# Model equations for long waves in nonlinear dispersive systems

@article{Benjamin1972ModelEF, title={Model equations for long waves in nonlinear dispersive systems}, author={Thomas Brooke Benjamin and Jerry L. Bona and J. J. Mahony}, journal={Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences}, year={1972}, volume={272}, pages={47 - 78} }

Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation ut+ux+uux−uxxt=0,(a), whose solution u(x,t) is considered in a class of real nonperiodic functions defined for ࢤ∞ <x< ∞,t≥0. As an approximation derived for moderately long waves of small but finite amplitude in…

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## References

SHOWING 1-10 OF 15 REFERENCES

The stability of solitary waves

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1972

The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary…

Internal waves of permanent form in fluids of great depth

- Environmental ScienceJournal of Fluid Mechanics
- 1967

This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these…

Integrals of Nonlinear Equations of Evolution and Solitary Waves

- Mathematics
- 1968

In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A…

Note on the undular jump

- PhysicsJournal of Fluid Mechanics
- 1967

A study is made of the asymptotic equations governing gravity waves on water which cause a transition from one surface level to a slightly higher one and approach a steady wave-form as time…

Some developments in the theory of vortex breakdown

- PhysicsJournal of Fluid Mechanics
- 1967

The primary aim of the analysis presented herein is to consolidate the ideas of the ‘conjugate-flow’ theory, which proposes that vortex breakdown is fundamentally a transition from a uniform state of…

Weakly non-linear waves in rotating fluids

- PhysicsJournal of Fluid Mechanics
- 1970

The Korteweg–de Vries equation is shown to govern formation of solitary and cnoidal waves in rotating fluids confined in tubes. It is proved that the method must fail when the tube wall is moved to…

Method for solving the Korteweg-deVries equation

- Mathematics
- 1967

A method for solving the initial-value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as…

A study of the motion of a cavity in a rotating liquid

- GeologyJournal of Fluid Mechanics
- 1964

Experiments are described in which a spinning tube was initially filled with water and closed at both ends; when the water had acquired uniform angular velocity the tube was suddenly opened at one…

Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion

- Mathematics
- 1968

With extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg‐de Vries and related…

XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves

- Environmental Science
- 1895

(1895). XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of…