Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

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Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

We review developments, issues and challenges in electrical impedance tomography (EIT) for the 4th Conference on Biomedical Applications of Electrical Impedance Tomography, held at Manchester in 2003. We focus on the necessity for three-dimensional data collection and reconstruction, efficient solution of the forward problem, and both present and future reconstruction algorithms. We also suggest common pitfalls or 'inverse crimes' to avoid.

https://iopscience.iop.org/article/10.1088/0967-3334/25/1/021/pdf

EIT reconstruction algorithms: pitfalls, challenges and recent developments

By means of a special variable separation approach, a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various (2+1)-dimensional models such as the Davey-Stewartson (DS) model, the Nizhnik-Novikov-Veselov (NNV) system, asymmetric NNV equation, asymmetric DS equation, dispersive long wave equation, Broer-Kaup-Kupershmidt system, long wave-short wave interaction model, Maccari system, and a general (N+M)-component Ablowitz-Kaup-Newell-Segur (AKNS) system. Selecting the arbitrary functions appropriately, one may obtain abundant stable localized interesting excitations such as the multidromions, lumps, ring soliton solutions, breathers, instantons, etc. It is shown that some types of lower dimensional chaotic patterns such as the chaotic-chaotic patterns, periodic-chaotic patterns, chaotic line soliton patterns, chaotic dromion patterns, fractal lump patterns, and fractal dromion patterns may be found in higher dimensional soliton systems. The interactions between the traveling ring type soliton solutions are completely elastic. The traveling ring solitons pass through each other and preserve their shapes, velocities, and phases. Some types of localized weak solutions, peakons, are also discussed. Especially, the interactions between two peakons are not completely elastic. After the interactions, the traveling peakons also pass through each other and preserve their velocities and phases, however, they completely exchange their shapes.

Localized excitations in (2+1)-dimensional systems.

This review is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters or buried objects such as cracks. These inverse problems are considered mainly for three-dimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e., fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.

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Inverse problems in elasticity

Scattering of localized solitons in the plane

A generalisation in 2+1 dimensions of the Korteweg-de Vries equation is related to the spectral problem ( delta x2- delta y2-p(x,y)) phi (x,y;k)=0. It can contain arbitrary functions of x+y or x-y and time. The Cauchy problem, associated with initial data decaying sufficiently rapidly at infinity, is linearised by an extension of the spectral transform technique to two spatial dimensions. The spectral data are explicitly defined in terms of the initial data and the inverse problem is formulated as a non-local Riemann-Hilbert boundary-value problem. The presence of arbitrary functions of x+y and x-y in the evolution equation implies that the time evolution of the spectral data is linear but non-local. Discrete spectral data are forbidden and, consequently, localised soliton solutions are not allowed.

https://iopscience.iop.org/article/10.1088/0266-5611/2/3/005/pdf

On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions

Nonlinear Evolution Equations and Dynamical Systems

A two-dimensional nonlinear evolution equation is solved in the inverse spectral transform scheme. It coincides, when reduced to one spatial dimension, with the dispersive long wave equation. The Backlund transformation, soliton solution and superposition formula are obtained. The spectral transform is explicitly defined and the corresponding linear evolution of the spectral data is given. The inverse spectral problem is formulated as a non-local Riemann-Hilbert boundary value problem and solved.

https://iopscience.iop.org/article/10.1088/0266-5611/3/3/007/pdf

Spectral transform for a two spatial dimension extension of the dispersive long wave equation

A two-dimensional generalisation of the sine- and the sinh-Gordon equations, which one refers to as the shine-Gordon equations, is obtained and solved through the inverse spectral transform (IST) method. The Backlund transformation and nonlinear superposition formula are constructed and explicit wave solitons are given. It is shown also that a slightly different procedure furnishes an IST-solvable extension in 2+1 dimensions of the dispersive long-wave equation.

Marco Boiti

Papers

Scattering of localized solitons in the plane

On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions

Nonlinear Evolution Equations and Dynamical Systems

Spectral transform for a two spatial dimension extension of the dispersive long wave equation

Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations