is large when compared to the undisturbed water depth h, and that the Stokes

Transcription

1 EXACT TRAVELING-WAVE SOLUTIONS TO BI-DIRECTIONAL WAVE EQUATIONS Min CHEN Department of Mathematics, Penn State University University Park, PA 1680 U.S.A. Abstract. In this paper, we present several systematic ways to nd exact travelingwave solutions of the systems t + ux +(u)x + auxxx, bxxt =0; ut + x + uux + cxxx, duxxt =0; where a; b; c and d are real constants. These systems, derived by Bona, Saut and Toland in [] for describing small-amplitude long waves in a water channel, are formally equivalent to the classical Boussinesq system (cf. [5]) and correct through rst order with regard to a small parameter characterizing the typical amplitude to depth ratio. Exact solutions for a large class of systems are presented. The existence of the exact traveling-wave solutions is in general extremely helpful in the theoretical and numerical study of the systems. 1. Introduction. We consider in this paper the model systems which describe two-way propagation of nonlinear dispersive waves in a water channel. Under the assumptions that the maximum deviation a of the free surface is small and a typical wave length is large when compared to the undisturbed water depth h, and that the Stokes number S = a =h is of order one, which means the eect of nonlinearity and dispersion is of the same order (S will be taken to be 1 in the rest of the paper for simplicity in notion), a restricted four-parameter family of systems was derived by Bona, Saut and Toland in [], having the form (1.1) t + u x +(u) x + au xxx, b xxt =0; u t + x + uu x + c xxx, du xxt =0; where x corresponds to distance along the channel (scaled by h) andt is the elapsed time scaled by (h=g) 1 where g denotes the acceleration of gravity, the variable (x; t) is the non-dimensional deviation of the water surface (scaled by h) from its Key words and phrases. water waves, Boussinesq system, exact solutions, traveling-waves. Typeset by AMS-TEX 1

2 undisturbed position and u(x; t) is the non-dimensional horizontal velocity (scaled by p gh) at a height h with 0 1above the bottom of the channel, and the real constants a, b, c and d satisfy (1.) a = 1 (, 1 ); c = 1 (1, ); b = 1 (, 1 )(1, ); d = 1 (1, )(1, ); where, are real numbers. These systems are formally equivalent and have the same formal status as the Kortweg-deVries equation for the uni-directional propagation of waves in a channel in the way that they are correct through rst order with regard to the small parameter = a=h. The reasons for the plethora of dierent, but formally equivalent Boussinesq system is owed to the fact that the lower-order relation can be used systematically to alter the higher-order terms without disturbing the formal level of approximation and to the considerable number of choices of dependent variables available (see [] in detail). Some interesting examples included in (1.1) are Whitham's system ( =0;=1;=0) (cf. [17], formula (1.101)): (1.) t + u x +(u) x, 1 6 u xxx =0; u t + x + uu x, 1 u xxt =0: Regularized Boussinesq system ( = ;=0;=0) (cf. []): (1.4) t + u x +(u) x, 1 6 xxt =0; u t + x + uu x, 1 6 u xxt =0: Coupled K-dV-regularized system ( = ;=1;=0): (1.5) t + u x +(u) x u xxx =0; u t + x + uu x, 1 6 u xxt =0: Boussinesq's original system ( = 1 ; arbitrary;=0) (cf. [5]): (1.6) t + u x +(u) x =0; u t + x + uu x, 1 u xxt =0: Coupled K-dV system ( = ;=1;=1): (1.7) t + u x +(u) x u xxx =0; u t + x + uu x xxx =0:

