scholarly journals PROPERTIES OF AN ALTERNATE LAX DESCRIPTION OF THE KdV HIERARCHY

1995 ◽  
Vol 10 (12) ◽  
pp. 931-939 ◽  
Author(s):  
J.C. BRUNELLI ◽  
ASHOK DAS
Keyword(s):  
Power Law ◽  
Recursion Relation ◽  
Recursion Operator ◽  
Conserved Quantities ◽  
Lax Equation ◽  
Hamiltonian Structures ◽  
Kdv Hierarchy ◽  
Commuting Flows ◽  
Definition Of

We study systematically the Lax description of the KdV hierarchy in terms of an operator which is the geometrical recursion operator. We formulate the Lax equation for the nth flow and construct the Hamiltonians which lead to commuting flows. In this formulation, the recursion relation between the conserved quantities follows naturally. We give a simple and compact definition of all the Hamiltonian structures of the theory which are related through a power law.

INTEGRABLE MODELS AND SPIN ALGEBRAS

1991 ◽  
Vol 06 (08) ◽  
pp. 1429-1445 ◽  
Author(s):  
ASHOK DAS ◽  
SHIBAJI ROY
Keyword(s):  
Recursion Relation ◽  
Boussinesq Equation ◽  
Conserved Quantities ◽  
Third Order ◽  
Kdv Hierarchy ◽  
The Third ◽  
Zero Curvature ◽  
Combined Flow ◽  
The Relationship ◽  
Independent Derivation

We show how the entire KdV hierarchy as well as the recursion relation between the conserved quantities can be obtained from a spin-2 flow. The relationship between this approach and the zero curvature formulation of the KdV system based on the group SL(2, R) is clarified. We show that the Boussinesq equation can be derived as a flow of the spin-3 transformations. The Boussinesq hierarchy, as well as the relevant Lenard relation, is derived from a combined flow of the spin-2 and spin-3 transformations. We have also given an independent derivation of the Lenard relation for the Boussinesq equation starting from the third order Schrödinger equation.

scholarly journals Observational evidence in favor of scale-free evolution of sunspot groups

2018 ◽  
Vol 618 ◽  
pp. A183
Author(s):  
A. Shapoval ◽  
J.-L. Le Mouël ◽  
M. Shnirman ◽  
V. Courtillot
Keyword(s):  
Weibull Distribution ◽  
Power Law ◽  
Large Range ◽  
Scale Free ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Spatio Temporal ◽  
Computational Procedures ◽  
Definition Of ◽  
Sunspot Groups

Context. The hypothesis stating that the distribution of sunspot groups versus their size (φ) follows a power law in the domain of small groups was recently highlighted but rejected in favor of a Weibull distribution. Aims. In this paper we reconsider this question, and are led to the opposite conclusion. Methods. We have suggested a new definition of group size, namely the spatio-temporal “volume” (V) obtained as the sum of the observed daily areas instead of a single area associated with each group. Results. With this new definition of “size”, the width of the power-law part of the distribution φ ∼ 1/Vβ increases from 1.5 to 2.5 orders of magnitude. The exponent β is close to 1. The width of the power-law part and its exponent are stable with respect to the different catalogs and computational procedures used to reduce errors in the data. The observed distribution is not fit adequately by a Weibull distribution. Conclusions. The existence of a wide 1/V part of the distribution φ suggests that self-organized criticality underlies the generation and evolution of sunspot groups and that the mechanism responsible for it is scale-free over a large range of sizes.

scholarly journals METHODS OF CFD MODELLING OF TWIN-SCREW PUMPS FOR NON-NEWTONIAN MATERIALS

2021 ◽  
Vol 2021 (6) ◽  
pp. 5366-5372
Author(s):  
MARIAN BOJKO ◽  
◽  
LUKAS HERTL ◽  
SYLVA DRABKOVA ◽  
◽  
...  
Keyword(s):  
Power Law ◽  
Food Industry ◽  
3D Model ◽  
Design Parameters ◽  
Cfd Modelling ◽  
Ansys Fluent ◽  
Screw Pump ◽  
Twin Screw ◽  
Definition Of ◽  
Power Law Parameters

