Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0807.5041

Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent

Katrin Grunert·Gerald Teschl
View original source ↗Read source document ↗Export metadata ↓

Abstract

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.

Explore related subjects

nlin.SImath-phmath.APmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Katrin Grunert, Gerald Teschl. 2009-07-13. Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent. https://doi.org/10.1007/s11040-009-9062-2

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the autonomisation of differential equations I: Sundman transformations for Painlevé equations I-IV

We consider the application of Sundman transformations to the first four Painlevé equations $P_I$-$P_{IV}$, with $P_V$ and $P_{VI}$ being left for future work. Since the Painlevé equations are nonautonomous, this requires first of all the construction of an equivalent autonomous system, a process which we discuss in detail, and which we characterize as being inverse to the use of Lie symmetries to obtain a reduction of order of an autonomous system. Secondly, we apply a Sundman transformation to this autonomous system, and identify the resulting equation with a corresponding nonautonomous equation. We give an explicit mapping between solutions of this nonautonomous equation and the original Painlevé equation. Interesting special cases of the nonautonomous equation found by Sundman transformation include, for $P_{II}$, an equation with coefficients expressed via Airy functions, and for $P_{IV}$, an equation with coefficients expressed via Weber-Hermite functions. We also discuss an alternative autonomisation process. Finally, a variety of possible directions for future work are identified, along with interesting open problems.

nlin.SI↗
arXiv2026

Integral Representations for Solutions of the Linearized Fourth Painleve Equation

We formulate the linearization of the fourth Painlevé equation as a linearized Hamiltonian system and transform the scalar variational equation to normal form, with the first-derivative term eliminated. From the canonical fundamental solutions of the Jimbo--Miwa Lax pair, we construct quadratic expressions $Q_{IV}$ and $R_{IV}$ that satisfy the identity $\partial_x^2Q_{IV}-U_{IV}Q_{IV}=\partial_λR_{IV}$. Here $U_{IV}$ is the coefficient in the normal-form equation. A rapid-decay cycle with a vanishing boundary term yields an integral solution of the linearized equation. Every nontrivial such cycle has asymptotic branches in at least two distinct Stokes sectors; otherwise the integral vanishes by Cauchy's theorem. A second rapid-decay homology class gives another integral solution. The Wronskian of the two integral solutions depends holomorphically on the monodromy data, so a single nonzero value implies generic linear independence. For one set of complex data, direct quadrature gives numerical evidence of a nonzero Wronskian. We also derive integral formulas for variations of the Stokes multipliers, the connection matrix, and the local monodromy, and verify the contour representation by substitution into the linearized equation.

nlin.SI↗
arXiv2026

Matrix Dressing Beyond Pfaff-Toda

We develop a matrix pseudodifference operator framework that places theAdler-Pfaff and Pfaff-Toda hierarchies in a common algebraic setting. The bi-infinite Adler-Pfaff hierarchy then becomes a $2\times2$ matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra $M_{2N}\bigl(\mathbb{C}[\mathcal{S},\mathcal{S}^{-1}]\bigr)$ using a Pfaff-Toda splitting of the matrix pseudodifference algebra. In the one-component case, its diagonal sector recovers the continuous Pfaff-Toda hierarchy, while the off-diagonal directions supply the missing odd flows, and the powers of a single bare operator reconstruct the full Adler-Pfaff hierarchy. For general $N$, on the regular factorization locus, the multicomponent Pfaff-Toda tau-functions of Savchenko and Zabrodin realize the commuting diagonal sector, with its dressing equations derived directly from the fermionic bilinear identity. Reductions to the block-diagonal and scalar cases recover multicomponent and scalar $2D$ Toda, identifying the even Pfaff hierarchy with an anti-diagonal Toda subhierarchy, and yielding the Krichever-Zabrodin C-Toda hierarchy.

nlin.SI↗