Sharp bounds on enstrophy growth in the viscous Burgers equation

We use the Cole--Hopf transformation and the Laplace method for the heat equation to justify the numerical results on enstrophy growth in the viscous Burgers equation on the unit circle. We show that the maximum enstrophy achieved in the time evolution is scaled as $\mathcal{E}^{3/2}$, where $\mathcal{E}$ is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as $\mathcal{E}^{-1/2}$. These bounds are sharp for sufficiently smooth initial conditions.Comment: 12 page

Collisions of solitons and vortex rings in cylindrical Bose-Einstein condensates

Interactions of solitary waves in a cylindrically confined Bose-Einstein condensate are investigated by simulating their head-on collisions. Slow vortex rings and fast solitons are found to collide elastically contrary to the situation in the three-dimensional homogeneous Bose gas. Strongly inelastic collisions are absent for low density condensates but occur at higher densities for intermediate velocities. The scattering behaviour is rationalised by use of dispersion diagrams. During inelastic collisions, spherical shell-like structures of low density are formed and they eventually decay into depletion droplets with solitary wave features. The relation to similar shells observed in a recent experiment [Ginsberg et al. Phys Rev. Lett. 94, 040403 (2005)] is discussed

Ablowitz-Ladik system with discrete potential. I. Extended resolvent

Ablowitz-Ladik linear system with range of potential equal to {0,1} is considered. The extended resolvent operator of this system is constructed and the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy

A Bi-Hamiltonian Structure for the Integrable, Discrete Non-Linear Schrodinger System

This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect to both a standard, local Poisson operator J and a new non-local, skew, almost Poisson operator K, on the appropriate space; (b) can be recursively generated from a recursion operator R (obtained by composing K and the inverse of J.) In addition, the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi-Hamiltonian structures for other families of discrete, integrable equations.Comment: 33 page

On integrability of the differential constraints arising from the singularity analysis

Integrability of the differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordinary differential equations are obtained in this way, which are integrable by quadratures in spite of very complicated branching of their solutions.Comment: arxiv version is already offcia

Yang-Baxter and reflection maps from vector solitons with a boundary

Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schr\"odinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on the half-line. Using dressing transformations combined with a mirror image technique, factorization of soliton-soliton and soliton-boundary interactions is proved. We discover a new object, which we call reflection map, that satisfies a set-theoretical reflection equation which we also introduce. Two classes of solutions for the reflection map are constructed. Finally, basic aspects of the theory of set-theoretical reflection equations are introduced.Comment: 29 pages. Featured article in Nonlinearit

Functional representation of the Ablowitz-Ladik hierarchy. II

In this paper I continue studies of the functional representation of the Ablowitz-Ladik hierarchy (ALH). Using formal series solutions of the zero-curvature condition I rederive the functional equations for the tau-functions of the ALH and obtain some new equations which provide more straightforward description of the ALH and which were absent in the previous paper. These results are used to establish relations between the ALH and the discrete-time nonlinear Schrodinger equations, to deduce the superposition formulae (Fay's identities) for the tau-functions of the hierarchy and to obtain some new results related to the Lax representation of the ALH and its conservation laws. Using the previously found connections between the ALH and other integrable systems I derive functional equations which are equivalent to the AKNS, derivative nonlinear Schrodinger and Davey-Stewartson hierarchies.Comment: arxiv version is already officia

A note on the integrable discretization of the nonlinear Schr\"odinger equation

We revisit integrable discretizations for the nonlinear Schr\"odinger equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the non-locality, can be overcome. Namely, we factorize the non-local difference scheme into the product of local ones. This must improve the performance of the scheme in the numerical computations dramatically. Using the equivalence of the Ablowitz--Ladik and the relativistic Toda hierarchies, we find the interpolating Hamiltonians for the local schemes and show how to solve them in terms of matrix factorizations.Comment: 24 pages, LaTeX, revised and extended versio

Integrable semi-discretization of the coupled nonlinear Schr\"{o}dinger equations

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 single-component discrete nonlinear Schr\"{o}dinger equation proposed by Ablowitz and Ladik. By means of the extension, the initial-value problem of the model is solved. Further, the integrals of motion and the soliton solutions are constructed within the framework of the extension of the inverse scattering method.Comment: 27 pages, LaTeX2e (IOP style