The inverse scattering transform for the focusing nonlinear Schrödinger equation is
presented for a general class of initial conditions whose asymptotic behavior at infinity …

In this paper we formulate the nonlocal dbar problem dressing method of Manakov and
Zakharov (Zakharov and Manakov, 1984, 1985; Zakharov, 1989) for the 4 scaling classes of …

We address two pressing questions in the theory of the Korteweg–de Vries (KdV) equation.
First, we show the uniqueness of solutions to KdV that are merely bounded, without any …

The concept of a primitive potential for the Schrödinger operator on the line was introduced
in Dyachenko et al.(2016, Phys. D, 333, 148–156), Zakharov, Dyachenko et al.(2016, Lett …

We show that the Cauchy problem for the KdV equation can be solved by the inverse
scattering transform (IST) for any initial data bounded from below, decaying sufficiently …

In this paper we show that all algebro-geometric finite gap solutions to the Korteweg–de
Vries equation can be realized as a limit of N-soliton solutions as N diverges to infinity (see …

In this paper, high-order compact-difference schemes involving a large number of mesh
points in the computational stencils are used to numerically solve partial differential …

In this short note, we discuss a new formula for solving the nonlocal ∂∂¯-problem, and
discuss application to the Manakov–Zakharov dressing method. We then explicitly apply this …

We survey recent results connected with constructing a new family of solutions of the
Korteweg-de Vries equation, which we call primitive solutions. These solutions are …

Recently, we introduced a new class of bounded potentials of the one-dimensional
stationary Schrödinger operator on the real axis, and a corresponding family of solutions of …