ADVANCES IN MECHAN ICS Aug125, ,, KdV. , KdV. KdV. (m KdV). (non - propagating soliton or standing soliton), (precursor soliton)

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1 8 Vol18 No ADVANCES IN MECHAN ICS Aug15, 1998 LNM,, ,,, 150, Russell,,,,,,, KdV,,, :, KdV,,, KdV (m KdV) Schrodinger (NL S ) KdV, KP, KP 80, KdV :, (non - propagating soliton or standing soliton), (precursor soliton) :,,, 1,, : , : China Academic Journal Electronic Publishing House All rights reserved

2 ,, 1918, Froude F = U gh ( U, h, g ) F 1, 90,, F < 1,, 195,, 198 Huang [1 ] ( F 1 ) Wu Wu [ ],, gh, ( 1), F 1,,, F > F c > 1, China Academic Journal Electronic Publishing House All rights reserved

3 F = F c (, (turning point) ),, F < F L < 1,,, F = F L ( cutoff point),, F L < F < F c, 1,,,,,,,,,,,, [,4 ] [5 ] WuD M [ ] [6 ] [7 ] Alberta, Wu Wu [ ] Boussinesq + [ ( h + ) u ] x = - h t (1) t u t + uu x + g x = - 1 P ax + 1 h[ h t + ( hu) x ] xt h u x xt + O (, ) (), x, t, x, t, g, = ( x, t), z = - h ( x, t) = - h 0 + b ( x, t), b ( x, t), h 0, u ( x, t) x,, P a ( x, t) Boussinesq, = a h ν 1,= h 0 at f f ν 1,= O ( ), a, P a ( x, t) b ( x, t),, h x, h t C 0, P ax g, P at gc 0 Φ O ( 1/ ) () C 0 = gh 0 (4), Boussinesq KdV (f KdV),, P a = P a ( x + U t), b = b ( x + Ut) (5) u, (1) (), f KdV 76 1 C 0 + U C 0 ie rv x - h 0 1 cei 6 h 0 x x x = 1 P a g + b x c m d b China Academic Journal Electronic Publishing House All rights reserved o (6)

4 U = 1 + O ( ), P a = O ( ), b = ( ), (), C 0,, (6) KdV, ( ), f KdV, (6) Wu [8 ] (6) (6),, (), Boussinesq (6) ( x,0) = - h 0 P( x,0), u ( x,0) = 0, ( x,0) = 0, u ( x,0) = 0, x t China Academic Journal Electronic Publishing House All rights reserved

5 , d E Tw = (1 - h d t 1 ) 1 = (17) C Ds = 1 gc D (18) w,,,,,, Wu,,, f KdV, f KdV T s m g v + m gg v + m 4 g vvv = 1 P v (19) P = p( ), [7 ] gc Dw = - m m 1 (0) = ( p / 4 m m 4 ) 1/ (1),,, Casciola Landrini [1 ] KdV, Boussinesq,, KdV Boussinesq F 1,,, Wu [8 ] (6) s = N n = 1 N P = n = 1 a n sech n kx () b n sech n kx b n = F n a n + k B n a n b n = - N 4 m = n - N F n = ( F - 1) - 78 n m = 1 a m a n - m ( n = N +,, N ) a m a n - m ( n = 1,,, N + 1) (a) (b) (c) k n (d) China Academic Journal Electronic Publishing House All rights reserved

6 B n = 1 ( n - 1) ( n - 1) (e) N + ( F, k, a 1,, a n ), N = 1 (1) s = asech kx, a = 4 k, b 1 = F 1 a = 0, b = 0, F 1 = 0, F = a, P = 0 (4) KdV () I s1 = asech kx, a = 4 k, b 1 = F 1 a, b = 0, F 1 = ( F - 1) - () II k, P = b 1 sech kx s1 = asech kx, b 1 = F 1 a = 0, F 1 = 0, b = ) a k - 4 a, P = b sech 4 kx (6) Wu,, ;,,, :???? (, ),,,,, [ ] 0 0 cm cm, cm,,,,, X Y,, Faraday,, [14, 1990 ],,, ; ( ),,,,,, g g= 0,, 1 = (5) i [ A ( Z + Y) - c1c ] (7) g (1 + ) China Academic Journal Electronic Publishing House All rights reserved

