Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields
Blackmore-Samulyak-Rosato (BSR) fields, originally developed as a means of obtaining reliable continuum approximations for granular flow dynamics in terms of relatively simple integro-differential equations, can be used to model a wide range of physical phenomena. Owing to results obtained for one-d...
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irk-123456789-321272012-04-10T12:17:29Z Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields Blackmore, D. Urban, K. Rosato, A. Blackmore-Samulyak-Rosato (BSR) fields, originally developed as a means of obtaining reliable continuum approximations for granular flow dynamics in terms of relatively simple integro-differential equations, can be used to model a wide range of physical phenomena. Owing to results obtained for one-dimensional granular flow configurations, it has been conjectured that BSR models of fields with perfectly elastic interactions are completely integrable infinite-dimensional Hamiltonian systems. This conjecture is proved for BSR models in one space dimension, and analogues of BSR fields involving fractional time derivatives are briefly investigated. Установлено, що BSR поля, початково розвинені як засіб одержання надійних континуумних апроксимацій для динаміки гранулярних потоків на мові відносно простих інтегро-диференціальних рівнянь, можна використовувати для моделювання широкого спектра фізичних явищ. Завдяки результатам, одержаним для конфігурацій одновимірного гранулярного потоку, зроблено припущення, що BSR моделі полів із ідеально пружними взаємодіями є повністю інтегровними нескінченновимірними гамільтоновими системами. Це припущення доведено для BSR моделей в одновимірному просторі, і аналоги BSR полів, що включають часткові похідні за часом, досліджені стисло. 2010 Article Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields / D. Blackmore, K. Urban, A. Rosato // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43403:1-7. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 45.70.Mg, 47.10.Df, 02.30.lk, 02.60.Nm http://dspace.nbuv.gov.ua/handle/123456789/32127 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Blackmore-Samulyak-Rosato (BSR) fields, originally developed as a means of obtaining reliable continuum approximations for granular flow dynamics in terms of relatively simple integro-differential equations, can be used to model a wide range of physical phenomena. Owing to results obtained for one-dimensional granular flow configurations, it has been conjectured that BSR models of fields with perfectly elastic interactions are completely integrable infinite-dimensional Hamiltonian systems. This conjecture is proved for BSR models in one space dimension, and analogues of BSR fields involving fractional time derivatives are briefly investigated. |
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Blackmore, D. Urban, K. Rosato, A. |
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Blackmore, D. Urban, K. Rosato, A. Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields Condensed Matter Physics |
| author_facet |
Blackmore, D. Urban, K. Rosato, A. |
| author_sort |
Blackmore, D. |
| title |
Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields |
| title_short |
Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields |
| title_full |
Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields |
| title_fullStr |
Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields |
| title_full_unstemmed |
Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields |
| title_sort |
integrability analysis of regular and fractional blackmore-samulyak-rosato fields |
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Інститут фізики конденсованих систем НАН України |
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2010 |
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| citation_txt |
Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields / D. Blackmore, K. Urban, A. Rosato // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43403:1-7. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT blackmored integrabilityanalysisofregularandfractionalblackmoresamulyakrosatofields AT urbank integrabilityanalysisofregularandfractionalblackmoresamulyakrosatofields AT rosatoa integrabilityanalysisofregularandfractionalblackmoresamulyakrosatofields |
| first_indexed |
2025-07-03T12:39:40Z |
| last_indexed |
2025-07-03T12:39:40Z |
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1836629498412400640 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 4, 43403: 1–7
http://www.icmp.lviv.ua/journal
Integrability analysis of regular and fractional
Blackmore-Samulyak-Rosato fields
D. Blackmore1∗, K. Urban2†, A. Rosato3‡
1 Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey
Institute of Technology Newark, NJ 07102–1982
2 Center for Solar-Terrestrial Research, New Jersey Institute of Technology, Newark, NJ 07102–1982
3 Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102–1982
Received October 11, 2010
Blackmore-Samulyak-Rosato (BSR) fields, originally developed as a means of obtaining reliable continuum
approximations for granular flow dynamics in terms of relatively simple integro-differential equations, can be
used to model a wide range of physical phenomena. Owing to results obtained for one-dimensional granular
flow configurations, it has been conjectured that BSR models of fields with perfectly elastic interactions are
completely integrable infinite-dimensional Hamiltonian systems. This conjecture is proved for BSR models in
one space dimension, and analogues of BSR fields involving fractional time derivatives are briefly investigated.