3 Coupled regularized- K-dV system ( = ;=0;=1): (1.8) t + u x +(u) x, 1 6 xxt =0; u t + x + uu x xxx =0: Integrable version of Boussinesq system ( =1;=1; arbitrary) (cf. [10]): (1.9) t + u x +(u) x + 1 u xxx =0; u t + x + uu x =0: Bona-Smith system ( =( 4, )=(, );=0;<0 arbitrary) (cf. [4]): (1.10) t + u x +(u) x, b xxt =0; u t + x + uu x + c xxx, bu xxt =0; where in the notation of (1.1) and (1.) b = 1, (, ) > 0 and c = (, ) < 0: In this paper, we will concentrate on nding exact traveling-wave solutions of (1.1). The existence of these solutions will be useful in several ways in the study of these model systems. In fact, one of the exact solution we found here for the regularized Boussinesq system (1.4) has been used in [] to demonstrate the convergence rate of a numerical algorithm. The structure of the paper is as follows. In section, we search for exact solutions ((x; t); u(x; t)) where (x; t) and u(x; t) are proportional to each other and they approach zero when x approaches 1. The solutions we found appear to have the form A sech, (x + x 0, C s t), where A,, x 0 and C s are constants. The explicit requirements on a; b; c; d (or on ; ; ) forsuch solutions to exist are presented. We then compare these solitary-wave solutions with those of the famous KdV equation which is a model equation describing uni-directional waves. In section, we search for exact traveling-wave solutions (; u) where u(x; t) is of the form u 1 + A sech, (x + x 0, C s t) and (x; t) is a function of u(x; t) and approaches a constant 1 at,1. It is shown that such exact solutions can be found by solving a system of nonlinear algebraic equations involving A;, C s, u 1 and 1. Section 4 is similar to section, but searching for the traveling-wave solutions where (x; t) is of the form 1 + A sech, (x + x 0, C s t).we conclude the paper in section 5.

4 . Solitary-wave solutions in the form of u(x; t) =u(x + x 0, C s t) and u(x; t) =B(x; t). Denoting = x + x 0, C s t with x 0 traveling-wave solution ((x; t); u(x; t)) as and C s being constants, one can write a (.1) (x; t)=() (x + x 0, C s t); u(x; t)=u() u(x + x 0, C s t): Substituting (.1) into (1.1), one nds (.), C s 0 + u 0 +(u) 0 + au bc s 000 =0;, C s u uu 0 + c dc s u 000 =0; where the derivatives are with respect to. Since we are searching for solitary-wave solutions, meaning that the solutions that are asymptotically small at large distance from their crest, so (();u())! 0as!1, system (.) can be integrated once to yield (.) (,C s + u) + bc s 00 =,u, au 00 ; + c 00 = C s u, 1 u, dc s u 00 : Suppose that () and u() are proportional to each other, namely u() = B() with B being a constant, one obtains (.4) (C s B, B ), (bc s B + ab ) 00 = B ; (C s B, ), (dc s B +c) 00 = B : In order for (.4) to have nontrivial solitary-wave solutions, it is necessary that the two equations are identical which implies (.5) B + C s B, =0; ab +(b, d)c s B, c =0: Above system is linear with respect to unknowns B and C s B and its solution depends on the values of a; b; c and d as follows, Case I: if a, b +d 6= 0, there is an unique solution B = (,b+c+d) a,b+d C s B =, B ; Case II: if a, b +d = 0 and a = c, there are innitely many solutions, which reads C s B =, B and B arbitrary; Case III: if a, b +d =0andc 6= a, there is no solution. In the cases that (.5) has a solution, taking derivative on one of the equations in (.4) and using (.5), one nds () satises (.6) (1, B ) 0, ((a, b)b +b) 000 =B 0 : The solution of (.6) can be easily obtained with the use of the following lemma which can be found in many textbooks (for example, see [11]). 4 and

5 Lemma 1. Let, be real constants, the equation 0 (), 000 () =() 0 () has a solitary-wave solution if > 0. Moreover, the solitary-wave solution is where 0 is an arbitrary constant. ()= sech 1 r ( + 0) In summary, one concludes that the conditions for (1.1) to have solitary-wave solutions which is of the form u() = B() are the following, (i) a; b; c and d are in the case I or II; (ii) the solution B of (.5) is non-negative and satises (.7) (B, 1)((b, a)b, b) > 0: Notice that B = 0 is not a solution of (.5), so the solitary-wave solutions can be expressed explicitly, s (x; t)= (1!, B ) 1 (1, B sech ) B (a, b)b +b (x + x 0, C s t) ; u(x; t) =B(x; t): Above solutions can be written in a more familiar form. Let one sees that 0 = (1, B ) B ; (.8) (x; t)= 0 sech, (x + x 0, C s t) ; r u(x; t)= sech, (x + x 0, C s t) ; where (.9) C s = + 0 p ( + 0 ) ; = 1 s 0 (a + b)+b 0 : After simple calculations, one can prove the following theorem which corresponds to the case I. 5

