TITLE

Analytic model for a weakly dissipative shallow-water undular bore

AUTHOR(S)
El, G. A.; Grimshaw, R. H. J.; Kamchatnov, A. M.
PUB. DATE
September 2005
SOURCE
Chaos;Sep2005, Vol. 15 Issue 3, p037102
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We use the integrable Kaup–Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by viscosity. This is derived in Riemann variables using a modified finite-gap integration technique for the Ablowitz–Kaup–Newell–Segur (AKNS) scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup–Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.
ACCESSION #
18669410

 

Related Articles

  • UNSTEADY BOUNDARY LAYERS: CONVECTIVE HEAT TRANSFER OVER A VERTICAL FLAT PLATE. Van Gorder, Robert A.; Vajravelu, K. // ANZIAM Journal;2009, Vol. 50 Issue 4, p541 

    In this paper, we extend the results in the literature for boundary layer flow over a horizontal plate, by considering the buoyancy force term in the momentum equation. Using a similarity transformation, we transform the partial differential equations of the problem into coupled nonlinear...

  • VARIABLE STEP-SIZE IMPLICIT-EXPLICIT LINEAR MULTISTEP METHODS FOR TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS. Dong Wang; Ruuth, Steven J. // Journal of Computational Mathematics;Nov2008, Vol. 26 Issue 6, p838 

    Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving partial differential equations (PDEs) with terms of different types. While fixed time-step versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general...

  • A method for topological optimization of structures with discrete variables under dynamic stress and displacement constraints. Shi Lian-shuan; Sun Huan-chun; Feng En-min // Applied Mathematics & Mechanics;Jul2001, Vol. 22 Issue 7, p781 

    A method for topological optimization of structures with discrete variables subjected to dynamic stress and displacement constraints is presented. By using the quasistatic method, the structure optimization problem under dynamic stress and displacement constraints is converted into one subjected...

  • Solution of the center-focus problem for a nine-parameter cubic system. Sadovskii, A.; Shcheglova, T. // Differential Equations;Feb2011, Vol. 47 Issue 2, p208 

    We present a solution of the center-focus problem for a nine-parameter cubic system that can be reduced to a Lienard type system.

  • Numerical solution of the Painlevé VI equation. Abramov, A.; Yukhno, L. // Computational Mathematics & Mathematical Physics;Feb2013, Vol. 53 Issue 2, p180 

    A numerical method for solving the Cauchy problem for the sixth Painlevé equation is proposed. The difficulty of this problem, as well as the other Painlevé equations, is that the unknown function can have movable singular points of the pole type; moreover, the equation may have...

  • Trajectory controllability of semilinear systems with multiple variable delays in control. Klamka, Jerzy; Niezabitowski, Michal // AIP Conference Proceedings;2014, Vol. 1637, p498 

    In this paper, finite-dimensional dynamical control system described by semilinear differential state equation with multiple variable delays in control are considered. The concept of controllability we extend on trajectory controllability for systems with multiple point delays in control....

  • Efficient Technique for Solving Stiff Delay Differential Equations in Variable Stepsize Variable Order. Isa, Nora Baizura Mohd; Ishak, Fuziyah; Othman, Khairil Iskandar // AIP Conference Proceedings;2014, Vol. 1605, p52 

    In this paper, we describe the development of predictor corrector variable stepsize variable order method based on backward differentiation formula. The formula is represented in divided difference form where the coefficients of differentiation are computed by a simple recurrence relation. This...

  • Boundary value problems for complete partial differential equations of variable order. Lomovtsev, F. // Differential Equations;Sep2010, Vol. 46 Issue 9, p1374 

    We prove the well-posedness of new boundary value problems for partially pseudodifferential complete nonclassical equations of variable order in space variables with higher derivatives of odd order in time.

  • Step Size Strategies Based On Error Analysis For The Linear Systems. Kizilkan, G�lnur �elik; Aydin, Kemal // Suleyman Demirel University Journal of Science;2011, Vol. 6 Issue 2, p149 

    In this paper, we have obtained the step size strategies for numerical integration of the linear differential equation systems. We have given the algorithms which calculate step sizes based on the given strategies and numerical solutions. These strategies and algorithms are generalized to...

Share

Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics