Related papers: Negative order KdV equation with both solitons and…
The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include…
An integrable dispersionless KdV hierarchy with sources (dKdVHWS) is derived. Lax pair equations and bi-Hamiltonian formulation for dKdVHWS are formulated. Hodograph solution for the dispersionless KdV equation with sources (dKdVWS) is…
The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are determined. A consequence of this study is the existence of classes of smooth, complex--valued…
We consider the following hypothesis: some of KdV equation shock-like waves are invariant with respect to the combination of the Galilean symmetry and KdV equation higher symmetries. Also we demonstrate our approach on the example of…
We establish the asymptotic stability of smooth solitons and multi-solitons for the Camassa-Holm (CH) equation in the energy space $H^1(\R)$. We show that solutions initially close to a soliton converge, up to translation, weakly in…
In this paper, we consider a second order nonlinear ordinary differential equation of the form $\ddot{x}+k_1\frac{\dot{x}^2}{x}+(k_2+k_3x)\dot{x}+k_4x^3+k_5x^2+k_6x=0$, where $k_i$'s, $i=1,2,...,6,$ are arbitrary parameters. By using the…
The construction of a solution of the perturbed KdV equation encounters obstacles to asymptotic integrability beyond the first order, when the zero-order approximation is a multiple-soliton wave. In the standard analysis, the obstacles lead…
The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for…
A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale…
We study stability of conservative solutions of the Cauchy problem for the periodic Camassa-Holm equation $u_t-u_{xxt}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$…
A non-isospectral Lax pair is first introduced from which a kind of non-isospectral integrable TD hierarchy is derived, whose reduction is an integrable system called the non-isospectral integrable TD system. Then by using the inverse…
In this paper we utilize a convex integration scheme to construct non-trivial solutions to the stationary KdV equation which lie in $L^p(\mathbb{T})$, $p < 2$. In addition, we demonstrate this result is sharp in the sense that if $u \in…
This paper is contributed to study the Cauchy problem of a new integrable two-component system with peaked soliton (peakon) and weak kink solutions. We first establish the local well-posedness result for the Cauchy problem in Besov spaces,…
The Hamiltonian-Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem $ J L u=\lambda u$, where $J$ is skew-symmetric and $L$ is self-adjoint. If $J$…
Fifth order, quasi-linear, non-constant separant evolution equations are of the form u_t=A\frac{\partial^5 u}{\partial x^5}+\tilde{B}, where A and \tilde{B} are functions of x, t, u and of the derivatives of u with respect to x up to order…
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…
In the multiple-soliton case, the freedom in the expansion of the solution of the perturbed KdV equation is exploited so as to transform the equation into a system of two equations: The (inte-grable) Normal Form for KdV-type solitons, which…
The inverse problem which arises in the Camassa--Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the…
Novel soliton structures are constructed for the Fokas-Lenells equation. In so doing, and after discussing the stability of continuous waves, a multiple scales perturbation theory is used to reduce the equation to a Korteweg-de Vries system…
An alternative way for the derivation of the new KdV-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It…