Hierarchy of negative order equation and its Lax pair

1995 ◽  
Vol 36 (8) ◽  
pp. 4220-4225 ◽  
Author(s):  
Ruguang Zhou
2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.


2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Fengfeng Dong ◽  
Lingjun Zhou

The negative order Camassa-Holm (CH) hierarchy consists of nonlinear evolution equations associated with the CH spectral problem. In this paper, we show that all the negative order CH equations admit peakon solutions; the Lax pair of the N-order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders -3 and -4 are obtained. Indeed, for N≤-2, the peakons of N-order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for N-order CH equation when N is odd are much different from the CH peakons; we present the case N=-3 as an example.


Author(s):  
V. V. Makarov ◽  
D. A. Lozovoy

  Enzootic bovine leucosis (EBL) has been known for more than a century and a half. Its occurrence and registration may have historically been associated with intensive breeding of dairy cattle in Western Europe to increase target productivity. It is known that any limiting intervention in the nature of the animal organism is always accompanied by an uncontrolled and unpredictable change in the genotype of a wider range than the required, particularly negative order. In particular, a decrease in the resistance to macroorganisms and the possibility of the new diseases emergence, including infectious ones (for example, immunodeficiencies such as BLAD syndrome of black-motley cattle and stress syndrome in pigs, the occurrence of scrapie and other slow sheep infections). In the last two decades of the last century, in many disadvantaged countries, primarily Western European, national programs for the eradication of EBL have been developed and subsequently successfully implemented. First of all the motivation was the economy of dairy cattle breeding (mainly the extension of productive age, as well as the tightening of requirements in international trade in cattle and bull products, breeding, pricing, etc.). In an analytical article are reviewed the elements of epizootology of EBL in the foreign countries with special attention to the situation in the USA, scenarios of various control programs, and promising methods for assessing the role of infected animals in the epizootic process. A critical assessment of the problem of EBL in the Russian Federation is given, the reasons for the ineffectiveness of against leucosis measures are discussed.


Author(s):  
S. G. Rajeev

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.


Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 224
Author(s):  
Ghaylen Laouini ◽  
Amr M. Amin ◽  
Mohamed Moustafa

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.


1994 ◽  
Vol 09 (12) ◽  
pp. 2103-2115 ◽  
Author(s):  
D.G. BARCI ◽  
L.E. OXMAN

We consider a fermionic field obeying a second order equation containing a pair of complex conjugate mass parameters. After obtaining a natural representation for the different degrees of freedom, we are able to construct a unique vacuum as the more symmetric state (zero energy-momentum, charge and spin). This representation, unlike the real mass case, is not holomorphic in the Grassmann variables. The vacuum eigenstate allows the calculation of the field propagator which turns out to be half advanced plus half retarded.


2017 ◽  
Vol 102 (1-2) ◽  
pp. 3-11 ◽  
Author(s):  
A. I. Aristov
Keyword(s):  

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