Integrable PDEs Research Articles

On soliton-type solutions of equations associated with N-component systems

The algebraic geometric approach to N-component systems of nonlinear integrable PDE’s is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.

On billiard solutions of nonlinear PDEs

This Letter presents some special features of a class of integrable PDEs admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discontinuities at peaks in their profiles. A connection is established between the peak locations and finite dimensional billiard systems moving inside n-dimensional quadrics under the field of Hooke potentials. Points of reflection are described in terms of theta-functions and are shown to correspond to the location of peak discontinuities in the PDEs weak solutions. The dynamics of the peaks is described in the context of the algebraic-geometric approach to integrable systems.

Near homoclinic orbits of the focusing nonlinear Schrödinger equation

The homoclinic orbits of the integrable nonlinear Schrödinger equation are of interest because of the chaotic behaviour of the equation when it is damped and driven near such an orbit (McLaughlin D W and Overman E A 1995 Whiskered tori for integrable Pde's: chaotic behaviour in near integrable Pde's Surveys in Applied Mathematics vol 1 ed Keller et al (New York: Plenum) ch 2). It is known that exact expressions can be obtained for these homoclinic orbits via Bäcklund transformations using squared eigenfunctions of the associated linear operator (see above reference). In this paper, the stationary equations of the NLS hierarchy are used to separate the temporal and spatial flows into completely integrable finite-dimensional systems which define the saturation of a given linear instability of a plane wave. Near the homoclinic orbits satisfying even boundary conditions, the phase space is a product of two-dimensional phase planes similar to the phase plane of the Duffing oscillator. The near homoclinic orbits of the plane wave are realized numerically by a simple Runge-Kutta scheme that solves the two systems along the characteristic directions of space and time.

The stationary equations of a coupled nonlinear Schrödinger system

The integrable coupled nonlinear Schrödinger (CNLS) equations under periodic boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases [M.G. Forest, D.W. McLaughlin, D. Muraki, O.C. Wright, Non-focussing instabilities in coupled, Integrable nonlinear Schrödinger PDEs, in preparation; D.J. Muraki, O.C. Wright, D.W. McLaughlin, Birefringent optical fibers: Modulational instability in a near-integrable system, Nonlinear Processes in Physics: Proceedings of III Postdam-V Kiev Workshop, 1991, pp. 242–245; O.C. Wright, Modulational stability in a defocussing coupled nonlinear Schrödinger system, Physica D 82 (1995) 1–10], whereas the scalar NLS equation is linearly unstable only in the focussing case [M.G. Forest, J.E. Lee, Geometry and modulation theory for the periodic Schrödinger equation, in: Dafermas et al. (Eds.), Oscillation Theory, Computation, and Methods of Compensated Compactness, I.M.A. Math. Appl. 2 (1986) 35–70]. These instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced Duffing oscillator [Y. Li, D.W. McLaughlin, Morse and Melnikov functions for NLS pde’s, Commun. Math. Phys. 162 (1994) 175–214; D.W. McLaughlin, E.A. Overman, Whiskered tori for integrable Pde’s: Chaotic behaviour in near integrable Pde’s, in: Keller et al. (Eds.), Surveys in Applied Mathematics, vol. 1, Chapter 2, Plenum Press, New York, 1995]. The homoclinic orbits and the near homoclinic tori that are connected to the unstable wave trains of the NLS and the CNLS reside in the finite-dimensional phase space of certain stationary equations [S.P. Novikov, Funct. Anal. Prilozen, 8 (3) (1974) 54–66] of the infinite hierarchy of integrable commuting flows. The correct stationary equations must be matched to the unstable torus through the analytic structure of the spectral curves [O.C. Wright, Near homoclinic orbits of the focussing nonlinear Schrödinger equation, preprint]. Thus, in this paper, the stationary equations of the CNLS are derived and the analytic structure of the trigonal spectral curve is examined, providing a basis for further study of the near homoclinic orbits of the CNLS system.