6 Theorem 1. Suppose a, b +d 6= 0and let p =,b+c+d, then the system (1.1) a,b+d hasapair of solitary-wave solution in the form of u(x; t) =B(x; t) if and only if p>0 and (p, 1 )((b, a)p, b) > 0. Moreover, the exact solitary-wave solution is in the form of (.8){(.9) with 0 = (1,p) p. Similarly, the situation corresponding to case II can be translated as that for a, b +d =0anda = c, the solitary-wave solutions exist for any B satises (B, 1)(dB, b) > 0andB > 0: More specically, the following theorem holds when one denotes the closed (or open) interval between and by[; ] (or (; )). For example, if =and =1,weuse[; 1] to denote the closed interval [1; ]. Theorem. (i) If a = b = c>0, d =0,thereare solitary-wave solutions in the form of (.8){(.9) for any 0 < 0 < +1. (ii) If a = b = c<0, d =0,thereare solitary-wave solutions in the form of (.8){(.9) for any, 0 < 0. (iii) If a, b +d =0, a = c, d>0, thereare solitary-wave solutions in the form of (.8){(.9) for any 0 >, and = [1; b 0 + d ]. (iv) If a, b +d =0, a = c, d<0, thereare solitary-wave solutions in the form of (.8){(.9) for any 0 >, and 0 + [1; b d ]. It is worth to note that that the phase velocity C s for these exact solitary-wave solutions depends on 0, the amplitude of the wave, but not on the constants a; b; c and d. The spread of the wave which is represented by does depend on the individual system. We now compare the phase velocity C s of (.8){(.9) with the phase velocity of the full Euler equations (.10) c =1+ 1 0, ; which is obtained by systematic expansion (cf. [5] and [7]) and is justied in some sense by Craig in [6]. Subtracting c from C s yields C s, c =0: higher-order terms in 0: If one also compare the phase velocity of the solitary-wave solutions of the KdV equation, which isc k =1+ 1 0, with c, one nds C K, c =0: higher-order terms in 0 : 6

7 Since the leading order in C s, c is smaller than that in C k, c, one expects that the exact solitary-wave solutions obtained in Theorem 1 and for systems in (1.1) are better approximations for small-amplitude waves. This fact is also observed numerically for other systems in (1.1) which wewere not able to nd exact solutions of the form (.8)-(.9) (cf. []). Comparisons with laboratory data also indicate that the Boussinesq systems capture far more accurately the general drift of amplitude-speed than does the KdV-equation even for somewhat larger amplitude waves. Theorem 1 and also demonstrate that although all the systems in (1.1)-(1.) are formally equivalent, they are dierent. Depending on the values of a; b; c and d, the system admits dierent number of sech -solitary-wave solutions. In the cases which satisfy the conditions stated in Theorem 1, there exists only one pair of sech -solitary-wave solutions (one travels to the left and one travels to the right). In the cases which satisfy the conditions stated in Theorem, many interesting situations occur. For instance the sech -solitary-wave solution can exists for 0 in a certain range and the waves can be in elevation or depression depending on the sign of 0. This is dierent from the result for KdV and BBM equation (cf. [1]) which admit an one-parameter family of solitary-wave solutions with elevation only (cf. [1]). The spread of the solitary-wave solutions are also dierent depending on the individual system (c.f. (.9)). Since the parameters a, b, c, d in (1.1) as model systems for water waves are not free, but form a restricted four-parameter family with the restrictions (1.), one can prove after tedious calculation that Corollary. Let R 1 =1+ 1 (, 1 )(, 1) ; R 1, =, (, 1 )(1, ) ; 1, the system (1.1) has nontrivial solitary-wave solutions, which are in the form of 7

8 (.8){(.9), if one of the following is satised: (1) 0 < 1 ; is arbitrary, <minf; R 1g; 8 1 () < < 7 >< < 1; <<R 1; 9 ; 1 >: ; >R ; > 1 ; >maxf; R g; () = 7 ; is arbitrary, =; >< < 1; R 1 <<; (4) 9 < < 1; 1 >: ; <R ; > 1; <minf; R g: We now applying Theorems 1 and to some of the systems included in (1.1). Example 1. Applying Theorem 1 to system (Whitham) (a =, 1 ;b= c =0;d= 6 ), one recovers the exact solitary-wave solution 1 (.11) (x; t)=, sech r 5 sech u(x; t) = 7 p 7, x + x0 1 p 15 t ; 1 p, 7 x + x0 p 1 t ; 15 obtained in [15] by using a homogeneous balance method. Example. Let = 4 ;=,1;=,4, the resulting system is 5 t + u x +(u) x, 7 0 u xxx, 7 15 xxt =0; u t + x + uu x, 5 xxx, 1 u xxt =0: According to Theorem 1, this system has an exact solitary-wave solution (x; t)= r! 1 5, 8 sech x + x0 5p 7 6 t ; u(x; t)= 1 p 1 sech r 5, x + x0 5p 7 6 t Example. For Bona-Smith system (1.10), p = b+c : Applying Theorem 1, one b nds the solitary-wave solutions in the form of (.8){(.9) exist in the cases (i) b>0 and c>1; (ii) b<0;c>0andb +c<0: Since,(b +c) 0 = (b + c) ; it is easy to check that 0 < 0 in both cases. Combining the results in [], one can nd systems which admit exact solitarywave solutions and have also other properties such as the linearized system is L - well-posed. Recalling from [], one has 8! :

