## Project Details

### Description

The existence of solitary waves on water (first observed by Scott Russell in 1834 on the Union Canal in Edinburgh, but regarded with scepticism for many years) and the existence of an idealised wave of greatest height with sharp crests (conjectured by Stokes in 1880) were generally accepted in the first half of the 20th century, but lacked mathematical proof. For solitary waves of small amplitude, a slightly dubious existence theorem appeared in 1946 and a convincing one in 1954, but a rigorous theory of water waves with no restriction on amplitude began to emerge only in the 1960's and 1970's with the help of modern mathematics (fixed-point theorems in Banach spaces). After J. F. Toland proved the existence of a wave of greatest height in 1978, C. J. Amick (who died in 1991) and Toland began to build a rigorous general theory of water waves (again using modern mathematics). With the help of Buffoni, Dancer, Plotnikov and others, this work has continued to the present. These developments have advanced so far that a workshop can now be held to celebrate the achievements and lend direction to the continuing research efforts of the senior and junior mathematical scientists who are expected to gather for it.

### Layman's description

The existence of solitary waves on water (first observed by John Scott Russell in 1834 on the Union Canal in Edinburgh, but regarded with scepticism for many years) and the existence of an idealised ``wave of greatest height'' with sharp crests (conjectured by Stokes in 1880) were generally accepted in the first half of the 20th century, but lacked mathematical proof. For solitary waves of small amplitude, a slightly dubious existence theorem appeared in 1946 and a convincing one in 1954, but a rigorous theory of water waves with no restriction on amplitude began to emerge only in the 1960's and 1970's with the help of modern mathematics (fixed-point theorems in Banach spaces). After J.F. Toland proved the existence of a wave of greatest height in 1978, C.J. Amick (who died in 1991) and Toland began to build a rigorous general theory of water waves (again using modern mathematics). With the help of Buffoni, Dancer, Plotnikov and others, this work has continued to the present. These developments had advanced so far that a workshop could be held in 2009 to celebrate the achievements and to lend direction to the continuing research efforts of the senior and junior mathematical scientists who were expected to gather for it.

### Key findings

Invited talks were given by Boris Buffoni (Lausanne), Walter Craig (Hamilton), Norman Dancer (Sydney), Gerard Iooss (Nice), Bryce McLeod (Oxford), Louis Nirenberg (New York), Pavel Plotnikov (Novosibirsk), Eugene Shargorodsky (London), Paul Rabinowitz (Madison), Charles Stuart (Lausanne), Neil Trudinger (Canberra) and David Williams (Swansea). There were about 6 contributions to the poster session and approximately 64 participants attended the workshop.

Status | Finished |
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Effective start/end date | 1/01/09 → 31/08/09 |

### Funding

- Engineering and Physical Sciences Research Council