Applicable Analysis wishes to become a leading journal for high-quality papers on all mathematical aspects of Completely Integrable Systems. As a first step in this direction, the Editorial Board has the honour of offering you this special issue dedicated to continuous and discrete integrable systems with applications.
Historically, a nonlinear partial differential equation (PDE) is called completely integrable if the initial value problem (with appropriate data) can be solved with the Inverse Scattering Transform (IST). The IST can be viewed as a nonlinear analogue of the Fourier transform. To apply the IST, one must first find a system of linear PDEs (Lax pair), so that the original nonlinear PDE comes about by expressing the compatibility of that Lax pair. Hence, the existence of a Lax pair is a necessary albeit insufficient condition for complete integrability.
Completely integrable models have a rich mathematical structure. For example, they can be formulated as infinite dimensional Hamiltonian systems and admit infinitely many conservation laws and higher-order symmetries; the latter connected via a recursion operator. Furthermore, completely integrable equations admit solitons of any order, i.e., pulse-like solutions which interact elastically upon collision. Any of the above rather unusual features could serve as an alternative definition of complete integrability.
To date, the best known completely integrable equations are PDEs that arise from physical phenomena, such as the Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), sine-Gordon, nonlinear Schrodinger (NLS), and Davey-Stewartson equations.
However, many of the features of complete integrability apply to nonlinear differential-difference equations (DDEs), where time is kept continuous but the space variable has been discretized. If all independent variables in a nonlinear PDE are discretized, one arrives at a nonlinear partial difference equation (PΔE). Discretization is at the core of numerical methods, and often one strives to preserve certain geometric or analytic properties of the differential equation, including its integrability. Apart from arising as discretizations of differential equations, nonlinear DDEs and PΔEs are used to model phenomena that are genuinely discrete. Therefore, in many field of mathematics and the applied sciences, difference equations are often more fundamental than differential equations.
The study of completely integrable PDEs peaked near the middle of the nineteen-eighties. Dozens of books have been written on the subject, and topics such as solitons, complete integrability, and IST, are now routinely included in scientific encyclopedias. The interested reader may consult Alwyn Scott's Encyclopedia of Nonlinear Science (Routledge, 2004), Encyclopedia of Mathematical Physics, edited by Jean-Pierre Françoise, Gregory Naber, and Tsou Sheung Tsun (Elsevier, 2007), or Encyclopedia of Complexity and System Science, edited by Robert A. Meyer (Springer, 2009).
In contrast, many of the research results related to nonlinear integrable difference equations were obtained during the last two decades or so and the research community involved with discrete equations is quite active. Indeed, the study of discrete systems has become one of the focal points in the integrable systems community. For example, during the first semester of 2009, the Isaac Newton Institute for Mathematical Sciences in Cambridge (U.K.) ran a programme devoted to Discrete Integrable Systems. Many of the lectures presented there are available in streaming video format. Novices and experts alike can also find a wealth of information (including topical review papers) in, e.g., the proceedings of the past eight ‘Symmetry and Integrability of Difference Equations (SIDE)’ conferences, held bi-annually since 1994.
The eleven papers in this special issue cover theoretical as well as computational aspects of the analysis of completely integrable systems, both PDEs and DDEs, as well as PΔEs. While the papers in this issue are ordered by their dates of acceptance, in the summary below I have grouped the papers by subject.
Starting with nonlinear PDEs, Poole and Hereman give explicit formulas for the homotopy operator, which facilitates integration by parts in several variables. They illustrate how the homotopy operator can be applied in the symbolic computation of conservation laws of the Zakharov-Kuznetsov equation.
The papers by Ma and Zhang, and Squires and Marí Beffa deal with geometrical aspects of integrable PDEs. Ma and Zhang derive component-trace identities for Hamiltonian structures. They apply their theory to the perturbation equations of the KdV hierarchy. Squires and Marí Beffa investigate the relationship between geometric Poisson brackets and integrable systems in flat Galilean and Minkowski spaces. They show how the evolutions, Poisson structures, and integrability of systems associated with the Lorentz group are connected with those of the Galilean group as the light speed goes to infinity.
Using the τ – function in Wronskian and Gramian determinant forms, Chakravarty, Lewkow, and Maruno construct line-solitons for the KP equation. Special attention is paid to the interaction properties of two distinct types of 2-soliton solutions.
With a focus on computational issues, Deconinck and Lovit in their contribution demonstrate that the stationary solutions of the NLS equation can be efficiently used as an orthonormal basis for representing square integrable functions with periodic boundary conditions. Using examples from water wave theory and nonlinear optics, they illustrate that the NLS basis sets offer an attractive alternative to commonly-used Fourier modes.
Turning to DDEs, Levi and Scimiterna use a multi-scale reduction technique to study the integrability of a class of discrete NLS equations involving constant parameters. The authors show that the Ablowitz-Ladik equation is the only equation in that class that passes their integrability test.
In their paper, Biondini and Hwang solve the initial-boundary value problem (IBVP) for the Ablowitz-Ladik lattice. They identify a new type of linearizable IBVP, characterize the symmetry properties of the discrete spectrum and norming constants, and investigate the behaviour of soliton solutions.
Four articles are dedicated to the analysis of PΔEs. In Hone's paper a set of problems relevant to integrable recurrence relations is discussed. Hone focusses on properties of Somos sequences, one of which arises from a bilinear discrete KP equation named after Hirota and Miwa. In particular, Hone constructs conserved quantities and an analytic solution for a family of quadratic Somos recurrences of order six.
In their paper, Hydon and Viallet present a systematic method to find conservation laws of a class of asymmetric quad-graph equations. Equations with an unusual number of conservation laws of low order are good candidates for integrability. By calculating their algebraic entropy, they subject these candidates to an additional diagnostic test for integrability.
Boll and Suris establish a relation between 3-dimensional consistent systems of quad-equations and all non-symmetric discrete relativistic Toda-type equations. They show that the known 3D consistent systems of quad-graph equations leading to both symmetric and non-symmetric discrete relativistic Toda systems are particular or limiting cases of one ‘master’ system. Their approach also allows for an algorithmic construction of the zero-curvature (Lax) representation.
Finally, a new discrete potential Boussinesq equation is proposed in the paper by Maruno and Kajiwara, together with its soliton solutions, and its ultra-discrete analogue. A comparison is made with two previously known discrete potential Boussinesq equations, one of which is analysed with the singularity confinement method.
I gratefully thank all those who have helped with the publication of this special issue. In particular, I thank Robert P. Gilbert and Yongzhi Steve Xu, Editors-in-Chief of Applicable Analysis, for their encouragement, support, and guidance. Last, but not least, I am grateful to the authors for their participation and the reviewers for their critical evaluations and suggestions for improvements.
The Editorial Board hopes that this special issue will further stimulate research of completely integrable systems and encourage researchers in this area to consider Applicable Analysis as a publication venue.
Willy A. Hereman
Colorado School of Mines, USA
22 February, 2010