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Register a new userSingular Miura type initial profiles for the KdV equation
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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.
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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 $t>0$.
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 paper discusses some general aspects and techniques associated with the long-time asymptotics of steplike solutions of the Korteweg-de Vries (KdV) equation via vector Riemann--Hilbert problems. We also elaborate on an ill-posedness of the matrix Riemann-Hilbert problems for the KdV case. To the best of our knowledge this is the first time such ill-posedness is discussed in applications of Riemann--Hilbert theory. Furthermore, we rigorously justify the asymptotics for the shock wave in the elliptic zone derived previously.
We present a multi-parameter family of rational solutions to the complex Korteweg-de Vries(KdV) equations. This family of solutions includes particular cases with high-amplitude peaks at the centre, as well as a multitude of cases in which high-order rogue waves are partially split into lower-order fundamental components. We present an empirically-found symmetry which introduces a parameter controlling the splitting of the rogue wave components into multi-peak solutions, and allows for nonsingular solutions at higher order in certain cases.
In this paper, we derive a B{a}cklund transformation for the supersymmetric Kortweg-de Vries equation. We also construct a nonlinear superposition formula, which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea, Ramani and Grammaticos.
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