Abstract

The work presented introduces correlation moment analysis. This technique can be employed to explore the growth of determinism from stochastic initial conditions in physical systems described by non-linear partial differential equations (PDEs) and is also applicable to wholly deterministic situations. Correlation moment analysis allows the analytic determination of the time dependence of the spatial moments of the solutions of certain types of non-linear partial differential equations. These moments provide measures of the growth of processes defined by the PDE, furthermore the results are obtained without requiring explicit solution of the PDE. The development is presented via case studies of the linear diffusion equation and the non-linear Kortweg de-Vries equation which indicate strategies for exploiting the various properties of correlation moments developed in the text. In addition, a variety of results have been developed which show how various classes of terms in PDEs affect the structure of a sequence of correlation moment equations. This allows results to be obtained about the behavior of the PDE solution, in particular how the presence of certain types of terms affects integral measures of the solution. It is also demonstrated that correlation moments provide a very simple, natural approach to determining certain subsets of conserved quantities associated with the PDEs.

1.
Budd
,
C. J.
, and
Peletier
,
M. A.
, 2000, “
Approximate Self-Similarity in Models of Geological Folding
,”
SIAM J. Appl. Math.
0036-1399,
60
(
3
), pp.
990
1016
.
2.
Morris
,
E. L.
,
Zienkiewicz
,
H. K.
, and
Belmont
,
M. R.
, 1998, “
Short Term Forecasting of the Sea Surface Shape
,”
Int. J. Shpbld. Prog.
,
45
(
444
), pp.
383
400
.
3.
Edgar
,
D. R.
,
Horwood
,
J. M. K.
,
Thurley
,
R.
, and
Belmont
,
M. R.
, 2000, “
The Effects of Parameters on the Maximum Prediction Time Possible in Short Term Forecasting of the Sea Surface Shape
,”
Int. J. Shpbld. Prog.
,
47
(
451
), pp.
287
302
.
4.
Belmont
,
M. R.
,
Bogsjo
,
K.
,
Horwood
,
J. M.
, and
Thurley
,
R. W. F.
, 2004, “
The Effect of Statistically Dependent Phases on Short Term Prediction of the Sea
,”
Int. J. Shipbiuld. Prog.
,
51
(
4
), pp.
314
339
.
5.
Îto
,
K.
, 1944, “
Stochastic Integral
,”
Proc. Imp. Acad. Tokyo
,
20
, pp.
519
524
.
6.
Îto
,
K.
, 1951, “
On Stochastic Differential Equations
,”
Mem. Am. Math. Soc.
0065-9266,
4
.
7.
Doob
,
J. L.
, 1953,
Stochastic Processes
,
Academic
, New York.
8.
Adomain
,
G.
, 1976, “
Nonlinear Stochastic Differential Equations
,”
J. Math. Anal. Appl.
0022-247X,
55
(
2
), pp.
441
452
.
9.
Adomain
,
G.
, 1983,
Stochastic Systems
,
Academic
, New York.
10.
Lee
,
Y. W.
, 1967,
Statistical Theory of Communication
,
Wiley
, New York.
11.
Wiener
,
N.
, 1959,
The Fourier Integral and Certain of its Applications
,
Dover
, New York.
12.
Townsend
,
A. A.
, 1976,
The Structure of Turbulent Shear Flow
,
Cambridge University Press
, Cambridge, UK.
13.
Vom
,
Scheidt
,
Starkloff
,
H.-J.
, and
Wunderlich
,
R.
, 2000, “
Asymptotic Expansions of Integral Functionals of Weakly Correlated Random Processes
,”
Z. Anal. ihre Anwend.
0232-2064,
19
(
1
), pp.
255
268
.
14.
Besicovitch
,
A. S.
, 1932,
Almost Periodic Functions
,
Cambridge University Press
, Cambridge, UK.
15.
Corduneau
,
C.
, 1968,
Almost Periodic Functions
,
Wiley Interscience
, New York.
16.
Crank
,
J.
, 1975,
The Mathematics of Diffusion
,
Oxford University Press
, Oxford, UK.
17.
Muira
,
R. M.
,
Garderner
,
C. S.
, and
Kruskal
,
M. D.
, 1968, “
Kortweg-de Vries Equation and Generalisations, II. Existence of Conservation Laws and Constants of Motion
,”
J. Math. Phys.
0022-2488,
9
, pp.
1204
1209
.
18.
Drazin
,
P. G.
, and
Johnson
,
R. S.
, 1989,
Solitons: An Introduction
,
Cambridge University Press
, Cambridge.
19.
Johnson
,
R. S.
, 1980, “
Water Waves and the Kortweg-deVries Equations
,”
J. Fluid Mech.
0022-1120,
97
, pp.
701
19
.
20.
Whitham
,
G. B.
, 1974,
Linear and Nonlinear Waves
,
Wiley
, New York.
21.
Gardener
,
C. S.
,
Greene
,
J. M.
,
Kruskal
,
M. D.
, and
Muira
,
R. M.
, 1967, “
Method for Solving the Kortweg-de Vries Equation
,”
Phys. Rev. Lett.
0031-9007,
19
, pp.
1095
1097
.
22.
Gardener
,
C. S.
,
Greene
,
J. M.
,
Kruskal
,
M. D.
, and
Muira
.
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