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On the radius of spatial analyticity for the higher order nonlinear dispersive equation

Aissa Boukarou, Kaddour Guerbati, Khaled Zennir (2022)

Mathematica Bohemica

In this work, using bilinear estimates in Bourgain type spaces, we prove the local existence of a solution to a higher order nonlinear dispersive equation on the line for analytic initial data u 0 . The analytic initial data can be extended as holomorphic functions in a strip around the x -axis. By Gevrey approximate conservation law, we prove the existence of the global solutions, which improve earlier results of Z. Zhang, Z. Liu, M. Sun, S. Li, (2019).

On the stability of compressible Navier-Stokes-Korteweg equations

Tong Tang, Hongjun Gao (2014)

Annales Polonici Mathematici

We consider the compressible Navier-Stokes-Korteweg (N-S-K) equations. Through a remarkable identity, we reveal a relationship between the quantum hydrodynamic system and capillary fluids. Using some interesting inequalities from quantum fluids theory, we prove the stability of weak solutions for the N-S-K equations in the periodic domain Ω = N , when N=2,3.

On the Transformations of Symplectic Expansions and the Respective Bäcklund Transformation for the KDV Equation

Khristov, E. (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 34B40; secondary: 35Q51, 35Q53By using the Deift–Trubowitz transformations for adding or removing bound states to the spectrum of the Schrödinger operator on the line we construct a simple algorithm allowing one to reduce the problem of deriving symplectic expansions to its simplest case when the spectrum is purely continuous, and vice versa. We also obtain the transformation formulas for the correponding recursion operator. As an illustration of...

Ondes progressives pour l’équation de Gross-Pitaevskii

Fabrice Béthuel, Philippe Gravejat, Jean-Claude Saut (2007/2008)

Séminaire Équations aux dérivées partielles

Cet exposé présente les résultats de l’article [3] au sujet des ondes progressives pour l’équation de Gross-Pitaevskii : la construction d’une branche d’ondes progressives non constantes d’énergie finie en dimensions deux et trois par un argument variationnel de minimisation sous contraintes, ainsi que la non-existence d’ondes progressives non constantes d’énergie petite en dimension trois.

PDE's for the Dyson, Airy and Sine processes

Mark Adler (2005)

Annales de l’institut Fourier

In 1962, Dyson showed that the spectrum of a n × n random Hermitian matrix, whose entries (real and imaginary) diffuse according to n 2 independent Ornstein-Uhlenbeck processes, evolves as n non-colliding Brownian particles held together by a drift term. When n → ∞ , the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend...

Periodic conservative solutions of the Camassa–Holm equation

Helge Holden, Xavier Raynaud (2008)

Annales de l’institut Fourier

We show that the periodic Camassa–Holm equation u t - u x x t + 3 u u x - 2 u x u x x - u u x x x = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u | t = 0 in H per 1 . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with μ ac = ( u 2 + u x 2 ) d x . The total energy is preserved by the solution.

Proof of the Treves theorem on the KdV hierarchy

Leonid Dickey (2005)

Annales de l’institut Fourier

A new, shorter, proof of the Treves theorem on an algebraic criterion for the first integrals of the KdV hierarchy is given, along with an addition to the theorem.

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