A Riemann–Hilbert problem for the finite-genus solutions of the KdV equation and its numerical solution

2013 ◽  
Vol 251 ◽  
pp. 1-18 ◽  
Author(s):  
Thomas Trogdon ◽  
Bernard Deconinck
Keyword(s):  
Numerical Solution ◽  
Hilbert Problem ◽  
Kdv Equation ◽  
The Kdv Equation ◽  
Finite Genus ◽  
Riemann Hilbert Problem

scholarly journals Rigorous Asymptotics of a KdV Soliton Gas

2021 ◽  
Author(s):  
M. Girotti ◽  
T. Grava ◽  
R. Jenkins ◽  
K. D. T.-R. McLaughlin
Keyword(s):  
Hilbert Problem ◽  
Kdv Equation ◽  
Broad Class ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Cnoidal Wave ◽  
Large Space ◽  
The Kdv Equation ◽  
Long Time ◽  
Riemann Hilbert Problem

AbstractWe analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann–Hilbert problem which we show arises as the limit $$N\rightarrow + \infty $$ N → + ∞ of a gas of N-solitons. We show that this gas of solitons in the limit $$N\rightarrow \infty $$ N → ∞ is slowly approaching a cnoidal wave solution for $$x \rightarrow - \infty $$ x → - ∞ up to terms of order $$\mathcal {O} (1/x)$$ O ( 1 / x ) , while approaching zero exponentially fast for $$x\rightarrow +\infty $$ x → + ∞ . We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

scholarly journals Periodic finite-genus solutions of the KdV equation are orbitally stable

2010 ◽  
Vol 239 (13) ◽  
pp. 1147-1158 ◽  
Author(s):  
Michael Nivala ◽  
Bernard Deconinck
Keyword(s):  
Kdv Equation ◽  
The Kdv Equation ◽  
Finite Genus

scholarly journals Numerical solution for the KdV equation based on similarity reductions

2009 ◽  
Vol 33 (2) ◽  
pp. 1107-1115 ◽  
Author(s):  
A.A. Soliman ◽  
A.H.A. Ali ◽  
K.R. Raslan
Keyword(s):  
Numerical Solution ◽  
Kdv Equation ◽  
The Kdv Equation ◽  
Similarity Reductions

scholarly journals A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

2019 ◽  
Author(s):  
Kenneth T-R McLaughlin ◽  
Patrik V Nabelek
Keyword(s):  
Vries Equation ◽  
Hilbert Problem ◽  
Kdv Equation ◽  
Unique Determination ◽  
Existence Of A Solution ◽  
Spectral Bands ◽  
Periodic Potentials ◽  
De Vries ◽  
Explicit Time Dependence ◽  
Riemann Hilbert Problem

Abstract We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials as a Riemann–Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann–Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potentials evolve according to the Korteweg–de Vries Equation (KdV) equation, we use integrability to derive an associated Riemann–Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann–Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann–Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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