The twin-screw pump is designed for pumping highly viscous materials in the food industry. Rheological characteristics of materials are important in the specification of design parameters of screw pumps. Analysis of flow in the twin-screw pumps with definition of non-newtonian materials can be made by numerical modelling. CFD generally oriented software ANSYS Fluent and ANSYS Polyflow has been used for modelling. In this study those software’s (ANSYS Fluent and ANSYS Polyflow) were defined for solution of flow in the twin-screw pumps. Results were compared for the same boundary conditions on the inlet and outlet of the 3D model. For definition of the viscosity were used the Nonnewtonian power law. Parameters as consistency coefficient and flow exponent for Nonnewtonian power law were analysed by software ANSYS Fluent and ANSYS Polyflow. Postprocessing form ANSYS Fluent and ANSYS Polyflow were made by contours of field and by graphs.

scholarly journals The Semidirect Sum of Lie Algebras and Its Applications to C-KdV Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xia Dong ◽  
Tiecheng Xia ◽  
Desheng Li
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Integrable Systems ◽  
The Other ◽  
Loop Algebra ◽  
Hamiltonian Structures ◽  
Kdv Hierarchy ◽  
Integrable Coupling

By use of the loop algebraG-~, integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by Tu scheme and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.

scholarly journals QUANTUM SINGULARITIES IN STATIC AND CONFORMALLY STATIC SPACE-TIMES

2011 ◽  
Vol 26 (22) ◽  
pp. 3878-3888 ◽  
Author(s):  
D. A. KONKOWSKI ◽  
T. M. HELLIWELL
Keyword(s):  
Power Law ◽  
Cosmic String ◽  
Higher Order ◽  
Space Time ◽  
Differential Invariants ◽  
Future Research ◽  
Quantum Singularities ◽  
Definition Of ◽  
Friedmann Robertson Walker ◽  
Static Space

The definition of quantum singularity is extended from static space-times to conformally static space-times. After the usual definitions of classical and quantum singularities are reviewed, examples of quantum singularities in static space-times are given. These include asymptotically power-law space-times, space-times with diverging higher-order differential invariants, and a space-time with a 2-sphere singularity. The theory behind quantum singularities in conformally static space-times is followed by an example, a Friedmann-Robertson-Walker space-time with cosmic string. The paper concludes by discussing areas of future research.

scholarly journals Entropy production and self-organized (sub)criticality in earthquake dynamics

2010 ◽  
Vol 368 (1910) ◽  
pp. 131-144 ◽  
Author(s):  
Ian G. Main ◽  
Mark Naylor
Keyword(s):  
Numerical Model ◽  
Entropy Production ◽  
Power Law ◽  
Subcritical State ◽  
Earthquake Dynamics ◽  
Broad Bandwidth ◽  
Rupture Area ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Definition Of

We derive an analytical expression for entropy production in earthquake populations based on Dewar’s formulation, including flux (tectonic forcing) and source (earthquake population) terms, and apply it to the Olami–Feder–Christensen numerical model for earthquake dynamics. Assuming the commonly observed power-law rheology between driving stress and remote strain rate, we test the hypothesis that maximum entropy production (MEP) is a thermodynamic driver for self-organized ‘criticality’ (SOC) in the model. MEP occurs when the global elastic strain is near-critical, with small relative fluctuations in macroscopic strain energy expressed by a low seismic efficiency, and broad-bandwidth power-law scaling of frequency and rupture area. These phenomena, all as observed in natural earthquake populations, are hallmarks of the broad conceptual definition of SOC (which has, to date, often included self-organizing systems in a near but strictly subcritical state). In the MEP state, the strain field retains some memory of past events, expressed as coherent ‘domains’, implying a degree of predictability, albeit strongly limited in practice by the proximity to criticality and our inability to map the natural stress field at an equivalent resolution to the numerical model.

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