Dmitry Arkhipov and Georgy Khabakhpashev New equations for

Dmitry Arkhipov and Georgy Khabakhpashev New equations for modeling nonlinear waves interaction on a free surface of fluid shallow layer Department of Physical Hydrodynamics Institute of Thermophysics SB RAS Novosibirsk, Russia

Moderately long nonlinear plane waves on a free surface of a liquid layer G. B. Whitham, Linear and Nonlinear Waves (1974) L. A. Ostrovsky, A. I. Potapov, Modulated Waves: Theory and Applications (1999) G. A. Khabakhpashev, Fluid Dynamics (1987) K. Y. Kim, R. O. Reid, R. E. Whitaker, J. Comp. Phys. (1988) D. Е. Pelinovsky, Yu. A. Stepanyants, JETP (1994) R. S. Johnson, J. Fluid Mechanics (1996) For plane perturbations above horizontal bottom Novosibirsk, Russia Arkhipov and Khabakhpashev 2

Nonlinear long disturbances of a free surface of the liquid layer above a gently sloping bottom Basic assumptions of the model Novosibirsk, Russia Arkhipov and Khabakhpashev 3

Main model equations for nonlinear waves, running at any angles between them D. G. Arkhipov, G. A. Khabakhpashev, Doklady Physics (2006) Novosibirsk, Russia Arkhipov and Khabakhpashev 4

New model equation for plane nonlinear waves in the shallow liquid layer If when In the nonlinear term Novosibirsk, Russia Arkhipov and Khabakhpashev 5

Propagation of moderately long nonlinear waves in the liquid layer above the horizontal bottom For perturbations running in one direction Novosibirsk, Russia Arkhipov and Khabakhpashev 6

Test calculations: Overtaking interaction of two plane solitary waves above the horizontal bottom h* = h / h h 1 = 3 h 2 x* = x / h t=0 t* = 650 t* = 472 Novosibirsk, Russia Arkhipov and Khabakhpashev 7

Test calculations: Exchange interaction of two plane solitary waves above the horizontal bottom h* = h / h h 1 = 2 h 2 x* = x / h t=0 t* = 1400 t* = 965 Novosibirsk, Russia Arkhipov and Khabakhpashev 8

Collision of two nonlinear plane solitary waves above the horizontal bottom h 1 = 2 h 2 t* = 51. 5 Novosibirsk, Russia t* = 103 h 1 = h 2 Arkhipov and Khabakhpashev t* = 56 9 t* = 112

Approximated analytical solutions to the problem of head-on collision of two solitary waves above the horizontal bottom -- modified Boussinesq equation Novosibirsk, Russia Arkhipov and Khabakhpashev 10

Inelastic interaction at head-on collision of two solitary waves above the horizontal bottom t* = 14 h i = hi 0 sech 2 [(x + x 0 – Uit ) /Li] h 10 = h 20 = 0. 15 h , x 0 = ± 15 h t* = 28 Novosibirsk, Russia Arkhipov and Khabakhpashev 11

Collision of two nonlinear plane solitary waves above the horizontal and uneven bottoms h 1 = h 2 t* = 20 Novosibirsk, Russia h (x) = h 0 [1– sech 2 (x / 2 L) / 2] Arkhipov and Khabakhpashev 12

New model equation for axisymmetrical nonlinear waves in the shallow liquid layer If when F. Calogero, A. Degasperis, Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations (1982) – Kd. V cylindrical eq. Novosibirsk, Russia Arkhipov and Khabakhpashev 13

Evolution of initially bell-type perturbation above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 14

Evolution of initially bell-type perturbation above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 15

Evolution of initially bell-type perturbation above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 16

Evolution of initially bell-type perturbation above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 17

Transformation of initially ring-type disturbance above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 18

Transformation of initially ring-type disturbance above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 19

Transformation of initially ring-type disturbance above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 20

Transformation of initially ring-type disturbance above two different bottom profiles r* = r / h 0 h = h 0 Novosibirsk, Russia Arkhipov and Khabakhpashev 21

Interaction of initially bell-type and ring-type perturbations above two different bottom profiles h = h 0 Novosibirsk, Russia r* Arkhipov and Khabakhpashev 22 = r / h 0

Principal results 1. New evolution differential equations for the dynamics description of moderately long plane and axiallysymmetrical nonlinear waves running towards each other are suggested. 2. A validity of new equations to the solution of a number of plane or axially-symmetrical problems of the nonlinear wave evolution including the case of a fluid with variable depth is shown with the help of numerical experiments. 3. An analytical solution for the problem of head-on collision of two solitons was constructed by the perturbation theory Novosibirsk, Russia Arkhipov and Khabakhpashev 23