Solution of the Korteweg-de Vries equation on the line with analytic initial potential

Abstract

Articles you may be interested inGeneralized Korteweg-de Vries equation induced from position-dependent effective mass quantum models and mass-deformed soliton solution through inverse scattering transform Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methodsWe present a theory of Sturm-Liouville non-symmetric vessels, realizing an inverse scattering theory for the Sturm-Liouville operator with analytic potentials on the line. This construction is equivalent to the construction of a matrix spectral measure for the Sturm-Liouville operator, defined with an analytic potential on the line. Evolving such vessels we generate Korteweg-de Vries (KdV) vessels, realizing solutions of the KdV equation. As a consequence, we prove the theorem as follows: Suppose that q(x) is an analytic function on R. Then there exists a closed subset ⊆ R 2 and a KdV vessel, defined on . For each x ∈ R one can find TThe potential q(x) is realized by the vessel for t = 0. Since we also show that if q(x, t) is a solution of the KdV equation on R × [0, t 0 ), then there exists a vessel, realizing it, the theory of vessels becomes a universal tool to study this problem. Finally, we notice that the idea of the proof applies to a similar existence of a solution for evolutionary nonlinear Schrödinger and Boussinesq equations, since both of these equations possess vessel constructions. C 2014 AIP Publishing LLC.differentiable potential by Gelfand-Levitan theory, 19 but not always it can be used to solve the KdV equation (1). For this situation in case var[dρ] < ∞ a solution of KdV in the first quadrant (x, t ≥ 0) is presented in Ref. 28.Although there is a good scattering theory of the SL equation (2) on the line (see Ref. 27, Chap. 2, p. 128, and Ref. 26, Chap. 2), even with arbitrary singularities, 13 the solutions of KdV, corresponding to them are not developed. In fact, the classes of initial potentials, for which solutions of (1) were presented using inverse scattering are as follows:Suppose that dρ = dρ + − dρ − for two positive measures dρ + , dρ − , creating two Hilbert spaces of column-functions H + , H − as above. We make the following definition. Definition 2.2. A Krein space Kρ, associated to the finite measure dρ is

Classification of KdV Vessels with Constant Parameters and Two Dimensional Outer Space

“…In other words, the multiplication by S(λ, x, t) serves as a Bäcklund transformations between equations of the wave type: if we carefully examine the system (12) it is easy to see that the first equation is a linear differential equation with constant coefficients and the second one is a wave equation. This fact can be found in [Mel14c,Mela,Mele], or alternatively an interested reader can prove the following, purely computational theorem.…”

“…It is shown in [Mel14c] in a more general setting that the operator X −1 (0, 0)X(x, t) possesses determinant. As a result, its invertability, and hence the invertability of the operator X(x, t) is equivalent to the determinant to be non-zero.…”

“…For example, the famous Korteweg-de Vries (KdV) equation [KdV95,GGKM67], which was first considered by the author in [Melb], was further significantly improved in [Mele] and finally culminated in a scattering theory of analytic parameters in [Mel14c], and KdV hierarchy [Melc]. Second example is a notable Evolutionary Non Linear Schrödinger (ENLS) equation [Kat89], which was inserted into the setting of vessels in [Mel14b].…”

“…The author showed in [Melb,Mel11,Mel14a,Mel14b,Mel14c] that in each particular case these equations are easily solvable. To see it, notice that the the system of equations for B(x, t)…”

“…Finally, we would like to mention some works of the author, originating the theory of vessels [Mel11,Mel14a,Mel14c,Mel14b] and joint works with collaborators [AMV09, AMV12, MV14].…”