If f(x, k) is the Jost solution and f(k) = f(0, k), then the I-function is I(k) := f ′(0, k)/f(k). It is proved that I(k) is in one-to-one correspondence with the scattering triple S := {S(k), kj , sj , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = −d/dx + q(x) on (0,∞) with the Dirichlet condition at x = 0 and q(x) ∈ L1,1 := {q | q = q, ∫∞ 0 (1 + x)|q(x)dx ...

In this paper, we study the inverse scattering transform for coupled modified Korteweg-de Vries (cmKdV) equation with nonzero boundary conditions (NZBCs) at infinity. order to deal direct and problems, a suitable uniformization variable is given on complex plane by transforming two-sheeted Riemann surface. problem, discuss analyticity, three symmetries asymptotic behaviors of Jost functions, ma...

We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is rep...

Under certain mode-matching conditions, small-amplitude waves can be trapped by coupling to solitons of nonlinear fields. We present a model for this phenomenon, consisting of a linear equation coupled to the Korteweg–de Vries (KdV) equation. The model has one parameter, a coupling constant κ. For one value of the coupling constant, κ = 1, the linear equation becomes the linearized KdV equation...