Eigenfunctions of a fourth order operator pencil

2014 ◽  
Author(s):  
Abdizhahan Sarsenbi ◽  
Makhmud Sadybekov
Keyword(s):  
Fourth Order ◽  
Operator Pencil ◽  
Order Operator

The Oscillation of Fourth Order Linear Differential Operators

1975 ◽  
Vol 27 (1) ◽  
pp. 138-145 ◽  
Author(s):  
Roger T. Lewis
Keyword(s):  
Fourth Order ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Oscillatory Behavior ◽  
The Self ◽  
Order Operator ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Even Order ◽  
Linear Differential

Define the self-adjoint operatorwhere r(x) > 0 on (0, ∞) and q and p are real-valued. The coefficient q is assumed to be differentiate on (0, ∞) and r is assumed to be twice differentia t e on (0, ∞).The oscillatory behavior of L4 as well as the general even order operator has been considered by Leigh ton and Nehari [5], Glazman [2], Reid [7], Hinton [3], Barrett [1], Hunt and Namb∞diri [4], Schneider [8], and Lewis [6].

scholarly journals The fourth order strongly noncanonical operators

2018 ◽  
Vol 16 (1) ◽  
pp. 1667-1674 ◽  
Author(s):  
Blanka Baculikova ◽  
Jozef Dzurina
Keyword(s):  
Canonical Form ◽  
Fourth Order ◽  
Canonical Representation ◽  
Order Operator

AbstractIt is shown that the strongly noncanonical fourth order operator$$\begin{array}{} \displaystyle \mathcal {L}\,y=\left(r_3(t)\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'\right)' \end{array}$$can be written in essentially unique canonical form as$$\begin{array}{} \displaystyle \mathcal {L}\,y = q_4(t)\left(q_3(t)\left(q_2(t)\left(q_1(t)\left(q_0(t)y(t)\right)'\right)'\right)'\right)'. \end{array}$$The canonical representation essentially simplifies examination of the fourth order strongly noncanonical equations$$\begin{array}{} \displaystyle \left(r_3(t)\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'\right)'+p(t)y(\tau(t))=0. \end{array}$$

Oscillations in the Interactions Among Multiple Solitons in an Optical Fibre

2016 ◽  
Vol 71 (12) ◽  
pp. 1079-1091 ◽  
Author(s):  
Wen-Qiang Hu ◽  
Yi-Tian Gao ◽  
Chen Zhao ◽  
Yu-Jie Feng ◽  
Chuan-Qi Su
Keyword(s):  
Optical Fibre ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Order Operator ◽  
Eighth Order ◽  
Spectral Parameters ◽  
The Third ◽  
Seventh Order

AbstractIn this article, under the investigation on the interactions among multiple solitons for an eighth-order nonlinear Schrödinger equation in an optical fibre, oscillations in the interaction zones are observed theoretically. With different coefficients of the operators in this equation, we find that (1) the oscillations in the solitonic interaction zones have different forms with different spectral parameters of this equation; (2) the oscillations in the interactions among the multiple solitons are affected by the choice of spectral parameters, the dispersive effects and nonlinearity of the eighth-order operator; (3) the second-, fifth-, sixth-, and seventh-order operators restrain oscillations in the solitonic interaction zones and the higher-order operators have stronger attenuated effects than the lower ones, while the third- and fourth-order operators stimulate and extend the scope of oscillations.

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