9 Proposition. The linearized system of (1.1) is L -well-posed if and only if one of the following condition hold. a<0 ; c<0 ; b>0 ; d>0; a = c ; b<0 ; d>0; b = d ; a<0 ; c>0; a = c ; b = d; a =0 ; b =0 ; c<0 ; d>0; c =0 ; d, 0 ; a<0 ; b>0; a =,b ; c<0 ; d>0; a =,b ; d =,c; d =,c ; b>0 ; a<0. Applying the proposition to the systems in Example 1 and, one nds that the linearized system from Example 1 is not L -well-posed while the linearized system from Example is L -well-posed. One might ask at this point that which system in (1.1) should be chosen in a concrete modeling situation. This issue is addressed both in paper [] and []. The primary criteria for the choice include but limited to that the system is mathematically well-posed, preserve energy or other physical quantities which was conserved by the full Euler equations, has stable solitarywave solutions and suitable for numerical simulations on a well-posed initial- and boundary-value problem. Due to the restrictions posed on the form of the exact solutions we are searching, it is clear from Theorem 1 and that there are many systems which do not admit solutions of the form (.8){(.9), including the systems listed in (1.4){(1.9). We therefore continuous our search in the next section.. Traveling-wave solutions with u(x; t) =u 1 + A sech, (x + x 0, C s t). In this section, we use a more general approach tosearch for more exact solutions. Denoting again that = x+x 0,C s t, the traveling-wave solution, u();() we search in this section are the ones that u() is of the form u 1 + A sech () where u 1 ;A;are constants and () tends to 1 as tends to,1. Notice that, u();() =(; ) are solutions of (1.1) for any constants and, one can restrict >0 and search only for non-constant solutions. The exact solutions found in section are in this form and can be recovered by using the method presented in this section. Solutions with non-zero 1 or u 1 are not physically relevant, but will be useful in the theoretical and numerical study of the corresponding systems. One of the exact solutions found in this section for the regularized Boussinesq system 9

10 (1.4) has been used to verify the convergence rate of a proposed numerical schemes (cf []). Starting from (.) and introducing functions h() =(), 1 and v() = u(), u 1, one obtains (.1), C s h 0 + v 0 +(vh) v 0 + u 1 h 0 + av bc s h 000 =0;, C s v 0 + h 0 + vv 0 + u 1 v 0 + ch dc s v 000 =0: Since h() and v() tends to zero as tends to,1, the system can be integrated once to obtain (.) h(,c s + v + u 1 )+bc s h 00 =,v, 1 v, av 00 ; h + ch 00 = C s v, 1 v, u 1 v, dc s v 00 : Assuming c is not zero and solving h and h 00 yields (.) h = g 1 (v)=f(v) and h 00 = g (v)=f(v); where (.4) (.5) f(v) =c(,c s + v + u 1 ), bc s ; g 1 (v) =c(,v, av 00, 1 v), bc s (C s v, 1 v, dc s v 00, u 1 v); g (v) =v + av v +(,C s + v + u 1 )(C s v, 1 v, dc s v 00, u 1 v): Dierentiating the rst equation in (.) twice with respect to and using the second equation, one obtains a 4th-order ordinary dierential equation with dependent variable v() f g = g 00 1 f, g 1 f 00 f, g 0 1 ff0 +g 1 (f 0 ) ; which is of the form after substituting f, g 1 and g into the equation (a 1 v + a v + a )v (a 4 v + a 5 )v 0 v 000 +(a 6 v + a 7 )(v 00 ) (.6) + a 8 (v 0 ) v 00 +(a 9 v + a 10 v + a 11 v + a 1 )v 00 + a 1 (v 0 ) + a 14 v 5 + a 15 v 4 + a 16 v + a 17 v + a 18 v =0; where a i ;i =1; ; 18 depend on a, b, c, d, C s, 1 and u 1. With the help of 10

