function techniques. This new Green's function method is illustrated by solving a variety of initial-boundary value problems for the diffusion equation, the linearized Schrödinger equation, and the linearized KdV equation.