ترغب بنشر مسار تعليمي؟ اضغط هنا

The Linearized Classical Boussinesq System on the Half-Line

262   0   0.0 ( 0 )
 نشر من قبل Dionyssios Mantzavinos
 تاريخ النشر 2020
  مجال البحث فيزياء
والبحث باللغة English




اسأل ChatGPT حول البحث

The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokass method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.



قيم البحث

اقرأ أيضاً

In this work, we revisit the study by M. E. Schonbek [11] concerning the problem of existence of global entropic weak solutions for the classical Boussinesq system, as well as the study of the regularity of these solutions by C. J. Amick [1]. We prop ose to regularize by a fractal operator (i.e. a differential operator defined by a Fourier multiplier of type $epsilon |xi |^lambda, , (epsilon,lambda) in,mathbb{R}_+times ] 0,2]$). We first show that the regularized system is globally unconditionally well-posed in Sobolev spaces of type $H^s(mathbb{R}),,s > frac {1}{2},$, uniformly in the regularizing parameters $(epsilon,lambda) in,mathbb{R}_+times ]0,2]$. As a consequence we obtain the global well-posedness of the classical Boussinesq system at this level of regularity as well as the convergence in the strong topology of the solution of the regularized system towards the solution of the classical Boussinesq equation as the parameter e goes to 0. In a second time, we prove the existence of low regularity entropic solutions of the Boussinesq equations emanating from $u_0 in H^1$ and $zeta_0$ in an Orlicz class as weak limits of regular solutions.
The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics where it describes waves in shallow water. It provides a multidimensional gen eralisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey-Stewartson equation cite{FDS2009}, in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d-bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial-value problems in 2+1 is that the d-bar formalism now involves a function in the complex plane which is discontinuous across the real axis.
A new method for the solution of initial-boundary value problems for textit{linear} and textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 cite{F1997}. This approach was subsequently exte nded to initial-boundary value problems for evolution PDEs in two spatial dimensions, first in the case of linear PDEs cite{F2002b} and, more recently, in the case of integrable nonlinear PDEs, for the Davey-Stewartson and the Kadomtsev-Petviashvili II equations on the half-plane (see cite{FDS2009} and cite{MF2011} respectively). In this work, we study the analogous problem for the Kadomtsev-Petviashvili I equation; in particular, through the simultaneous spectral analysis of the associated Lax pair via a d-bar formalism, we are able to obtain an integral representation for the solution, which involves certain transforms of all the initial and the boundary values, as well as an identity, the so-called global relation, which relates these transforms in appropriate regions of the complex spectral plane.
513 - Enno Lenzmann 2019
We review the current state of results about the half-wave maps equation on the domain $mathbb{R}^d$ with target $mathbb{S}^2$. In particular, we focus on the energy-critical case $d=1$, where we discuss the classification of traveling solitary waves and a Lax pair structure together with its implications (e.,g.~invariance of rational solutions and infinitely many conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also comment on the one-dimensional space-periodic case. Finally, we list some open problem for future research.
77 - Peter C. Gibson 2021
Motivated by applications to acoustic imaging, the present work establishes a framework to analyze scattering for the one-dimensional wave, Helmholtz, Schrodinger and Riccati equations that allows for coefficients which are more singular than can be accommodated by previous theory. In place of the standard scattering matrix or the Weyl-Titchmarsh $m$-function, the analysis centres on a new object, the generalized reflection coefficient, which maps frequency (or the spectral parameter) to automorphisms of the Poincare disk. Purely singul
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا