=Paper=
{{Paper
|id=Vol-1987/paper13
|storemode=property
|title=Traveling Waves and Functional Differential Equations of Pointwise Type. What Is Common?
|pdfUrl=https://ceur-ws.org/Vol-1987/paper13.pdf
|volume=Vol-1987
|authors=Levon A. Beklaryan,Armen L. Beklaryan
}}
==Traveling Waves and Functional Differential Equations of Pointwise Type. What Is Common?==
https://ceur-ws.org/Vol-1987/paper13.pdf
Traveling Waves and Functional Differential Equations of
Pointwise Type. What Is Common?
Levon A. Beklaryan
Central Economics and Mathematics Institute RAS
Nachimovky prospect 47,
117418 Moscow, Russia.
beklar@cemi.rssi.ru, lbeklaryan@outlook.com
Armen L. Beklaryan
National Research University Higher School of Economics
Kirpichnaya Ulitsa 33/5,
105187 Moscow, Russia.
abeklaryan@hse.ru
Abstract
For equations of mathematical physics, which are the Euler-Lagrange
equation of the corresponding variational problem, an important class
of solutions are traveling wave solutions (soliton solutions). In turn,
soliton solutions for finite-difference analogs of the equations of math-
ematical physics are in one-to-one correspondence with solutions of
induced functional differential equations of pointwise type (FDEPT).
The presence of a wide range of numerical methods for constructing
FDEPT solutions, as well as the existence of appropriate existence
and uniqueness theorems for the solution, a continuous dependence
on the initial and boundary conditions, the “rudeness” of such equa-
tions, allows us to construct soliton solutions for the initial equations
of mathematical physics. Within the framework of the presented work,
on the example of a problem from the theory of plastic deformation
the mentioned correspondence between solutions of the traveling wave
type and the solutions of the induced functional differential equation
will be demonstrated.
1 Introduction
In the theory of plastic deformation, the following infinite-dimensional dynamical system is studied
mÿi = yi+1 − 2yi + yi−1 + ϕ(yi ), i ∈ Z, yi ∈ R, t ∈ R, (1)
where potential ϕ(·) is given by a smooth periodic function. The equation (1) is a system with the Frenkel-
Kontorova potential [Frenkel, 1938]. Such a system is a finite difference analog of the nonlinear wave equation.
Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of
the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org
81
It simulates the behavior of a countable number of balls of mass m placed at integer points of the numerical line,
where each pair of adjacent balls is connected by an elastic spring, and describes the propagation of longitudinal
waves in an infinite homogeneous absolutely elastic rod. The study of such systems with different potentials is
one of the intensively developing directions in the theory of dynamical systems. For these systems, the central
task is to study solutions of the traveling wave type as one of the observed wave classes.
Definition 1 We say that the solution {yi (·)}+∞ −∞ of the system (1), defined for all t ∈ R, has a traveling wave
type, if there is τ > 0 , independent of t and i, that for all i ∈ Z and t ∈ R the following equality holds
yi (t + τ ) = yi+1 (t).
The constant τ will be called a characteristic of a traveling wave.
One of the methods for studying such systems is the construction of solutions using the explicit form of the
potential and further, by the methods of perturbation theory, the establishment of the existence of solutions of
the traveling wave type for nearby potentials. Another frequently used method is the applying of the presence
of symmetries in the original equations.
At the same time, it is important not only the question of the existence of solutions of the traveling wave
type, but also the question of their uniqueness. For this, one of the approaches is the localization of solutions in
the space of infinitely differentiable functions or analytic functions. As a rule, it is possible to show the existence
of a solution in the space of infinitely differentiable functions, and uniqueness in the space of analytic functions
[Pustyl’nikov, 1997].
The proposed approach is based on the existence of a one-to-one correspondence of solutions of traveling
wave type for infinite-dimensional dynamical systems with solutions of induced FDEPT [Beklaryan, 2007]. To
study the existence and uniqueness of solutions of traveling wave type, it is proposed to localize solutions of
induced FDEPT in spaces of functions, majorized by functions of a given exponential growth. This approach
is particularly successful for systems with Frenkel-Kontorova potentials. In this way, it is possible to obtain a
“correct” extension of the concept of a traveling wave in the form of solutions of the quasi-traveling wave type,
which is related to the description of processes in inhomogeneous environments for which the set of traveling
wave solutions is trivial [Beklaryan, 2010, Beklaryan, 2014].
