Certain explicit solutions to the Korteweg-de Vries equation in the first quadrant of the xt-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schrödinger equation. In the reflectionless case such solutions reduce to pure N-soliton solutions. An illustrative example is provided.

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable x and the temporal variable t, and they are exponentially asymptotic to integer multiples of 2π as x → ±∞. The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation using a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation. C

A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values q ± ≡ q 0 e iθ ± as x → ±∞ is presented. The direct problem is shown to be well posed for potentials q such that q − q ± ∈ L 1,2 (R ± ), for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann-Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrödinger (NLS) equation with non-zero boundary values ql/r(t)=Al/re2iAl/r2t+il/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl θ r. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that (q(x,t)-q_{l/r}(t) L1,1(R±) with respect to x for all t ≥ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables λ _{l/r}=\sqrt{k2+A2 t/r where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations

The inverse scattering transform (IST) with non-zero boundary conditions at infinity is presented for a matrix nonlinear Schrödinger-type equation which has been proposed as a model to describe hyperfine spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and antiferromagnetic spin-exchange interactions (self-defocusing case), or attractive interatomic interactions and ferromagnetic spin-exchange interactions (self-focusing case). Both the direct and the inverse problems are formulated in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the one-soliton solutions is analyzed in details in the self-focusing case, including some special cases not previously discussed in the literature.