On asymptotic solutions of integrable wave equations

Asymptotic 'soliton train' solutions of integrable wave equations described by inverse scattering transform method with second-order scalar eigenvalue problem are considered. It is shown that if asymptotic solution can be presented as a modulated one-phase nonlinear periodic wavetrain, then the corresponding Baker-Akhiezer function transforms into quasiclassical eigenfunction of the linear spectral problem in weak dispersion limit for initially smooth pulses. In this quasiclassical limit the corresponding eigenvalues can be calculated with the use of the Bohr Sommerfeld quantization rule. The asymptotic distributions of solitons parameters obtained in this way specify the solution of the Whitham equations. (C) 2001 Elsevier B.V. B.V. All rights reserved.

Integrable hierarchy for multidimensional Toda equations and topological-anti-topological fusion

The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous A(m)-hierarchy and its (gl) over cap (m + 1, C) extension.A loop group automorphism of order two is used to define a sub-hierarchy of (gl) over cap (m + 1, C) hierarchy containing only the odd symmetry flows. The positive and negative flows of the +/-1 grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion. (C) 2002 Elsevier B.V. B.V. All rights reserved.

Integrable aspects of the scaling q-state Potts models I: bound states and bootstrap closure

We discuss the q-state Potts models for q less than or equal to 4, in the scaling regimes close to their critical or tricritical points. Starting from the kink S-matrix elements proposed by Chim and Zamolodchikov, the bootstrap is closed for the scaling regions of all critical points, and for the tricritical points when 4 > q greater than or equal to 2. We also note a curious appearance of the extended last line of Freudenthal's magic square in connection with the Potts models. (C) 2003 Elsevier B.V. B.V. All rights reserved.

T-duality in 2D integrable models

The non-conformal analogue of Abelian T-duality transformations relating pairs of axial and vector integrable models from the non-Abelian affine Toda family is constructed and studied in detail.

Integrable aspects of the scaling q-state Potts models II: finite-size effects

We continue our discussion of the q-state Potts models for q less than or equal to 4, in the scaling regimes close to their critical and tricritical points. In a previous paper, the spectrum and full S-matrix of the models on an infinite line were elucidated; here, we consider finite-size behaviour. TBA equations are proposed for all cases related to phi(21) and phi(12) perturbations of unitary minimal models. These are subjected to a variety of checks in the ultraviolet and infrared limits, and compared with results from a recently-proposed non-linear integral equation. A non-linear integral equation is also used to study the flows from tricritical to critical models, over the full range of q. Our results should also be of relevance to the study of the off-critical dilute A models in regimes 1 and 2. (C) 2003 Elsevier B.V. B.V. All rights reserved.

Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation

In this Letter we investigate Lie symmetries of a (2 + 1)-dimensional integrable generalization of the Camassa-Holm (CH) equation. Through the similarity reductions we obtain four different (1 + 1)-dimensional systems of partial differential equations in which one of them turns out to be a (1 + 1)-dimensional CH equation. We establish their integrability by providing the Lax pair for all of them. Further, we present a brief analysis for some types of particular solutions which include the cuspon, peakon and soliton solutions for the two-dimensional generalization of the CH equation. (C) 2000 Published by Elsevier B.V. B.V.

On the relationship between a 2 x 2 matrix and second-order scalar spectral problems for integrable equations

A simple proof is given that a 2 x 2 matrix scheme for an inverse scattering transform method for integrable equations can be converted into the standard form of the second-order scalar spectral problem associated with the same equations. Simple formulae relating these two kinds of representation of integrable equations are established.

Classical integrable super sinh-Gordon equation with defects

The introduction of defects is discussed under the Lagrangian formalism and Backlund transformations for the N = 1 super sinh-Gordon model. Modified conserved momentum and energy are constructed for this case. Some explicit examples of different Backlund soliton solutions are discussed. The Lax formulation within the space split by the defect leads to the integrability of the model and henceforth to the existence of an infinite number of constants of motion.