11 Mathematica, one sees a 1 = bc dcs, ac ; a =c(,ac + bdcs ),,(b + c)c s + cu 1 ; a =(,ac + bdcs)((b + c)c s, cu 1 ) ; a 4 =,bc dcs +ac ; a 5 =c(,ac + bdcs)((b + c)c s, cu 1 ); a 6 = a 4 =; a 7 = a 5 =; a 8 =,a 4 ; a 9 = c (d + b=)c s ; (.7) a 10 =,c(cd +bd +bc=+b =)C s, ac +c C s u 1 (b=+d); a 11 =(,(b + c)c s + cu 1 )(,ac, c (1 + 1 ), (b +bc + bd +cd)c s + c(b +d)c su 1 ); a 1 =(,(b + c)c s + cu 1 ) (,a, c(1 + 1 ), (b + d)c s (,C s + u 1 )); a 1 =(,(b + c)c s + cu 1 )(c (1 + 1 ), (b, c)bc s, bcc s u 1 ); a 14 = c =; a 15 =,bcc s, 5c =C s +5c u 1 =; a 16 =,c (1 + 1 )+(b =+4bc +9c =)C s, c(4b +9c)C s u 1 +9c u 1=; a 17 =((b + c)c s, cu 1 )(4c(1 + 1 ), (b +7c)C s +(b +14c)C s u 1, 7cu 1))=; a 18 =(,(b + c)c s + cu 1 ) (,1, 1 +(C s, u 1 ) ): We therefore established the fact that in order to nd a traveling-wave solution of (1.1), it is suce to nd a solution of the ordinary dierential equation (.6). Theorem. For given a; b; c 6= 0;d;C s, u 1 and 1, any solution v() of (.6) will provide a traveling-wave solution u(x; t)=u 1 + v(), (x; t)= 1 + g 1(v()) f (v()), where = x + x 0, C s t, f and g 1 are dened in(.4) and (.5). On the other hand, any traveling-wave solution ((x; t); u(x;t)) ((); u()) of system (1.1) with c 6= 0 which approaches constants ( 1 ;u 1 ) as approaches,1 has the property that u(), u 1 satises (.6). Instead of solving v() from (.6) directly, which is very dicult if it is not impossible, the technique used by Kichennassamy and Olver in [9] for a single 5th-order equation is adopted here. In the present case, the ordinary dierential equation has coecients depending on C s ;u 1 and 1 which are part of the unknowns. Assuming that v() can be reconstructed as the solution of a simple rst-order ordinary dierential equation (.8) w(v) =(v 0 ) : 11

12 Once the function w(v) is known, v() can be solved by a simple quadrature: Z v ds p w(s) = + C: Examples of solutions that have this form are the soliton and cnoidal wave solutions of the KDV equation [17] where the function w(v) is a cubic polynomial. Solutions corresponding to w(v) in other forms can be found in [18]. Using (.8), one nds that for v 0 6=0 (v 0 ) = w; v 00 = 1 w0 ; (.10) v 0 v 000 = 1 ww00 ; v 0000 = 1 ww w0 w 00 ; where the primes on w indicate derivatives with respect to v. Substituting above relationships into (.6), one nds w must satisfy a third-order ordinary dierential equation (a 1 v + a v + a )( 1 ww w0 w 00 )+ 1 (a 4v + a 5 )ww 00 (.11) (a 6v + a 7 )(w 0 ) + 1 a 8ww (a 9v + a 10 v + a 11 v + a 1 )w 0 + a 1 w + a 14 v 5 + a 15 v 4 + a 16 v + a 17 v + a 18 v =0: For a solitary-wave solution of the form (.1) v()=a sech () ; >0; the corresponding function w(v) must be a cubic polynomial: w(v)=4 (v, 1 A v ) v + v ; where =4 > 0 and =, 4. Substituting w(v) into (.11), the left-hand side A become a degree-ve homogeneous polynomial in v. In order for v to be a nontrivial solution, all the coecients have tobezerowhich yields that,, C s, u 1 and 1 have to satisfy the following algebraic equations: a 18 + a 1 + a =0; a 17 +(a 11 + a 1 ) +(a + a 5 + a 7 ) +a 1 =+15a ==0; a 16 + a 10 +(a 1 + a 4 + a 6 + a 8 ) +(a 11 =+a 1 ) (.1) +(15a =+4a 5 +a 7 ) +15a ==0; a 15 + a 9 +a 10 =+(15a 1 =+4a 4 a 6 +5a 8 =) + (15a =+a 5 +9a 7 =4) =0; 4a 14 +6a 9 +(0a 1 +1a 4 +9a 6 +6a 8 ) =0; 1

