اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe matrix Lax representation of the generalized Riemann equations and its conservation Laws
197
0
0.0
(
0
)
تأليف
Ziemowit Popowicz
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.
قيم البحث
اقرأ أيضاً
Starting from positive and negative helicity Maxwell equations expressed in Riemann-Silberstein vectors, we derive the ten usual and ten additional Poincar{e} invariants, the latter being related to the electromagnetic spin, i.e., the intrinsic rotat
ion, or state of polarization, of the electromagnetic fields. Some of these invariants have apparently not been discussed in the literature before.
This note is designed to show some classes of differential-difference equations admitting Lax representation which generalize evolutionary equations known in the literature.
We present direct methods and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).The methods are applied to nonlinear PDEs in (1+1) dimensions wit
h polynomial nonlinearities which include the Korteweg-de Vries (KdV), Boussinesq, and Drinfeld-Sokolov-Wilson equations. An adaptation of the methods is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville equations. With respect to nonlinear DDEs, our methods are applied to Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. To overcome the shortcomings of the undetermined coefficients method, we designed a new direct method which uses leading order analysis. That method is applied to discretizations of the KdV and modified KdV equations, and a combination thereof. Additional examples include lattices due to Bogoyavlenskii, Belov-Chaltikian, and Blaszak-Marciniak. The undetermined coefficient methods for PDEs and DDEs have been implemented in Mathematica. The code TransPDEDensityFlux.m computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code DDEDensityFlux.m does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library discrete computes densities and fluxes of nonlinear DDEs.
We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers $kgeq 0$ and $sgeq k+1$. We describe the first integrals for these two classes
in terms of special discrete polynomials. We show an equivalence of two difference equations belonged to different classes corresponding to the same pair $(k, s)$. We show that solution spaces $mathcal{N}^k_s$ of different ordinary difference equations with fixed value of $s+k$ are organized in chain of inclusions.
We study the Veronese web equation $u_y u_{tx}+ lambda u_xu_{ty} - (lambda+1)u_tu_{xy} =0$ and using its isospectral Lax pair construct two infinite series of nonlocal conservation laws. In the infinite differential coverings associated to these seri
es, we describe the Lie algebras of the corresponding nonlocal symmetries. Finally, we construct a recursion operator and explore its action on nonlocal shadows. The operator provides a new shadow which serves as a master-symmetry.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
