Integrable Nonlinear Evolution Partial Differential Equations in4+2and3+1Dimensions

The solution of the Cauchy problem, an infinite set of conserved quantities, a variational formulation, the auto-Backlund transformations and the soliton phenomenology (featuring singly and doubly humped solitons, as well as breathers) are surveyed briefly for the integrable non-linear evolution partial differential equation (PDE) ut+(uxx-2 eta u3-23uux3/( eta +u2))x=0, eta =+or-1.

Overview of the Kadomtsev–Petviashvili-hierarchy reduction method for solitons

A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE’s) in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: (a) The classical integral transform method for solving linear PDE’s and several of its variations such as the Wiener–Hopf technique. (b) The integral representation of the solution of linear PDE’s in terms of the Ehrenpreis fundamental principle. (c) The inverse spectral (scattering) method for solving the initial value problem for nonlinear integrable evolution equations. The detailed implementation of the method is presented for: (a) An arbitrary linear dispersive evolution equation on the half line. (b) The nonlinear Schrödinger equation on the half line. (c) The Laplace, Helmholtz and modified Helmholtz equations in an arbitrary convex polygon. In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed.

By using the self-similar method for obtaining localized solutions of nonlinear evolution partial differential equations, we found analytical breather solutions to the nonlinear Schrödinger equation with longitudinally variable coefficients and an arbitrary transversely linear potential. The Ma and the second-order breather solutions are derived by choosing the parameters appropriately. We discuss the influence of different parameters on the characteristics of the solutions found. We demonstrate that the parameters not only control the propagation direction of the breather, but also influence its shape and the period.

Using an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, a new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained starting from the Kadomtsev–Petviashvili equation. We apply the reduction technique to the Lax pair of the Kadomtsev–Petviashvili equation and demonstrate the integrability property of the new equation, because we obtain the corresponding Lax pair. The new equation reduces to the Hirota equation in the 1+1-dimensional limit.

A new integrable nonlinear partial differential equation (PDE) in 2+1 dimensions is derived starting from the Konopelchenko–Dubrovsky equation. We use an asymptotically exact reduction method based on Fourier expansion and spatio-temporal rescaling and obtain a new integrable Davey–Stewartson-type equation. In order to demonstrate the integrability of the new equation by the inverse scattering method, we apply the reduction technique to the Lax pair of the Konopelchenko–Dubrovsky equation and find the corresponding Lax pair of the new equation. The new equation reduces to the Davey–Stewartson or the nonlinear Schrodinger equation by appropriate limits.

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, such as the heavenly equation of Plebanski, the dispersionless Kadomtsev–Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one-dimensional vector fields, such as the Pavlov equation. We also showed that the associated Riemann–Hilbert inverse problems are powerful tools to establish if the solutions of the Cauchy problem break at finite time, to construct their long-time behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, (i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; (ii) constructing the long-time behaviour of the solutions of their Cauchy problems; (iii) characterizing a distinguished class of implicit solutions of the heavenly equation.

Exact solutions of a novel integrable partial differential equation

Recently a new method for solving linear and integrable nonlinear partial differential equations (PDEs) in two dimensions has been introduced. For linear PDEs, this method involves the construction of appropriate integral representations for both the solution and the associated spectral function. Here we present an alternative approach for constructing these integral representations. This approach is based on the introduction of what we call a fundamental differential form, and on a reformulation of Green's formula. The new approach is illustrated for the Laplace and the modified Helmholtz equations in a convex polygon, for a dispersive evolution equation with spatial derivatives of arbitrary order on the half-line, and for the heat equation in a domain involving a moving boundary. In addition we show that the new approach can be used to construct appropriate integral representations for multidimensional linear PDEs.

In shallow water theory, completely integrable nonlinear partial differential equations arise at various levels of approximations/simulations. Such equations possess traveling wave solutions, called solitons, which themselves are self-phase-modulated signals. Wavelets are traditionally applied in signal processing but it has limitations to deal effectively with geometric features with line and surface singularities. Due to its poor orientation selectivity, wavelet can only capture limited directional information in signal processing. A multi-resolution geometric analysis, named curvelet transform overcomes the drawbacks of the conventional wavelet transform. Such a geometric characterization provides better understanding of cascade mechanism and dissipation range dynamics. In the present work, a multi-scale geometric analysis with curvelets as basis functions is proposed for analyzing higher dimensional lump-like solitons- that arise in the evolution of lump solutions for the Zakharov-Kuznetsov equation and the surface electromigration equation. Moreover, the application of curvelet thresholding technique for de-noising the contaminated signals is presented.

A novel heuristic technique that enhances the search facility of the standard genetic programming (GP) algorithm is presented. The method provides a dynamic sniffing facility to optimize the local search in the vicinity of the current best chromosomes that emerge during GP iterations. Such a hybrid approach, that combines the GP method with the sniffer technique, is found to be very effective in the solution of inverse problems where one is trying to construct model dynamical equations from either finite time series data or knowledge of an analytic solution function. As illustrative examples, some special function ordinary differential equations (ODEs) and integrable nonlinear partial differential equations (PDEs) are shown to be efficiently and exactly recovered from known solution data. The method can also be used effectively for solution of model equations (the direct problem) and as a tool for generating multiple dynamical systems that share the same solution space.

Boundary Conditions for Infinite Conservation Laws

We investigate two interesting (1+1)-dimensional nonlinear partial differential evolution equations (NLPDEEs), namely the nonlinear dispersion equation with compact structures and the generalized Camassa–Holm (CH) equation describing the propagation of unidirectional shallow water waves on a flat bottom, and arising in the study of a certain non-Newtonian fluid. Using an interesting technique known as the sine-cosine method for investigating travelling wave solutions to NLPDEEs, we construct many new families of wave solutions to the previous NLPDEEs, amongst which the periodic waves, enriching the wide class of solutions to the above equations.

We propose a method for finding the Lax pairs and rational solutions of integrable partial differential equations. That is, when an equation possesses the Painlevé property, a Bäcklund transformation is defined in terms of an expansion about the singular manifold. This Bäcklund transformation obtains (1) a type of modified equation that is formulated in terms of Schwarzian derivatives and (2) a Miura transformation from the modified to the original equation. By linearizing the (Ricati-type) Miura transformation the Lax pair is found. On the other hand, consideration of the (distinct) Bäcklund transformations of the modified equations provides a method for the iterative construction of rational solutions. This also obtains the Lax pairs for the modified equations. In this paper we apply this method to the Kadomtsev–Petviashvili equation and the Hirota–Satsuma equations.

Surfaces in the four-space and the Davey–Stewartson equations