Related papers: On the (Non)-Integrability of KdV Hierarchy with S…
We consider KdV-type equations with $C^1$ nonhomogeneous nonlinearities and small dispersion $\varepsilon$. The first result consists in the conclusion that, in the leading term with respect to $\varepsilon$, the solitary waves in this…
Nonisospectral integrable systems can describe solitary waves in nonuniform media. In this paper, we apply the Cauchy matrix approach to construct three types of nonisospectral matrix modified Korteweg-de Vries (mKdV) eqautions and present…
We study a particular deformation of the potential KdV model (pKdV) and construct the quasi-conservation laws by a direct method. The charge densities, differing from their integrable counterpart with homogeneous degree terms, exhibit mixed…
In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather…
Sardar et al. [Phys. Plasmas 23, 073703 (2016)] have studied the stability of small amplitude dust ion acoustic solitary waves in a collisionless unmagnetized electron - positron - ion - dust plasma. They have derived a Kadomtsev…
Integrable equations with second order Lax pair like KdV and Camassa-Holm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy…
In order to study the invariant measures of discrete KdV- and Toda-type systems, this article focusses on models, discretely indexed in space and time, whose dynamics are deterministic and defined locally via lattice equations. A detailed…
We consider the logarithmic Korteweg-de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV…
Uniform estimates for the decay structure of the $n$-soliton solution of the Korteweg-deVries equation are obtained. The KdV equation, linearized at the $n$-soliton solution is investigated in a class $\WW$ consisting of sums of travelling…
We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as…
Under investigation in this paper is the fractional integrable and non-integrable discrete modified Korteweg-de Vries hierarchies. The linear dispersion relations, completeness relations, inverse scattering transform, and fractional soliton…
We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of shock waves for the Korteweg-de Vries (KdV) equation in the soliton region. In particular, we improve the results…
We study the integrability of mappings obtained as reductions of the discrete Korteweg-de Vries (KdV) equation and of two copies of the discrete potential Korteweg-de Vries equation (pKdV). We show that the mappings corresponding to the…
In this work, we mainly study the general $N$-soliton solutions of the nonlocal modified Korteweg-de Vries (mKdV) equation by utilizing the Riemann-Hilbert (RH) method. For the initial value belonging to Schwarz space, we firstly obtain the…
In this study, we give a survey of derivations of KdV-type equations with an uneven bottom for several cases when small (perturbation) parameters $\alpha, \beta, \delta$ are of different orders. Six different cases of such ordering are…
We give a complete characterisation of the reflectionless Schr\"odinger operators on the line with integrable potentials, solve the inverse scattering problem of reconstructing such potentials from the eigenvalues and norming constants, and…
It is well known that the linear stability of solutions of partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the…
We show that the discrete Kadomtsev-Petviashvili (KP) equation with sources obtained recently by the "source generalization" method can be incorporated into the squared eigenfunction symmetry extension procedure. Moreover, using the known…
Lie symmetry method is applied to investigate symmetries of the combined KdV-nKdV equation, that is a new integrable equation by combining the KdV equation and negative order KdV equation. Symmetries which are obtained in this article, are…
We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation $\partial_t u = -\partial_x (\partial_x^2 u + 3u^2-bu)$, where $b(x,t) = b_0(hx,ht)$, $h\ll 1$ is a slowly varying, but not small, potential. We option an…