For the infinite-dimensional dynamical system under consideration, the study of solutions of the traveling
wave type with the characteristic τ , i.e. solutions of the system
ÿi = m−1 (yi+1 − 2yi + yi−1 + ϕ(yi )), i ∈ Z, t ∈ R,
yi (t + τ ) = yi+1 (t)
turns out to be equivalent to the study of a solution space of the induced FDEPT
ẍ(t) = m−1 (x(t + τ ) − 2x(t) + x(t − τ ) + ϕ(x(t))), t ∈ R. (2)
In this case, the corresponding solutions are related as follows: for any t ∈ R
x(t) = y[tτ −1 ] (t − [tτ −1 ]),
where [·] means the integer part of a number. In fact, the described connection between solutions of the travel-
ing wave type of the infinite-dimensional dynamical system and solutions of the induced functional-differential
equation is a fragment of a more general scheme that goes beyond the scope of this article.
2 Existence and Uniqueness Theorem
We assume that the nonlinear potential ϕ satisfies the Lipschitz condition with constant L. Thus, we should
study solutions of the functional-differential equation (2) with a quasilinear right-hand side. A solution of the
FDEPT with a quasilinear right-hand side will be sought in a one-parameter family of Banach spaces of functions
that have at most exponent growth. The exponent is the parameter of the selected family of functions, which is
defined as follows
{ }
Lnµ C (k) (R) = x(·) : x(·) ∈ C (k) (R, Rn ) , max sup ∥x(r) (t)µ|t| ∥Rn < +∞ . (3)
0≤r≤k t∈R
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In our approach for the initial infinite-dimensional dynamical system (1) we will study traveling wave solutions
having a given growth (exponential) both in time and space. To this end, we define a vector space
∏
K n = i∈Z Rin , Rin = Rn , i ∈ Z
(κ ∈ K n , κ = {xi }+∞
−∞ )
with the standard topology of the complete direct product (metrizable space).
In particular, the elements of the space K 2 are infinite sequences
′
κ = {(ui , vi ) }+∞
−∞ , ui , v i ∈ R
(prime means transposition).
In the space K n we define a family of Hilbert subspaces K2µ
n
, µ ∈ (0, 1)
{ }
∑
+∞
K2µ = κ : κ ∈ K ;
n n
∥xi ∥Rn µ < +∞
2 2|i|
i=−∞
with the norm
( +∞ ) 21
∑
∥κ∥2µ = ∥xi ∥2Rn µ2|i| .
i=−∞
Here µ is a free parameter, due to which the solution space will be selected.
Let’s consider a transcendental equation with respect to two variables τ ∈ (0, +∞) and µ ∈ (0, 1)
( )
Cτ 2µ−1 + 1 = ln µ−1 , (4)
where √
C = max{1; 2m−1 L2 + 2}.
The set of solutions of the equation (4) is described by functions µ1 (τ ), µ2 (τ ) given in Figure 1.
Figure 1: Graphs of Functions µ1 (τ ), µ2 (τ )
Let’s formulate the theorem of existence and uniqueness of a solution of traveling wave type.
Theorem 1 For any initial values ī ∈ Z, a, b ∈ R, t̄ ∈ R and characteristics τ > 0 satisfying the condition
0 < τ < τ̂ ,
for the initial system of differential equations (1) there exists a unique solution of the traveling wave type
+∞
{yi (·)}−∞ with characteristic τ such that it satisfies the initial conditions yī (t̄) = a, ẏī (t̄) = b. For any pa-
rameter µ ∈ (µ1 (τ ), µ2 (τ )) the vector function ω(t) = {(yi (t), ẏi (t))T }+∞
−∞ belongs to the space K2µ for any t ∈ R,
2
and the function ρ(t) = ∥ω(t)∥2µ belongs to the space L τ µ C (R). Such a solution depends continuously on the
1√ (1)
initial values a, b ∈ R, as well as on the mass m.
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Theorem 1 not only guarantees the existence of a solution but also determines the limitation of its possible
growth both in time t and in coordinates i ∈ Z (over space). It is obvious that for each 0 < τ < τ̂ the space
2 2
K2µ , at µ < µ2 (τ ) but close to µ2 (τ ), is much narrower than the space K2µ , at µ > µ1 (τ ) but close to µ1 (τ ).