Key words: BSR model, bi-Hamiltonian, completely integrable, fractional derivative
PACS: 45.70.Mg, 47.10.Df, 02.30.lk, 02.60.Nm
1. Introduction
Blackmore et al. [5] introduced an (averaged-limiting) integro-partial differential equation con-
tinuum approximation for the momentum equation in particulate dynamics that we shall designate
as a BSR model ; the associated infinite-dimensional vector field shall be called a BSR field, and
the method of employing these models shall be referred to as the BSR approach. The primary
goal in the development of the BSR models was to find relatively simple and mathematically more
tractable models for analyzing and predicting particle dynamics, but they can be readily general-
ized to obtain a variety of plausible approximate mathematical models for physical fields ranging
from micro to subatomic scales.
Another goal of the BSR approach is to provide alternatives for other approximate continuum
methods for investigating nearest-neighbor interaction fields comprised of large numbers of com-
ponents, such as the long-wave approximation , which occupies a special place in the creation and
application of soliton theory via a continuum limiting model for Fermi-Pasta-Ulam chains [12] in
terms of the Korteweg-de Vries (KdV) equation by Zabusky & Kruskal [26]. Therefore, it seems
likely that analogous connections with completely integrable Hamiltonian systems should also be
possible for BSR models, and it is our intention to explore such links in considerable detail in this
paper.
The intimate link between approximate continuum models for many-component nonlinear inter-
action phenomena and infinite-dimensional completely integrable Hamiltonian dynamical systems
(with their associated soliton solutions) established for the long-wave method and indicated for
the BSR approach has manifested itself in numerous applications, several of which have arisen in
the investigation of granular flows. For example, Nesterenko [17] showed that certain long-wave ap-
proximations of perfectly elastic (Hertzian) one-dimensional particle dynamics can be transformed
∗E-mail: deblac@m.njit.edu
†E-mail: kdu2@njit.edu
‡E-mail: anthony.rosato@njit.edu
c© D. Blackmore, K. Urban, A. Rosato 43403-1
http://www.icmp.lviv.ua/journal
D. Blackmore, K. Urban, A. Rosato
into KdV equations. On the other hand, Blackmore & Dave [4] proved that perfectly elastic one-
dimensional particle dynamics modeled by a BSR field can be approximated by Burgers’ equation,
which is also completely integrable. In particular, the results in [4] led to the conjecture that BSR
field approximations to perfectly elastic (non-dissipative) granular flows are completely integrable.
And as BSR approximations can be generalized to apply to a much broader class of physical phe-
nomena, the conjecture should be extended to include BSR continuum models of more general
non-dissipative fields. A proof of this conjecture is the main result of this paper.
When it comes to field equations of practically every kind, a recent – and rather productive
– trend has been to reformulate the associated classical differential equation models in terms of
fractional derivatives, and compare the solutions of the fractional and classical versions. These
fractional versions of classical field equations have yielded some very interesting and useful results,
which include certain types of intrinsic gauge invariance and systemic dissipation effects in frac-
tional forms of non-dissipative classical fields. With this in mind, we shall briefly consider analogues
of BSR fields having fractional Caputo time derivatives. Our main observation is that fractional
near integral derivatives with respect to time tend to introduce dissipation in classically inelastic
systems – an observation seen in many fractionalized classical field equations.
Our treatment begins in section 2 with a terse introduction to generalized BSR field equations.
In section 3, we follow this with our main result that the BSR fields satisfying certain mild as-
sumptions are completely integrable infinite-dimensional Hamiltonian dynamical systems, which is
proved by demonstrating that the fields in question have a rather simple bi-Hamiltonian structure.
Then in section 4, we consider fractional derivative reformulations of our BSR fields and show by
example that the resulting dynamical systems include dissipative effects that render them non-
integrable. We conclude our investigation in section 5 with some remarks on the results of our
analysis here and a brief indication of related future research.
2. BSR field equations
Our original derivation of the BSR model [5] was formulated in the context of granular flows
in one-, two- or three-dimensional Euclidean space, denoted as usual by R, R2 or R3, respectively,
where the particle-particle and particle-boundary forces were postulated to be of a form general
enough to subsume the usual Hertz-Mindlin and Walton-Braun models (cf. [6, 17, 23, 23–25]). It
took the form of the following integro-partial differential equation (IPDE):
ut +
∑m
j=1
ujuxj
= E(x, t) +
∫
Rm
Θ(x, y, t, u(x+ y, t)− u(x, t);µ) dy, (1)
where x, y ∈ R
m, m = 1, 2, 3, t represents time, u = (u1, u2, u3) = ẋ is the velocity of a particle
moving throughR
m, the variable subscripts denote partial derivatives, (x, t) is a point in the (m+1)-
dimensional spacetime R
m × R, E represents the external force on the flow configuration that is
often a constant, the integral is the standard Lebesgue integral, the kernel Θ is a smooth function
that vanishes for |y| sufficiently large uniformly in the other variables in accordance with the
assumed nearest-neighbor or nearest-boundary interaction assumptions for the particle dynamics,
and µ represents a vector of parameters defined by the physical properties of the constituents of
the granular flow.