Non-autonomous Hénon-Heiles systems

Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable Hénon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivatives of the tau-functions (inherited from the original PDEs). The ODEs for the similarity solutions also have inherited Bäcklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlevé transcendent.

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.

Algorithmic determination of infinite-dimensional symmetry groups for integrable systems in 1 + 1 dimensions

We present algorithms and describe CA-packages to compute the infinitesimal generators of infinite-dimensional symmetry groups for integrable PDEs (evolution equations) in one space and one time dimension. Here, integrable is meant in the sense that the vector field defining the equation is a member of the abelian part of some infinite-dimensional Virasoro algebra. The method of computation is completely different from the usual prolongation method, no determining equations are solved. Instead, all necessary generators of the finitely generated Virasoro algebra are computed from one given element by direct Lie algebra methods. The implementation of the algorithms in MuPAD is described. A sample session is included in which the recursion structures of the KdV and the Krichever-Novikov equations are computed.

Statistical Mechanics of Classical Particles with Gravitational Interactions: Exactly Solvable (for N≤∞) in d=1 and d>2

This paper is concerned with a curious gap in a string of exactly solvable models, a gap that is suggestively related to a completely integrable nonlinear PDE in d=2 known as Liouville's equation. This PDE emerges in a limit N→∞ from the equilibrium statistical mechanics of classical point particles with gravitational interactions (SMGI) in dimension d=2 which, accordingly, is an exactly solvable continuum model in this limit. Interestingly, in d=1 and all d>2, the SMGI can be, and partly has been, exactly evaluated for all N≤∞. This entitles one to suspect that the SMGI for d=2 is likewise exactly solvable for N>∞, but currently this is an unproven hypothesis. If this conjecture can be answered in the affirmative, spin-offs in various areas associated with Liouville's equation, such as vortex gases, superfluidity, random matrices, and string theory can be expected.

Deformations of nonassociative algebras and integrable differential equations

A new class of nonassociative algebras related to integrable PDE's and ODE's is introduced. These algebras can be regarded as a noncommutative generalization of Jordan algebras. Their deformations are investigated. Relationships between such algebras and graded Lie algebras are established.

Toda chain and nonlinear Schrördinger equation. Nonlocalities

Abstract The problem of non-localities in the theory of 2D Toda chain is discussed. Our approach is based on direct generalization of the notion of first integrals to the 2D case. We formulate also special geometrical properties of the coordinates system formed by non-local dynamical variables. Applications to 2+1 integrable pde's are considered.

On the link between umbilic geodesics and soliton solutions of nonlinear PDEs

In this paper we describe a new class of soliton solutions, called umbilic solitons, for certain nonlinear integrable PDEs. These umbilic solitons have the property that as the space variable x tends to infinity, the solution tends to a periodic wave, and as x tends to minus infinity, it tends to a phase shifted wave of the same shape. The equations admitting solutions in this new class include the Dym equation and equations in its hierarchy. The methods used to find and analyse these solutions are those of algebraic and complex geometry. We look for classes of solutions by constructing associated finite-dimensional integrable Hamiltonian systems on Riemann surfaces. In particular, in this setting we use geodesics on n -dimensional quadrics to find the spatial, or x -flow, which, together with the commuting t -flow given by the equation itself, defines new classes of solutions. Amongst these geodesics, particularly interesting ones are the umbilic geodesics, which then generate the class of umbilic soliton solutions. This same setting also enables us to introduce another class of solutions of Dym-like equations, which are related to elliptic and umbilic billiards.

A Weakly Nonlinear Analysis of Elasto-plastic-Microstructure Models

At certain critical values of the hardening modulus, the governing equations of elasto-plastic flow may lose their hyperbolicity and exhibit two modes of ill-posedness: shear-band and flutter ill-posedness. These modes of ill-posedness are characterized by the uncontrolled growth of modes at infinitely fine scales, which ultimately violates the continuum assumption. In previous work [L. An and A. Peirce, SIAM J. Appl. Math., 54(1994), pp. 708–730], a continuum model accounting for microscale deformations was built. Linear analysis demonstrated the regularizing effect of the microstructure and provided a relationship between the width of the localized instabilities and the microlength scale. In this paper a weakly nonlinear analysis is used to explore the immediate post-critical behavior of the solutions. For both one-dimensional and anti-plane shear models, post-critical deformations in the plastic regions are shown to be governed by the Boussinesq equation (one of the completely integrable PDEs having so...

