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Register a new userSpatial Analyticity of solutions to integrable systems. I. The KdV case
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Alexei Rybkin
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We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles qs which are, in a certain sense, essentially bounded from below and q(x)=O(e^{-cx^{{epsilon}}}),xrightarrow+infty, with some positive c and {epsilon}. Using the inverse scattering transform, we show that the KdV flow turns such initial data into a function which is (1) meromorphic (in the space variable) on the whole complex plane if {epsilon}>1/2, (2) meromorphic on a strip around the real line if {epsilon}=1/2, and (3) Gevrey regular if {epsilon}<1/2. Note that qs need not have any decay or pattern of behavior at -infty.
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In this short note we reconsider the integrable case of the Hamiltonian N-species Volterra system, as it has been introduced by Vito Volterra in 1937. In the first part, we discuss the corresponding conserved quantities, and comment about the solutions of the equations of motion. In the second part we focus our attention on the properties of the simplest model, in particular on period and frequencies of the periodic orbits. The discussion and the results presented here are to be viewed as a complement to a more general work, devoted to the construction of a global stationary state model for a sustainable economy in the Hamiltonian formalism.
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.
begin{abstract} We show that if the initial profile $qleft( xright) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $int^{infty }x^{5/2}leftvert qleft( xright) rightvert dx<infty,$ (no decay at $-infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. end{abstract}
In this paper we present a reduction technique based on bilinearization and double Wronskians (or double Casoratians) to obtain explicit multi-soliton solutions for the integrable space-time shifted nonlocal equations introduced very recently by Ablowitz and Musslimani in [Phys. Lett. A, 2021]. Examples include the space-time shifted nonlocal nonlinear Schrodinger and modified Korteweg-de Vries hierarchies and the semi-discrete nonlinear Schrodinger equation. It is shown that these nonlocal integrable equations with or without space-time shift(s) reduction share same distributions of eigenvalues but the space-time shift(s) brings new constraints to phase terms in solutions.
We study higher order KdV equations from the GL(2,$mathbb{R}$) $cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/elliptic $N$-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find $N$-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.
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