Page 694 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 694
POSTER SESSION
Positive periodic solutions for nonlinear delay dynamic equations on time
scales
Kamel Ali Khelil, kamel.alikhelil@univ-annaba.org
University of Annaba, Algeria
Coauthor: Faycal Bouchelaghem
In this work, we use fixed point theorem to study the existence of positive periodic solutions
for delay dynamic equation on time scales. Transforming the equation to an integral equation
enables to show the existence of positive periodic solutions by appealing to Krasnoselskii’s fixed
point theorem. The obtained integral equation is the sum of two mappings; one is a contraction
and the other is compact.
Integration the loaded KdV equation in the class of steplike function
Iroda Baltaeva, iroda-b@mail.ru
Urgench State University, Uzbekistan
Coauthor: Gayrat Urazboev
2010 Mathematics Subject Classification: 39A23, 35Q51, 34K13, 34K29.
Keywords: loaded Korteweg-de Vries equation, inverse scattering problem, the class of steplike
function, scattering data, Lax pair, eigenvalue, eigenfunction.
It is known, that the Korteweg-de Vries equation can be integrated with Inverse Scattering
Method [1]. In the works [2,3], the Korteweg-de Vries equations with a self-consistent source
were integrated for a class of initial data of “step” type; in particular, laws of evolution of the
scattering data were established. In applications of the method of inverse scattering transfor-
mation one looks for pairs of operators B and L such that the equation has some interesting
nonlinear evolution equation for functions u(x, t) that occur as potentials in the operator L. For
the successful application of the method two further ingredients are needed: 1. the inverse scat-
tering problem must be solved so that the potentials u(x, t) can be reconstructed from scattering
data; 2. and that one must be able to determine the evolution of the scattering data witht.
In this paper, we will consider the loaded Korteweg-de Vries equation
ut − 6uux + uxxx + γ(t)u(0, t)ux = 0, (1)
where u = u(x, t), x ∈ R, t ≥ 0, γ(t) - is an arbitrary, continuous function.
The function u = u(x, t) is a sufficiently smooth and tending to its limits steplike (c > 0)
0∞ 3 ∞ ∂ku(x, t)
∂xk
(1 − x)|u(x, t)|dx + (1 + x) u(x, t) − c2 dx + dx < ∞ (2)
−∞ 0 k=1−∞
The equation (1) is considered with initial condition
u|t=0 = u0(x), x ∈ R1, (3)
where u0(x) function satisfies the conditions (c > 0):
0∞
1. (1 − x)|u0(x)|dx < ∞, (1 + x)|u0(x) − c2|dx < ∞,
−∞ 0
692
Positive periodic solutions for nonlinear delay dynamic equations on time
scales
Kamel Ali Khelil, kamel.alikhelil@univ-annaba.org
University of Annaba, Algeria
Coauthor: Faycal Bouchelaghem
In this work, we use fixed point theorem to study the existence of positive periodic solutions
for delay dynamic equation on time scales. Transforming the equation to an integral equation
enables to show the existence of positive periodic solutions by appealing to Krasnoselskii’s fixed
point theorem. The obtained integral equation is the sum of two mappings; one is a contraction
and the other is compact.
Integration the loaded KdV equation in the class of steplike function
Iroda Baltaeva, iroda-b@mail.ru
Urgench State University, Uzbekistan
Coauthor: Gayrat Urazboev
2010 Mathematics Subject Classification: 39A23, 35Q51, 34K13, 34K29.
Keywords: loaded Korteweg-de Vries equation, inverse scattering problem, the class of steplike
function, scattering data, Lax pair, eigenvalue, eigenfunction.
It is known, that the Korteweg-de Vries equation can be integrated with Inverse Scattering
Method [1]. In the works [2,3], the Korteweg-de Vries equations with a self-consistent source
were integrated for a class of initial data of “step” type; in particular, laws of evolution of the
scattering data were established. In applications of the method of inverse scattering transfor-
mation one looks for pairs of operators B and L such that the equation has some interesting
nonlinear evolution equation for functions u(x, t) that occur as potentials in the operator L. For
the successful application of the method two further ingredients are needed: 1. the inverse scat-
tering problem must be solved so that the potentials u(x, t) can be reconstructed from scattering
data; 2. and that one must be able to determine the evolution of the scattering data witht.
In this paper, we will consider the loaded Korteweg-de Vries equation
ut − 6uux + uxxx + γ(t)u(0, t)ux = 0, (1)
where u = u(x, t), x ∈ R, t ≥ 0, γ(t) - is an arbitrary, continuous function.
The function u = u(x, t) is a sufficiently smooth and tending to its limits steplike (c > 0)
0∞ 3 ∞ ∂ku(x, t)
∂xk
(1 − x)|u(x, t)|dx + (1 + x) u(x, t) − c2 dx + dx < ∞ (2)
−∞ 0 k=1−∞
The equation (1) is considered with initial condition
u|t=0 = u0(x), x ∈ R1, (3)
where u0(x) function satisfies the conditions (c > 0):
0∞
1. (1 − x)|u0(x)|dx < ∞, (1 + x)|u0(x) − c2|dx < ∞,
−∞ 0
692