7 = gk T (1 + ), T = tanh kh,= k / (g),, b, k = / b, Z = exp (i ( ky - t), Y = exp [ - i ( ky + t) ], x, y, c c, x, A x 1, x, t 1, t, 0 - = [15 ] A + i A t + k A x 1 x + A r A + (1 + ) A = 0 (8), = d / d k > 0, r = 4 0 a/ g, a, 0 = k4 ( T - 1) (9 - T ) +( - T ) ( T - 7) T - ( - T, + 9T ) 1 + = 0 (0) kh = 110,= 0 (9) = k4 [6 T4-5 T T - ] (0) (8),, (8) > 0 < 0 > 0, (8) sech 4 A = i ( k T 0 a ) sech k ( k T 0 a ) x (1) (1) :, < 0, (8) A = - ( k T 0 a) tanh k, ( k T 0 a) x () (8), (9), kh > 0 < 0, > 0, (1), < 0 kh Φ 1, (), kh Φ 1,, 5 %, (8), (9), [17, ], (8) ( ), (0) [18, ] :,,, (8) [19, (8), ] (8), (8),, [0 ], (8) China Academic Journal Electronic Publishing House All rights reserved [16 ] -

8 ia, [1 ],,,,,, (8) (1), (), A = A = i - 4 ( - t 0 - k T 0 a) sech k ( k T 0 a) x () ( k T 0 a) th k 4 ( k T 0 a) x (4),, (), (4),, [ ], [,4, ] [5,6 ] [7 ] KP A tx + B ( ) x x + C x x x x + D zz + E= 0 (5) KdV, E, E = 0 ( ), KdV KP z x y, [8 Grimshaw (5) ],, 80, 1 Huang D B, et al Ship moving in transcritical range In : Proc Conf on Behavior of Ships in Restricted Waters Varna, Bulgaria, 198 V II, Wu D M, Wu T Y Three - dimensional nonlinear long waves due to moving surface pressure In : Proc 14th Symp Naval Hydrodyn, Grimshaw R, Smyth N F Resonant flow of a stratified fluid over a topograph J Fluid Mech, 1986, 169 : Shen S S Forced solitary waves and hydraulic falls in two - layer flows J Fliud Mech, 199, 4 : : 60 ( ), 1985, Zhu J, Wu T Y, Yates G T Internal solitary waves generated by moving disturbances In : Third Intl Symp on stratified flows Feb 5, 1987, CIT, Pasadena, CA, USA 7 Xu Z, Wu K, Shen S S Theoretical mean wave resistance of precursor soliton generation in two layer flow Acta Mech Sinica, 1997, 1 (1) : 19 8 Wu T Y Generation of upstream advancing solitons by moving disturbances J Fluid Mech, 1987, 184 : Ertekin R C, et al Waves caused by a moving disturbance in a shallow channel of finite width J Fluid Mech, 1986, 169 : China Academic Journal Electronic Publishing House All rights reserved

9 10 Mei C C, Choi H S Forces on a slender ship advancing near the critical speed in a wide canal J Fluid Mech, 1987, 179 : Chen X, Sharma S D Aslender ship moving at a near - critical speed in a shallow channel J Fluid Mech, 1995, 91 : Casciola C M, Landrini M Nonlinear long waves generated by a moving pressure disturbance J Fluid Mech, 1996, 5 : Wu J, et al Observation of a non - propagating hydrodynamic soliton Phys Rev L ett, 1984, 5 : Denardo B, et al Observation of a kink soliton on the surface of a liquid Phys Rev L ett, 1990, 64 : , (A), 199, (1) : Miao G, Wei R The effect of surface tension on nonpropagating hydrodynamic solitons Phys L ett ( A ), 1996, 0 : , 1998, 18, 1988 : (1) : Zhou X, et al The stability of a standing soliton In : Proc nd Int Conf of Nonlinear Mech, Beijing, Miles J W Parametrically excited solitary waves J Fluid Mech, 1984, 148 : Laedke E W, Spatschek K H On localized solutions in nonlinear Faraday resonance J Fluid Mech, 1991, : Yan J R, Mei Y P Interaction between two Wu s solitons Europhys L ett, 199, : 540 Chen X, Wei R Dynamic behaviour of a non - propagating soliton under a periodically modulated oscillation J Fluid Mech, 1994, 59 : Chen W Z, Wei R J, Wang B R Chaos in nonpropagating hydrodynamics solitons Phys Rev 1996, 5 : Chen W Z Experimental observation of self - localized structure in granular material Phys L ett ( A ), 1995, 196 : 15 6 Chen W Z, Wei R J, Wang B R Nonpropagating interface solitary waves in fluid Phys L ett ( A ), 1995, 08 : ,, 1988, 7 : Grimshaw R Evolution equations for weakly nonlinear long internal waves in a rotating fluid S tudies in A ppl M ath, 1985, 7 (1) : 14 FORCED SOL ITON IN FL UID MECHANICS Zhou Xianchu Institute of Mechanics, Chinese Academy of Sciences, Beijing , China Abstract Forced solitons in fluid mechanics representing by precursor solitons and nonpropagating solitons found in t he 1980s are briefly discussed in t his paper Production met hods, mechanical mod els, governing equations, the - state - of - art and the problems to be solved on these solitons are discussed Keywords forced soliton, precurcor soliton, nonpropagating soliton China Academic Journal Electronic Publishing House All rights reserved