13 which is the ultimate equations one has to solve to nd solutions of the form (.1). In the cases that c = 0, one sees from the second equation of (.) that h = C s v, 1 v, u 1 v, dc s v 00 : Substituting h into the rst equation of (.), one again obtains an ordinary differential equation on v. The same technique used on (.6) can then be used again. We therefore can prove the following theorem. Theorem 4. For a specied system, that is for a given a; b; c and d, let a i ;i = 1; ; 18 be asin(.7), (i) if c 6= 0and ( 0;;C s ;u 1 ; 1 ) is a solution of (.1), or (ii) if c =0and ( 0;;C s ;u 1 ; 1 ) is a solution of (a 18 + a 1 + a )=b =0; (.14) (a 17 +(a 11 + a 1 ) +a 1 +15a )=b =0; (a 16 +(a 11 +a 1 ) +15a )=b =0; then system (1.1) has a traveling-wave solution which reads (.15) u(x; t) u() =u 1 + v(); 1 + g 1 (v())=f(v()); if c 6= 0 (x; t) ()= 1 + C s v, 1 v, dc s v 00, u 1 v; if c =0 where v() =, sech, 1 p, = x + x0, C s t, f and g 1 are denedin(.4) and (.5). We nowpresent the exact solutions found for the examples listed in the Introduction, except for the integrable version of Boussinesq system (1.9) which does not possess solutions in the form of (.15). Since (.1) and (.14) are systems of nonlinear algebraic equations on ; ; C s, u 1 and 1, one can solve them with the help of Mathematica. The solutions we found may only be a part of the whole set of solutions. The equalities sech (x)(cosh(x)+1)=; sech 4 (x)(cosh(4x) + 4 cosh(x)+)=8 are used to simplifying the solutions. We will also give examples of the exact solutions where 1 = 0 and (or) u 1 =0. The notation = x + x 0, C s t is used where x 0 and C s are real constants. When it is not specied, x 0 and C s are arbitrary constants and is a non-negative constant. 1

14 Example 4. More exact solutions of Whitham's system (a =,1=6; b = 0;c = 0; d=1=). Substituting a; b; c and d into (.14) and solving the nonlinear algebraic system, one nds u 1 = 1 6C s + C s, 1 C s; 1 =,1+ 1 6C s which leads to the exact solutions u(x; t)=u C p s sech ; (x; t)= 1, p sech : + 1 ; =, C s ; Setting 1 =0andu 1 = 0 yields C s = 1 15 and = 7, which recovers the exact solution (.11) found in Example 1. Example 5. Exact solutions of the regularized Boussinesq system (a = 0;b = 1=6;c=0;d=1=6). Using Theorem 4, one nds two sets of exact traveling-wave solutions u(x; t) =C s (1, 1 6 )+ 1 1 C p s sech ; (x; t)=,1; and u(x; t)=c s (1, 5 18 )+ 5C s 1 6 sech (x; t)=, C s C s p ; 1 p 1 p sech, sech 4 : Setting =18=5 andc s = 5 4, one nds 1 = u 1 = 0, which leads to the exact traveling-wave solution u(x; t) = 15 sech p (x + x t) ; (x; t)= 15 4 sech p 10 (x + x 0 5 t), sech 4 p 10 (x + x 0 5 t) ; which we used in [] to demonstrate numerically the rate of the convergence of a numerical method. Example 6. Exact solutions of Coupled K-dV-regularized system (b = c = 0;a = d =1=6). 14

15 Theorem 4 yields u(x; t) =, 1 C s + C s, 1 6 C s + 1 C s sech (x; t)=,1+ 1 4C s where setting = 6 and C s = =0 r u(x; t) =, sech 1 1 p : p ; gives a pair of exact traveling-wave solutions with r sech (x; t)= sech r (x + x 0 1 p 6 t) r! (x + x 0 p 1 t) ; 6! : Example 7. Exact solutions of Boussinesq's original system (a = 0;b = 0;c = 0; d= 1 6 ). One can easily verify using Theorem 4 that u(x; t) =(1, 1 6 )C s C p s sech ; (x; t)=,1; are exact solutions. Example 8. Exact solutions of Coupled K-dV system (a = 1=6; b = 0;c = 1=6; d=0). Thanks to Theorem 4 again, one nds the exact solutions u(x; t) = p ( )+C s 1 (x; t)=, 1 ( )+1 4 sech 1 1 p ; p : p sech Setting = 6 and C s = p, one nds u(x; t) =p r sech ( (x + x 0 p t); (x; t)=,1+ r sech ( (x + x 0 p t): Example 9. Exact solutions of Coupled regularized-k-dv system (a = 0;b = 1=6; c=1=6; d= 0). 15

