Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNew Two-Componet Coupled KdV Equation and its Connection with the Generalized Harry Dym Equation
99
0
0.0
(
0
)
Authors
Ziemowit Popowicz
Ask ChatGPT about the research
No Arabic abstract
It is shown that, two different Lax operators in the Dym hierarchy, produce two generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component KdV system. The first equation gives us new integrable two-component KdV system while the second reduces to the known symmetrical two-component KdV equation. For this new two-component coupled KdV system the Lax representation and Hamiltonian structure is defined.
rate research
Read More
It is shown that, three different Lax operators in the Dym hierarchy, produce three generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component KdV system. The first equation gives us known integrable two-component KdV system while the second reduces to the known symmetrical two-component KdV equation. The last one reduces to the Drienfeld-Sokolov equation. This approach gives us new Lax representation for these equations.
The soliton resolution for the Harry Dym equation is established for initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})$. Combining the nonlinear steepest descent method and $bar{partial}$-derivatives condition, we obtain that when $frac{y}{t}<-epsilon(epsilon>0)$ the long time asymptotic expansion of the solution $q(x,t)$ in any fixed cone begin{equation} Cleft(y_{1}, y_{2}, v_{1}, v_{2}right)=left{(y, t) in R^{2} mid y=y_{0}+v t, y_{0} inleft[y_{1}, y_{2}right], v inleft[v_{1}, v_{2}right]right} end{equation} up to an residual error of order $mathcal{O}(t^{-1})$. The expansion shows the long time asymptotic behavior can be described as an $N(I)$-soliton on discrete spectrum whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone and the second term coming from soliton-radiation interactionson on continuous spectrum.
It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock type singularities are presented.
We present a multi-parameter family of rational solutions to the complex Korteweg-de Vries(KdV) equations. This family of solutions includes particular cases with high-amplitude peaks at the centre, as well as a multitude of cases in which high-order rogue waves are partially split into lower-order fundamental components. We present an empirically-found symmetry which introduces a parameter controlling the splitting of the rogue wave components into multi-peak solutions, and allows for nonsingular solutions at higher order in certain cases.
In this paper, we study the generalized Heisenberg ferromagnet equation, namely, the M-CVI equation. This equation is integrable. The integrable motion of the space curves induced by the M-CVI equation is presented. Using this result, the Lakshmanan (geometrical) equivalence between the M-CVI equation and the two-component Camassa-Holm equation is established. Note that these equations are gauge equivalent each to other.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
