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

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 describe slightly dissipative wavepropagation for Ursell numbers ( h 1 L 2 / h 0 3 ) up to 800. This is an input-output experiment, where the initial condition for the KdV equation is obtained from upstream (station 1) data. At a downstream location, the number of crests and troughs and their phases (or relative locations within a period) agree quantitatively with numerical solutions. The crest-to-trough amplitudes disagree somewhat, as they are more sensitive to dissipative forces. This work firmly establishes the soliton concept as necessary for treating the propagation of shallow-water waves of moderate amplitude in a low-dissipation environment.

[1]  Ryogo Hirota,et al.  Studies on Lattice Solitons by Using Electrical Networks , 1970 .

[2]  N. Zabusky.  A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction , 1967 .

[3]  R. Taylor,et al.  Formation and Interaction of Ion-Acoustic Solitions , 1970 .

[4]  F. Tappert,et al.  Asymptotic Theory of Self-Trapping of Heat Pulses in Solids , 1970 .

[5]  Norman J. Zabusky,et al.  Solitons and Bound States of the Time-Independent Schrödinger Equation , 1968 .

[6]  Chiang C. Mei,et al.  The transformation of a solitary wave over an uneven bottom , 1969, Journal of Fluid Mechanics.

[7]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

[8]  The Evolution of Time-Periodic Long Waves of Finite Amplitude, , 1970 .

[9]  C. S. Gardner,et al.  Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion , 1968 .

[10]  H. Washimi,et al.  ON THE PROPAGATION OF ION ACOUSTIC SOLITARY WAVES OF SMALL AMPLITUDE. , 1966 .

[11]  D. Peregrine.  Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[12]  Galvin.  FINITE-AMPLITUDE, SHALLOW WATER-WAVES OF PERIODICALLY RECURRING FORM , 1970 .

[13]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[14]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[15]  Robert M. Miura,et al.  Korteweg‐deVries Equation and Generalizations. V. Uniqueness and Nonexistence of Polynomial Conservation Laws , 1970 .

[16]  C. S. Gardner,et al.  Korteweg‐de Vries Equation and Generalizations. III. Derivation of the Korteweg‐de Vries Equation and Burgers Equation , 1969 .