16 Using Theorem 4 yields the exact solutions u(x; t)=c s ( 1, 1 1 )+1 4 C s sech 1 p ; (x; t)=,1+c s ( 1 4, 1 4 )+1 8 C s sech 1 p : Setting = 6(C s,4) C s for jc s j, one obtains u(x; t)= 1 C s + C s (C s, 4)sech (x; t)= 4 (C s, 4)sech s, 6 C s s, 6! : C s! ; Example 10. Exact solutions to one of the Bona-Smith systems (a =0;b= d = 1 ;c=, 1 ). Substituting a; b; c and d into (.1) and using Theorem 4, one nds twosetsof the exact solutions, which are (.16) u(x; t) =(1, 1 1 )C s + C s sech p ; (x; t)=,1; and (.17) u(x; t) = 1 C s(1, 1 )+ 1 1 C p s sech ; (x; t)=, C s (1, 1 )+ 1 1 p 4 C s sech : Setting = (C s,4) C s for jc s j in (.17), one obtains u(x; t)= 1 C s + C s (C s, 4)sech (x; t)= (C s, 4) p sech 4 p p! C s, 4 ; C s! p C s, 4 C s ) which recovers the exact solution found by Bona (cf. [14]). For any systems with a = 0 and d>0, which includes the Boussinesq's original system and the Bona-Smith system in Example 7 and 10, the following theorem provides a one-parameter family of exact traveling-wave solutions. 16 ;

17 Theorem 5. If a =0,then 1 u(x; t)=(1, d)c s +dc s sech p (x + x0, C s t) ; (x; t)=,1; are exact solutions, where x 0 and C s are arbitrary constants and 0. Proof. Substituting a =0and(x; t)=,1, (i.e. h = 0 and 1 =,1), into (.), the rst equation is true for any u (i.e. for any v and u 1 ) and the second equation becomes after being integrated once (C s, u 1 )v, 1 v, dc s v 00 =0: The same technique used on (.6) can then be used (or applying the Lemma 1 directly) to nd u 1 =(1, d)c s and A =dc s. The theorem therefore holds. A similar procedure can be performed to nd solutions (u; ) where is of the form 1 +Asech (). We will show in next section that such technique will enable us to nd exact solutions for the integrable version of Boussinesq system (1.9). 4. Traveling-wave solutions with (x; t)= 1 + Asech ((x + x 0, C s t)). Instead of searching exact traveling-wave solutions ((x; t); u(x;t)) where u(x; t) has the form u 1 + Asech ((x + x 0, C s t)), we now search for the exact solutions where (x; t) has the form 1 + Asech ((x + x 0, C s t)). Similar to that in section, one needs rst to nd an ordinary dierential equation for (). In the cases that a 6= 0, multiplying the rst equation in (.) by dc s and subtracting a times the second equation, one obtains a quadratic equation of v, where, 1 av + y(h)v + z(h) =0; y(h) =dc s (1 + h + 1 )+ac s, au 1 ; z(h) =,dc s h + dc su 1 h + bdc s h00, ah, ach 00 : Denoting g(h) =y(h) +az(h) and solving for v, one nds (4.1) v = y(h) a 1 a g(h) 1 ; which is then being substituted into (.) to yield (1 + h + 1 ) y(h) a, C sh + u 1 h + bc s h 00 + dc s h 00 =, 1 a (1 + h + 1)g(h) g(h),= g 0 (h), 1 g(h),1= g 00 (h) : 17