Symmetry analysis of an integrable reaction-diffusion equation

In this paper, we investigate the invariance and integrability properties of an integrable two-component reaction-diffusion equation. We perform Painleve analysis for both the reaction-diffusion equation modelled by a coupled nonlinear partial differential equations and its general similarity reduced ordinary differential equation and confirm its integrability. Further, we perform Lie symmetry analysis for this model. Interestingly our investigations reveals a rich variety of particular solutions, which have not been reported in the literature, for this model. (C) 2000 Elsevier B.V. Ltd. All rights reserved.

Integrable field theories with defects

The structure of integrable field theories in the presence of defects is discussed in terms of boundary functions under the Lagrangian formalism. Explicit examples of bosonic and fermionic theories are considered. In particular, the boundary functions for the super sinh-Gordon model is constructed and shown to generate the Backlund transformations for its soliton solutions.

Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers

In this paper, we explicitly construct an infinite number of Hopfions (static, soliton solutions with nonzero Hopf topological charges) within the recently proposed (3 + 1)-dimensional, integrable, and relativistically invariant field theory. Two integers label the family of Hopfions we have found. Their product is equal to the Hopf charge which provides a lower bound to the soliton's finite energy. The Hopfions are explicitly constructed in terms of the toroidal coordinates and shown to have a form of linked closed vortices.

A new approach to integrable theories in any dimension

The zero curvature representation for two-dimensional integrable models is generalized to spacetimes of dimension d + 1 by the introduction of a d-form connection. The new generalized zero curvature conditions can be used to represent the equations of motion of some relativistic invariant field theories of physical interest in 2 + 1 dimensions (BF theories, Chern-Simons, 2 + 1 gravity and the CP1 model) and 3 + 1 dimensions (self-dual Yang-Mills theory and the Bogomolny equations). Our approach leads to new methods of constructing conserved currents and solutions. In a submodel of the 2 + 1-dimensional CP1 model, we explicitly construct an infinite number of previously unknown non-trivial conserved currents. For each positive integer spin representation of sl(2) we construct 2j + 1 conserved currents leading to 2j + 1 Lorentz scalar charges. (C) 1998 Elsevier B.V. B.V.

T-duality of axial and vector dyonic integrable models

A general construction of affine nonabelian (NA)-Toda models in terms of the axial and vector gauged two loop WZNW model is discussed. They represent integrable perturbations of the conformal sigma -models (with tachyons included) describing (charged) black hole type string backgrounds. We study the off-critical T-duality between certain families of axial and vector type integrable models for the case of affine NA-Toda theories with one global U(1) symmetry. In particular we find the Lie algebraic condition defining a subclass of T-selfdual torsionless NA-Toda models and their zero curvature representation. (C) 2001 Academic Press.

General soliton matrices in the Riemann-Hilbert problem for integrable nonlinear equations

We derive the soliton matrices corresponding to an arbitrary number of higher-order normal zeros for the matrix Riemann-Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order solitons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann-Hilbert problem. We have applied these general results to the three-wave interaction system, and derived new classes of higher-order soliton and two-soliton solutions, in complement to those from our previous publication [Stud. Appl. Math. 110, 297 (2003)], where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to nonelementary zeros generically describe the simultaneous breakup of a pumping wave (u(3)) into the other two components (u(1) and u(2)) and merger of u(1) and u(2) waves into the pumping u(3) wave. The two-soliton solutions corresponding to two simple zeros generically describe the breakup of the pumping u(3) wave into the u(1) and u(2) components, and the reverse process. In the nongeneric cases, these two-soliton solutions could describe the elastic interaction of the u(1) and u(2) waves, thus reproducing previous results obtained by Zakharov and Manakov [Zh. Eksp. Teor. Fiz. 69, 1654 (1975)] and Kaup [Stud. Appl. Math. 55, 9 (1976)]. (C) 2003 American Institute of Physics.

Noncommutative integrable field theories in 2d

We study the noncommutative generalization of (Euclidean) integrable models in two dimensions, specifically the sine- and sinh-Gordon and the U(N) principal chiral models. By looking at tree-level amplitudes for the sinh-Gordon model we show that its naive noncommutative generalization is not integrable. on the other hand, the addition of extra constraints, obtained through the generalization of the zero-curvature method, renders the model integrable. We construct explicit nonlocal nontrivial conserved charges for the U(N) principal chiral model using the Brezin-Itzykson-Zinn-Justin-Zuber method. (C) 2003 Elsevier B.V. All rights reserved.