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Register a new userThe inverse scattering transform for the KdV equation with step-like singular Miura initial profiles
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We develop the inverse scattering transform for the KdV equation with real singular initial data $q(x)$ of the form $q(x) = r(x) + r(x)^2$, where $rin L^2_{textrm{loc}}$ and $r=0$ on $mathbb R_+$. As a consequence we show that the solution $q(x,t)$ is a meromorphic function with no real poles for any $t>0$.
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We show that the KdV flow evolves any real singular initial profile q of the form q=r+r^2, where rinL_{loc}^2, r|_{R_+}=0 into a meromorphic function with no real poles.
The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include discrete soliton solutions, Backlund transformations and an associated linear problem, called a Lax pair, for which it provides the compatibility condition. In this paper, we solve the initial value problem for the LKdV equation through a discrete implementation of the inverse scattering transform method applied to the Lax pair. The initial value used for the LKdV equation is assumed to be real and decaying to zero as the absolute value of the discrete spatial variable approaches large values. An interesting feature of our approach is the solution of a discrete Gelfand-Levitan equation. Moreover, we provide a complete characterization of reflectionless potentials and show that this leads to the Cauchy matrix form of N-soliton solutions.
This is the second in a series of papers on scattering theory for one-dimensional Schrodinger operators with Miura potentials admitting a Riccati representation of the form $q=u+u^2$ for some $uin L^2(R)$. We consider potentials for which there exist `left and `right Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev--Marchenko potentials in $L^1(R,(1+|x|)dx)$ generating positive Schrodinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients $r$ and justify the algorithm reconstructing $q$ from $r$.
This is the first in a series of papers on scattering theory for one-dimensional Schrodinger operators with highly singular potentials $qin H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive Schrodinger operators that admit a Riccati representation $q=u+u^2$ for a unique $uin L^1(R)cap L^2(R)$. Such potentials have a well-defined reflection coefficient $r(k)$ that satisfies $|r(k)|<1$ and determines $u$ uniquely. We show that the scattering map $S:umapsto r$ is real-analytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.
The inverse scattering transform is extended to investigate the Tzitz{e}ica equation. A set of sectionally analytic eigenfunctions and auxiliary eigenfunctions are introduced. We note that in this procedure, the auxiliary eigenfunctions play an important role. Besides, the symmetries of the analytic eigenfunctions and scattering data are discussed. The asymptotic behaviors of the Jost eigenfunctions are derived systematically. A Riemann-Hilbert problem is constructed to study the inverse scattering problem. Lastly, some novel exact solutions are obtained for reflectionless potentials.
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