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

Arkhipov and Khabakhpashev 2 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 3 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 4 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 5 New model equation for plane nonlinear waves in the shallow liquid layer Ifwhen In the nonlinear term

Novosibirsk, Russia Arkhipov and Khabakhpashev 6 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 77 Test calculations: Overtaking interaction of two plane solitary waves above the horizontal bottom t* = 650 t* = 472 t = 0   =  / h x  = x / h  1 = 3  2

Novosibirsk, Russia Arkhipov and Khabakhpashev 88 Test calculations: Exchange interaction of two plane solitary waves above the horizontal bottom t* = 1400 t* = 965 t = 0   =  / h x  = x / h  1 = 2  2

Novosibirsk, Russia Arkhipov and Khabakhpashev 99 Collision of two nonlinear plane solitary waves above the horizontal bottom t* = 56t* = 51.5t* = 103t* = 112  1 = 2  2  1 =  2

Novosibirsk, Russia Arkhipov and Khabakhpashev 10 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 11 Inelastic interaction at head-on collision of two solitary waves above the horizontal bottom  i =  i0 sech 2 [(x + x 0 – U i t ) /L i ] t* = 28  10 =  20 = 0.15 h, t* = 14 x 0 = ± 15 h

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

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

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

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

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

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

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

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

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

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

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

Novosibirsk, Russia Arkhipov and Khabakhpashev 23 Principal results 1. New evolution differential equations for the dynamics description of moderately long plane and axially- symmetrical 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