Related papers: Enlarged spectral problems and nonintegrability
We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral…
We delve into the inverse scattering transform of the real-valued vector modified Korteweg--de Vries equation, emphasizing the challenges posed by $N$ pairs of higher-order poles in the transmission coefficient and the enhanced spectral…
A new integration scheme, combining the stability and the precision of usual pseudo-spectral codes with the locality of finite differences methods, is introduced. It turns out to be particularly suitable for the study of front and…
The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained.…
A system of semi-discrete coupled nonlinear Schr\"{o}dinger equations is studied. To show the complete integrability of the model with multiple components, we extend the discrete version of the inverse scattering method for the…
We address the question of whether integrable models allow for PT-symmetric deformations which preserve their intgrability. For this purpose we carry out the Painleve test for PT-symmetric deformations of Burgers and the Korteweg-De Vries…
An integrable field theory, due to path-independence on the space-time plane, should yield together with an infinite set of independent conserved charges also similar dual charges determining the boundary and defect contributions. On the…
An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations: hereditary recursion operators, master symmetries,…
Quasi-monochromatic complex reductions of a number of physically important equations are obtained. Starting from the cubic nonlinear Klein-Gordon (NLKG), the Korteweg-deVries (KdV) and water wave equations, it is shown that the leading…
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential…
This is to review some recent progress in PDE. The emphasis is on (energy) supercritical nonlinear Schr\"odinger equations. The methods are applicable to other nonlinear equations.
We have derived the extended Korteweg-de Vries equation describing the long gravity waves without limitation to surface deviation. The only restriction to the surface deviation is connected with the stability condition for appropriate…
A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.
We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV…
Self-induced decoherence formalism and the corresponding classical limit are extended from quantum integrable systems to non-integrable ones.
A linearizable version of multidimensional system of $n$-wave type nonlinear PDEs is proposed. This system is derived using the spectral representation of its solution via the procedure similar to the dressing method for the ISTM-integrable…
We consider the relation between the discrete coupled nonlinear Schr\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\"{o}dinger equation. In the same way…
An application of the Exp-function method to search for exact solutions of nonlinear differential equations is analyzed. Typical mistakes of application of the Exp-function method are demonstrated. We show it is often required to simplify…
In this article we study an integrable deformation of the Kapustin-Witten equations. Using the Weyl-Wigner-Moyal-Groenewold description an integrable $\star$-deformation of a Kapustin-Witten system is obtained. Starting from known solutions…
The Schr\"odinger-like equations for the marginal and conditional probability amplitudes resulting from the exact factorization of the wavefunction of a two-component system are derived in a form that is invariant to gauge and coordinate…