The theorem guarantees the existence of a solution in narrower spaces and uniqueness in wider spaces.
The full text of the proof of theorem 1, as well as a detailed description of the proposed approach, is given in
the papers [Beklaryan, 2007, Beklaryan, 2010, Beklaryan, 2014].
3 Numerical Experiments
Next, the results of the computational experiments on the study of boundary value problems for systems of
FDEPT using OPTCON-F software will be presented. The software complex OPTCON-F is designed to obtain
a numerical solution of boundary value problems, parametric identification problems and optimal control for
dynamical systems described by FDEPT [Gornov et al., 2013]. The proposed technology for solving boundary
value problems is based on the Ritz method and spline collocation approaches. To solve the problem we dis-
cretized system trajectories on the grid with a constant step and formulate the generalized residual functional,
including both weighted residuals of the original differential equation and residuals of boundary conditions
[Zarodnyuk et al., 2016].
Let’s consider the FDEPT of the following form
ẍ(t) = m−1 (x(t + τ ) − 2x(t) + x(t − τ ) + A sin Bx(t)), t ∈ R, (5)
where A, B ∈ R, m, τ ∈ R+ . Using a time-variable transformation the equation (5) can be rewritten in the form
of the following system of equations of the first order:
ż1 (t) = τ z2 (t),
ż2 (t) = τ m−1 (z1 (t + 1) − 2z1 (t) + z1 (t − 1) + A sin(Bz1 (t))).
Under this system, we have the following real parameters: A, B, τ, m. In the following example, for a given
system, we consider such parameter values that conditions of the existence theorem are satisfied.
3.1 Example
We consider dynamical system in the following form:
{
ż1 (t) = 0.15z2 (t),
t ∈ R,
ż2 (t) = 0.15 × 100−1 (z1 (t + 1) − 2z1 (t) + z1 (t − 1) + 500 sin(0.1z1 (t))),
initial conditions (6)
{
z1 (0) = 30,
z2 (0) = 0.
In this case, the equality (4) takes the form
√
0.003 2502(2µ−1 + 1) = ln µ−1 ,
that has on the interval (0, 1) two solutions with approximate values 0.22 and 0.424191 (the exact values is
expressed in terms of the Lambert W-function and is not written out in quadratures).
Taking into account the impossibility of considering the numerical solution of the system on an infinite interval,
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we introduce the parameter k and the corresponding family of expanding initial-boundary value problems
{
ż1 (t) = 0.15z2 (t),
t ∈ [−k, k],
ż2 (t) = 0.15 × 100−1 (z1 (t + 1) − 2z1 (t) + z1 (t − 1) + 500 sin(0.1z1 (t))),
boundary conditions
{
ż1 (t) = 0,
t ∈ (−∞, −k] ∪ [k, +∞), (7)
ż2 (t) = 0,
initial conditions
{
z1 (0) = 30,
z2 (0) = 0.
According to the [Beklaryan, 2007], the solution of the system (7) converges (according to the metric of the space
(3) with µ ∈ (µ1 (τ ), µ2 (τ ))) to the solution of the system (6) as k → ∞. The graphs of the solution of the system
(7) at different values of k are shown in Figure 2.
(a) k = 300 (b) k = 500
(c) k = 1000 (d) k = 5000
Figure 2: Trajectories of the System (7) at Different k.
Since the equation (5) is autonomous, the solution space of such equation is invariant with respect to time-
variable shifts. On the other hand, from the periodicity of the right-hand side with respect to the phase variables
85
it follows that the solution space of such equation is invariant with respect to a shift in phase variables for a
period equal to 2π B . Therefore, it suffices to consider a family of solutions of the initial problem (6) with a
value of z1 (0) from zero to the value of the period. Nevertheless, the stationary solutions are repeated each
π
half-period B . Since the right-hand side of the equation is an odd function of its arguments, the solution space
of such equation can withstand the reflection transformation with respect to the axis t. Hence, it is sufficient to
construct trajectories in the strip from zero to the half-period. Figure 3 shows the integral curves for different
values of the parameter c = z1 (0) for both the system (7) and the original system (6) (the values of c are reduced
to half-period).
(a) Trajectories of the System (7)
(b) Trajectories of the System (6)
Figure 3: Integral Curves at Different c.
Acknowledgements
This work was supported by Russian Science Foundation, Project 17-71-10116.
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