Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The good equation initial condition is investigated Riemann–Hilbert method. Starting three-order pair of this equation, inverse scattering transform formulated and problem established, properties reflection coefficients presented. Then, formulas long-time asymptotics to its versio...

We are concerned with the long-time asymptotic behavior of solution for focusing Hirota equation (also called third-order nonlinear Schrödinger equation) symmetric, non-zero boundary conditions (NZBCs) at infinity. Firstly, based on Lax pair NZBCs, direct and inverse scattering problems used to establish oscillatory Riemann-Hilbert (RH) problem distinct jump curves. Secondly, Deift-Zhou steepes...

Liftings of endofunctors on sets to relations are commonly used capture bisimulation coalgebras. Lax versions have been in those cases where strict lifting fails bisimilarity, as well modeling other notions simulation. This paper provides tools for defining and manipulating lax liftings. As a central result, we define notion distributive law functor over the powerset monad, show that there is a...

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

We reconstruct a rational Lax matrix of size R+ 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy–like kernel (a bi-differential of bi-weight (1−ν, ν)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights ν, 1−ν respectively. All objects are described in...

We consider some natural connections which arise between right-flat (p, q) para-conformal structures and integrable systems. We find that such systems may be formulated in Lax form, with a " Lax p-tuple " of linear differential operators, depending a spectral parameter which lives in (q − 1)-dimensional complex projective space. Generally, the differential operators contain partial derivatives ...

Four explicit finite difference schemes, including Lax-Friedrichs, Nessyahu-Tadmor, Lax-Wendroff and Lax-Wendroff with a nonlinear filter are applied to solve water hammer equations. The schemes solve the equations in a reservoir-pipe-valve with an instantaneous and gradual closure of the valve boundary. The computational results are compared with those of the method of characteristics (MOC), a...

In this thesis I will describe an explicit method for performing an 8-descent on elliptic curves. First I will present some basics on descent, in particular I will give a generalization of the definition of n-coverings, which suits the needs of higher descent. Then I will sketch the classical method of 2-descent, and the two methods that are known for doing a second 2-descent, also called 4-des...

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E.Cartan. Especially, the E.Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and CartanEhresmann connection theory on fibered spaces. General structu...

We have introduced, in a previous paper, the fundamental lax 2-category of a ‘directed space’ X. Here we show that, when X has a T1-topology, this structure can be embedded into a larger one, with the same objects (the points of X), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells a...