# The KdV Equation on the Half-Line: Time-Periodicity and Mass Transport

@article{Bona2019TheKE, title={The KdV Equation on the Half-Line: Time-Periodicity and Mass Transport}, author={Jerry L. Bona and Jonatan Lenells}, journal={SIAM J. Math. Anal.}, year={2019}, volume={52}, pages={1009-1039} }

The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that generated unidirectional waves of relatively small amplitude and long wavelength when compared with the undisturbed depth. It was observed that the wave profile at any point down the channel very quickly became periodic in time with the same period as that of the…

## 2 Citations

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## References

SHOWING 1-10 OF 42 REFERENCES

### A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

- Mathematics
- 2001

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial…

### Long Wave Approximations for Water Waves

- Mathematics
- 2005

In this paper, we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions. These systems are symmetric and conservative. Rigorous convergence…

### Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

- Mathematics
- 2003

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes…

### Shallow-water waves, the Korteweg-deVries equation and solitons

- PhysicsJournal of Fluid Mechanics
- 1971

A comparison of laboratory experiments in a shallow-water tank driven by an oscillating piston and numerical solutions of the Korteweg-de Vries (KdV) equation show that the latter can accurately…

### An evaluation of a model equation for water waves

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1981

This study assesses a particular model for the unidirectional propagation of water waves, comparing its predictions with the results of a set of laboratory experiments. The equation to be tested is a…

### Comparison of quarter-plane and two-point boundary value problems: The KdV-equation

- Mathematics
- 2007

This paper is concerned with the Korteweg-de Vries equation which models unidirectional propagation of small amplitude long waves in dispersive media. The two-point boundary value problem wherein the…

### Integrable Nonlinear Evolution Equations on the Half-Line

- Mathematics
- 2002

Abstract: A rigorous methodology for the analysis of initial-boundary value problems on the half-line, is applied to the nonlinear §(NLS), to the sine-Gordon (sG) in laboratory coordinates, and to…

### TEMPORAL GROWTH AND EVENTUAL PERIODICITY FOR DISPERSIVE WAVE EQUATIONS IN A QUARTER PLANE

- Mathematics
- 2008

Studied here is the
large-time behavior and eventual periodicity of solutions of
initial-boundary-value problems for the BBM equation and the KdV
equation, with and without a Burgers-type…

### Model equations for long waves in nonlinear dispersive systems

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1972

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.…

### A Comparison of Solutions of Two Model Equations for Long Waves.

- Mathematics
- 1983

Abstract : This paper is concerned with mathematical models representing the unidirectional propagation of weakly nonlinear dispersive waves. Interest will be directed toward two particular models…