If one examines the limiting argument used to obtain (1) in the granular flow context, it is easy
to see that it generalizes quite naturally to obtain approximate continuum models for any physical
dynamical system on any smooth finite-dimensional manifold M comprised of a very large number
of particles – possibly acting upon one another and their environment in both near-neighbor and
at a distance action modes. Then, the resulting general BSR field model takes the form
ut +∇uu = E(x, t) +
∫
M
Θ
(
x, y, t, u\x(y, t)− u(x, t);µ
)
dy, (2)
where ∇u denotes the usual covariant derivative (for fixed t) on the Riemannian manifold M , dy
is the standard Lebesgue measure based differential associated with integration on M and u\x
43403-2
Integrability analysis of BSR fields
is the parallel transport of the vector u(y, t) along a geodesic from y to x on M . Everything in
(2) is assumed to be smooth (i.e. C∞) with the integrand and all of its derivatives going to zero
exponentially fast as the (geodesic) distance from y to any fixed x, denoted as ̺(x, y), increases
without bound. To simplify matters, we shall assume that the field is defined on the flat manifold
R
m, so that the field equations that we are going to consider in the sequel have the form
ut +∇uu = E(x, t) +
∫
Rm
Θ(x, y, t, u(y, t)− u(x, t);µ) dy, (3)
where
∇uu =
∑m
j=1
ujuxj
for any natural number m in N and dy denotes the standard m-dimensional Lebesgue measure.
It should be noted that (3) can also be approximated as a partial differential equation (PDE)
obtained by expanding Θ in Taylor series (about (x, t)), integrating and then truncating the series
after a suitable finite number of terms (cf. [4, 5]), but we shall not consider these approximations
in any detail in the sequel.
3. Integrability analysis of BSR system
Integrability analysis has played a vital role in the development and investigation of approxi-
mate continuum models for systems of very large numbers of interacting components. In fact, one
may argue persuasively that soliton theory and integrability of infinite-dimensional Hamiltonian
dynamical systems is a direct result of such approximations. In particular, the work of Zabusky
& Kruskal [26] in showing that the long-wave limit of an FPU chain [12] of interacting nonlinear
oscillators is equivalent to the completely integrable KdV equation served as the catalyst for the
spectacular advances in the theory and applications of infinite-dimensional Hamiltonian systems
during the last half-century, recorded and applied in such works as [1, 7–11, 14, 17–21].
From the perspective of the origins of (3), the demonstration of soliton-like behavior for long-
wave limits of one-dimensional particle chains interacting in a perfectly elastic manner [17] and the
integrability of PDE approximations of the BSR model [4] for the same kinds of chains strongly
suggest that the corresponding dynamical systems are completely integrable Hamiltonian systems.
We shall prove this here for (3) in one-space dimension (m = 1) when the interactions are per-
fectly elastic (non-dissipative). Analogous integrability results can be obtained for any finite space
dimension, but the modifications necessary to deal with the higher dimensional equations are quite
extensive, so we shall defer a more general treatment to a forthcoming study.
Our investigations [4–6] have revealed that perfectly elastic particle-particle and particle-boundary
interactions are synonymous with the kernel in (3) being independent of u(y, t)− u(x, t), so that
the BSR equation assumes the simpler form
ut +∇uu = F (x, t;µ) := E(x, t) +
∫
Rm
Θ(x, y, t;µ) dy, (4)
where F is a smooth function. Hence, for our investigation here, we take our (space dimension one)
perfectly elastic BSR continuum model to be the above equation with m = 1.
To see the plausibility of the complete integrability of (4), let us look for a traveling-wave
solution of the following special case of the governing equation:
∂u
∂t
+ u
∂u
∂x
= ϕ (x− V t) , (5)
where ϕ : R → [0,∞) is a smooth (nonnegative) absolutely (Lebesgue) integrable function on R.
