Search results for "Riemann–Hilbert problem"

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Large-x Analysis of an Operator-Valued Riemann–Hilbert Problem

2015

International audience; The purpose of this paper is to push forward the theory of operator-valued Riemann-Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method a la Deift-Zhou. In this paper, we demonstrate that the operator-valued Riemann-Hilbert problem arising in the characterization of so-called c-shifted integrable integral operators allows one to extract the large-x asymptotics of the Fredholm determinant associated with such operators.

Pure mathematicsIntegrable systemNonlinear schrodinger-equationMathematics::Complex VariablesGeneral Mathematics010102 general mathematicsMathematicsofComputing_NUMERICALANALYSIS[ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph]Fredholm determinantImpenetrable bose-gas[ MATH.MATH-FA ] Mathematics [math]/Functional Analysis [math.FA][MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]01 natural sciencessymbols.namesakeRiemann hypothesisOperator (computer programming)[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]0103 physical sciencesHilbert's problemssymbolsMethod of steepest descentRiemann–Hilbert problem010307 mathematical physics0101 mathematicsMathematics
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Multiparametric Rational Solutions of Order N to the KPI Equation and the Explicit Case of Order 3

2021

We present multiparametric rational solutions to the Kadomtsev-Petviashvili equation (KPI). These solutions of order N depend on 2N − 2 real parameters. Explicit expressions of the solutions at order 3 are given. They can be expressed as a quotient of a polynomial of degree 2N(N +1)−2 in x, y and t by a polynomial of degree 2N(N +1) in x, y and t, depending on 2N − 2 real parameters. We study the patterns of their modulus in the (x,y) plane for different values of time t and parameters.

symbols.namesakesymbolsOrder (group theory)Applied mathematicsRiemann–Hilbert problemGeneral MedicineKadomtsev–Petviashvili equationMathematicsArchives of Current Research International
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