We prove that the KdV-Burgers is globally well-posed in H−1(T) with a solution-map that is analytic fromH−1(T) to C([0, T ];H−1(T)) whereas it is ill-posed in Hs(T), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(T) to even D′(T) at any fixed t > 0 small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dis...

We study the relation between the centro-affine geometry of starshaped planar curves and the projective geometry of parametrized maps into RP. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwa...

This paper studies the problem of optimal control of the viscous KdV-Burgers’ equation. We develop a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the viscous KdV-Burgers’ equation. While the viscous KdV-Burgers’ equation is transformed into a simpler linear equation, the performance index is transformed to a complicated rational expression. We show t...

We construct a new variety of N = 2 supersymmetric integrable systems by junction of pseudo-differential superspace Lax operators for a = 4, N = 2 KdV and multi-component N = 2 NLS hierarchies. As an important particular case, we obtain Lax operator for N = 4 super KdV system. A similar extension of one of N = 2 super Boussinesq hierarchies is given. We also present a minimal N = 4 supersymmetr...

The KdV equation arises in the framework of the Boussinesq scaling as a model equation for waves at the surface of an inviscid fluid. Encoded in the KdV model are relations that may be used to reconstruct the velocity field in the fluid below a given surface wave. In this paper, velocity fields associated to exact solutions of the KdV equation are found, and particle trajectories are computed n...

We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether’s ...

The N = 2 supersymmetric α = 1 KdV hierarchy in N = 2 superspace is considered and its rich symmetry structure is uncovered. New nonpolynomial and nonlocal, bosonic and fermionic symmetries and Hamiltonians, bi-Hamiltonian structure as well as a recursion operator connecting all symmetries and Hamiltonian structures of the N = 2 α = 1 KdV hierarchy are constructed in explicit form. It is observ...

Elzaki transform and Adomian polynomial is used to obtain the exact solutions of nonlinear fifth order Korteweg-de Vries (KdV) equations. In order to investigate the effectiveness of the method, three fifth order KdV equations were considered. Adomian polynomial is introduced as an essential tool to linearize all the nonlinear terms in any given equation because Elzaki transform cannot handle n...

First, starting from two hierarchies of autonomous Stäckel ordinary differential equations (ODEs), we reconstruct the hierarchy Korteweg de Vries (KdV) stationary systems. Next, deform considered systems to nonautonomous Painlevé ODEs. Finally, related KdV respective

Some types of coupled Korteweg de-Vries (KdV) equations are derived from an atmospheric dynamical system. In the derivation procedure, an unreasonable yaverage trick (which is usually adopted in literature) is removed. The derived models are classified via Painlevé test. Three types of τ -function solutions and multiple soliton solutions of the models are explicitly given by means of the exact ...