We seek a solution of the form
u = f (ξ) = f (x− V t) ,
which we substitute in (4) to obtain
−V f ′ + ff ′ = ϕ (ξ) .
43403-3
D. Blackmore, K. Urban, A. Rosato
Then one integration yields the solutions
u = V ±
√
V 2 + 2Φ (x− V t),
where Φ is simply the integral of ϕ. We also see that the form of the wave is essentially completely
determined by the force field ϕ (x− V t).
3.1. Hamiltonian form of the nondissipative field for m = 1
As mentioned above, we shall investigate the Hamiltonian structure and integrability of the
perfectly elastic BSR field in one space dimension, which assumes the form
ut + uux = F (x, t;µ), (6)
where (x, t) ∈ R× [0,∞), µ is a parameter (vector) in R
k, and F is a smooth function on its domain
of variables and parameters. First, we attend to the Hamiltonian structure and the identification
of the associated Hamiltonian function. The simplest Hamiltonian representation for (6) is
ut = θ ◦ ∇H, (7)
where
H :=
∫
R
(
−
1
6
u3 + u∂−1
x F
)
dx,
∇ is the usual (variational) gradient, and θ = ∂x, where ∂x is the standard partial derivative
operator with respect to x. For our proof of integrability we are going to find a different compati-
ble Hamiltonian representation that guarantees that there is an infinite hierarchy of independent
invariants for (7) in virtue of Magri’s theorem and its various improvements (cf. [7, 8, 16]), which
we take as complete integrability. In particular, we have the following result with a very straight-
forward proof.
Theorem 1 Suppose that all of the functions in (6) are in the usual Schwarz space of functions
(depending on the number of their variables and parameters) and that F does not depend explicitly
on time, i.e. F (x, t;µ) = F (x;µ). Then the infinite-dimensional Hamiltonian dynamical system
(7) representing (6) is completely integrable in the sense that it possesses an infinite hierarchy of
constants of motion
H0 := H,H1, H2, . . . (8)
that are in involution with respect to the usual Poisson bracket induced by θ, namely
{f, g}(u) = {f, g}θ(u) := 〈∇g, θ ◦ ∇f〉 (u) = Lθ◦∇fg(u),
where 〈·, ·〉 is the standard scalar (inner) product and L is the Lie derivative (cf. [8]).
Proof. It follows from refinements of Magri’s theorem (see [7, 8]) that it suffices to find another
Hamiltonian representation of the system (7) in the form
ut = θ̂ ◦ ∇Ĥ, (9)
where the Poisson (implectic) operator pair
(
θ, θ̂
)
is compatible. For completely integrable systems
there is naturally a denumerable choice of compatible bi-Hamiltonian representations, and it suffices
to find just one of the form (9) having the necessary properties.
For (9) we choose
θ̂ := −u∂xu+ F and Ĥ := u.
Then it is a simple matter to verify the compatibility of the representations (7) and (9), from which
the complete integrability follows, thereby concluding the proof. �
43403-4
Integrability analysis of BSR fields
One can use the details of the proof to construct an infinite hierarchy of integrals of (6) using
the recursion techniques developed to deal with multi-Hamiltonian systems, but there is in fact a
much simpler way to find integrals: Start by simply writing (6) in the conservation form
(u)t +
(
−
1
2
u2 + ∂−1
x F
)
x
= 0, (10)
which, under the Schwarz space assumption, leads immediately to the conclusion that
∫
R
udx
is a constant of motion, which is – not surprisingly – tantamount to conservation of mass. Then
multiply (6) by u to obtain the following equation written in conservation form:
(
u2
2
)
t
+
(
−
1
3
u3 + ∂−1
x (uF )
)
x
= 0,
from which we obtain the constant of motion corresponding to conservation of momentum, namely
∫
R
u2
2
dx.
Continuing in this fashion, we find that all of the functions
∫
R
un
n+ 1
dx, n = 1, 2, . . . (11)
are constants of motion.
4. Fractional BSR models
We shall now touch briefly on a particularly simple fractional derivative analogue of the BSR
field model involving only Reisz-Caputo fractional derivatives in time. As mentioned in the intro-
duction, there is a very active current trend of revisiting fractional derivative forms of classical
dynamical systems representations of phenomena arising in physics and other fields in order to
compare the behaviors of the classical and fractional models (examples of which can be found in
[2, 3, 13]).