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.

Class of nonsingular exact solutions for Laplacian pattern formation.

We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions are free of finite-time singularities (cusps) for quite general initial conditions and may well describe real fingering instabilities. At long times the interface consists of N separated moving Saffman-Taylor fingers, with ``stagnation points'' in between, in agreement with numerous observations. This evolution resembles the N-soliton solution of classical integrable PDE's.

Long-time behavior of the N-finger solution of the Laplacian growth equation

The non-singular N-finger solutions of the Laplacian Growth Equation, Im [ ovbar| ∂ f( x, t)∂ t) (∂ f( x, t)/∂ x)] = 1, describing the motion of the interface in numerous non-equilibrium processes, such as dendritic growth, flows through porous media, electrodeposition, etc., is analyzed. The motion of the interface is described by N + 1 moving singularities (simple poles) in the upper-half of an auxiliar “mathematical plane”. In the long-time limit these singularities tend to the real axis, following an exponential law. Meanwhile, the physical interface develops at most N separated fingers. In the case of enough separation, each of the gaps between fingers corresponds to one singularity while each finger is locally similar to the Saffman-Taylor one. The analogy with the N-soliton solutions of exactly integrable PDE's, such as Korteweg-de Vries, Nonlinear Schrödinger, and sine-Gordon equations, is discussed. Using the asymptotic properties of the N-finger solution, canonical variables of “action-angle”-type are introduced.

Group classification and symmetry reductions of the non-linear diffusion-convection equation ut = ( D( u) ux) x − K'( u) ux

A non-linear diffusion-convection equation arising from the theory of transport in porous media is analyzed using the Lie group technique. A complete classification of the functional forms of the transport coefficients is presented for which different symmetry groups are admitted. For a number of interesting cases, symmetry reductions are performed leading to exact group-invariant solutions. In particular, the special case of the (integrable) Fokas-Yortsos equation is examined in the light of the “Painlevé conjecture” concerning the symmetry reductions of integrable PDEs.

Strongly nonlinear modal equations for nearly integrable PDEs

The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations ofN-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to anyN-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation orderN. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.

Chaotic transport and integrable instabilities in a near-integrable, many-particle, Hamiltonian lattice

This paper is a report of detailed numerical experiments on a large-dimensional Hamiltonian lattice which arises as a nearly integrable discretization of the periodic sine-Gordon equation. Through four different one-parameter families of initial data, we focus on the interplay between numerically observable near-integrable behavior, quantified by modern statistical diagnostics, and the integrable PDE solutions and phase space geometry associated with the initial data. Transitions are observed within each family from quasiperiodic behavior to chaotic motion, which is sometimes accompanied by transport toward equipartition. These transitions in the near-integrable system are correlated with integrable features of the initial data, in particular, transitions from neutrally stable to exponentially unstable sine-Gordon solutions and associated homoclinic orbits. Geometrically, these correspond to transitions from neutrally stable tori to whiskered tori. Our results indicate that: (1) Positive Lyapunov exponents (our messure of chaos) in near-integrable orbits coincide one-to-one with integrable PDE instabilities and homoclinic orbits. (2) The statistical transition to equipartition of energy on a long, fixed timescale occurs only if (but not necessarily if) there are integrable instabilities and associated whiskered tori. The rate of transport appears to depend on the amplitude of the homoclinic degrees of freedom. (3) Isolated cases of discreteness-induced instabilities and action transport are observed; these return to quasiperiodic flow at finer discretizations. (4) A simple measure, such as energy of the initial data, is not an accurate predictor of chaotic or recurrent behavior; one family of initial data exhibits, as the energy increases, alternating windows of quasiperiodic and chaotic orbits.