The fractional derivative analogue of (6), when F does not depend explicitly on time, that we
consider is
d
α
t u+ uux = F (x;µ), (12)
where dαt is the (partial) Riesz-Caputo derivative of order α (taken to be any real number close to
1) defined as
d
α
t f(x, t) :=
1
Γ(n− α)
∫ ∞
0
|t− τ |n−α−1
(
∂
∂τ
)n
f(τ)dτ, (13)
and Γ is the usual Gamma function. To show that such an equation can actually be extracted from
first principles of a sort, we can consider a fractional variational approach. Following the approach
for obtaining the Euler-Lagrange equations and then the corresponding Hamiltonian formulation
for a Lagrangian involving fractional derivatives (as in [2, 3]) – in this case only a fractional Riesz-
Caputo derivative of u with respect to time – one can obtain (12) in a more natural way consistent
with the fundamental principles of mechanics.
Our preliminary analysis of (12) shows that non-integer values of the time derivative inherently
produce dissipative effects, just as can readily be shown via an exact solution in the case of the
fractional linear oscillator
D
α
t ϕ+ ϕ = 0
for non-integral values of α near two. Here Dα
t represents the ordinary Riesz-Caputo derivative of
order α. The inference (based so far only on limited evidence) is that truly integrable behavior may
not be possible for (12) when α is not an integer. However, it may be possible to obtain solutions
in traveling wave type form for some cases.
43403-5
D. Blackmore, K. Urban, A. Rosato
5. Epilogue
We have shown that the BSR equation in one space dimension is an infinite-dimensional com-
pletely integrable Hamiltonian dynamical system when it does not depend explicitly on time.
This also appears to be the case for higher space dimensions, and we intend to address this in a
forthcoming paper, where we shall present a detailed investigation of solitons and other wave-like
phenomena. These results will be compared with the experimental findings on various granular flow
regimes such as the types in Rosato et al. [22, 23] in accordance with our continuing project on the
development of dynamical systems based methods for predicting particulate dynamics (cf. [6]).
Another area of possible research along the lines presented here concerns the case where the BSR
model involves additional terms that represent small perturbations from a completely integrable
form. If this perturbed system is still Hamiltonian, we shall want to see which of the existing
infinite-dimensional KAM theorems are applicable (see e.g. [14]) and perhaps prove some new
results of this kind that can be used to gain further insight into the nature of more general BSR
dynamics. When the perturbation is non-Hamiltonian, it may still be possible to extend some of
the methods employed in Prykarpatsky et al. [20] and other related studies to characterize certain
invariant aspects of BSR dynamics. We plan to look into each of these questions in the near future.
The preliminary findings on fractional analogues of BSR fields suggest that a more thorough
study of such systems is in order. We shall in the near future investigate (12) in much greater
detail, and also delve into what are perhaps more realistic fractional analogues in which the actual
interaction forces in the particle fields are fundamentally fractional in nature as for example in [15].
Acknowledgement
The authors wish to thank the National Science Foundation for partial support of the research
in this paper under NSF Grant CMMI–1029809. Thanks are also due to Anatoliy Prykarpatsky
for his constant support and indispensable advice on all matters relating to integrability analysis
– and especially for the work here.
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Аналiз iнтегровностi регулярних i функцiональних BSR полiв
Д. Блекмор1, К. Урбан2, А. Росато3
1 Факультет математичних наук i Центр прикладної математики i статистики, Iнститут технологiї Нью
Джерсi, Ньюарк, США
2 Центр сонячно-земних дослiджень, Iнститут технологiї Нью Джерсi, Ньюарк, США
3 Iнженерно-механiчний факультет, Iнститут технологiї Нью Джерсi, Ньюарк, США
BSR поля, початково розвинутi як засiб отримання надiйних континуумних апроксимацiй для
динамiки гранулярних потокiв на мовi вiдносно простих iнтегро-диференцiальних рiвнянь, можуть
бути використанi для моделювання широкого спектру фiзичних явищ. Завдяки результатам,
отриманим для конфiгурацiй одновимiрного гранулярного потоку, було зроблено припущення що
BSR моделi полiв з iдеально пружними взаємодiями є повнiстю iнтегровними нескiнченновимiрними
гамiльтоновими системами. Це припущення є доведеним для BSR моделей в одновимiрному
просторi, i аналоги BSR полiв, що включають частковi похiднi за часом є стисло дослiдженнi.
Ключовi слова: BSR модель, бi-гамiльтонiан, повнiстю iнтегровний, часткова похiдна
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Introduction
BSR field equations
Integrability analysis of BSR system
Hamiltonian form of the nondissipative field for m=1
Fractional BSR models
Epilogue
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