18 Squaring both sides of the equation and multiplying g(h), one nds a 4th-order ordinary dierential equation for h(), which reads (4.) where g(h) f(1 + h + 1 ) y(h) a +(u 1, C s )h +(b + d)c s h 00 g = f, 1 a (1 + h + 1)g(h) g0 (h), 1 g(h)g00 (h)g g 0 (h) =dc s y(h)h 0 +afd(,c s + u 1 )C s h 0 + bdc s h 000, ah 0, ach 000 g; g 00 (h) =dc s y(h)h 00 +(dc s h 0 ) +afd(,c s + u 1 )C s h 00 + bdc sh 0000, ah 00, ach 0000 g: Hence one can obtain a traveling-wave solution by solving the ordinary dierential equation (4.). where In the cases that a = 0, one nds from the rst equation of (.) that v(h) =, ~z(h) ~y(h) ~y(h) =1+h + 1 ; ~z(h) =,C s h + u 1 h + bc s h 00 : Substituting v into the second equation of (.) and multiplying ~y(h), one obtains again a 4th-order ordinary dierential equation on h() (4.) (C s, u 1 )~z(h)~y(h) +(h + ch 00 )~y(h) + 1 ~z(h) ~y(h), dc s (h 00 + bh 0000 )~y(h) +dc sh 0 ~y(h)(,h + bh 000 ), dc s ~z(h)h 00 =0: The following theorem provides the relationship between the solutions of ordinary dierential equations (4.) or (4.) and the traveling-wave solution of (1.1). Theorem 6. For given (a; b; c; d; C s, u 1, 1 ), any solution h() of (4.) (or (4.) if a =0)willprovide a traveling-wave solution (x; t) = 1 + h(), u(x; t) = u 1 + v(), where v() is dened by(4.1) (or v() =, ~z(h()) ~y(h()) if a = 0). On the other hand, any traveling-wave solution ((x; t); u(x; t)) ((); u()) of (1.1) which approaches constants (u 1 ; 1 ) as approaches,1 has the property that (), 1 satises (4.) (or (4.) if a =0). Similar technique can be employed to nd the solution h of (4.) (or (4.)) in the form of Asech (). Setting =4 ; =,4 =A, and using the formulas similar to (.10), one nds that (4.4) (h 0 ) = h + h ; h 00 = h + h ; h 0 h 000 = h +4h + h 4 ; h 0000 = h + 15 h + 15 h : 18

19 Substituting these into ordinary dierential equation (4.) (or (4.)), one obtains a rather complicated homogeneous polynomial equation of degree ten (or degree ve if a = 0). For the equation to have a nontrivial solution, all the coecients have to be zero which leads to a system of nonlinear algebraic equations involving a; b; c; d;u 1 ; 1 ;; and C s. A solution of the algebraic system leads to a solution of the ordinary dierential equation. Theorem 6 can then be used to nd a travelingwave solution. Example 11. Exact solutions of Integrable version of Boussinesq system (b = c = d = 0). With the procedure described above, one nds the exact solutions u() =C s p 1 p,a sech ; ()=, a a p sech : if a<0; and u() =C s p 1 p a tanh ; ()=, a p sech : if a>0 As an example, one solution for a = 1 (setting = C s )is (x; t)=,1+ 1 C s sech Cs (x + x 0, C s t) ; Cs u(x; t) =C s 1 tanh (x + x 0, C s t) : Remark. The system with a = 1 has been studied using dierent methods. For example, a homogeneous balance method was used in [15] and [16]. In addition, a rational solution was found by Sach using Hirota form (cf. [1]). For the system with a =1=, a solution of the form u() =Asech =(1 + Bsech )was found using Jacobin Cosine elliptic function (cf. [10]). 5. Conclusion. In this paper we have showed that in order to nd the exact traveling-wave solutions of a given system of partial dierential equations of the form (1.1), it is sucient to nd a solution of a nonlinear fourth-order ordinary dierential equation. In consequence, any method for nding exact solutions of ordinary dierential equation can be used to generate exact traveling-wave solutions of (1.1). For instance, one can use Weierstrass elliptic function to construct periodic wave solutions 19

20 [8], inverse scattering theory to nd N-soliton solutions if they exist, and use certain ansatz equations, for instance the ones in [18], to nd traveling wave solutions in some prescribed form. We have found in this paper all the solutions ((x; t);u(x; t)) where either u(x; t) is of the form u 1 +A sech ((x+x 0,C s t)) or (x; t) is of the form 1 +A sech ((x+ x 0, C s t)) for each system listed in the Introduction. The method presented here can also be used for the systems ~ t +(u~) x + au xxx, b~ xxt =0; u t +~ x + uu x + c~ xxx, du xxt =0; by observing that under the transformation ~(x; t) = (x; t) + 1 these systems become (1.1). Acknowledgments. This work was supported in part by NSF grants DMS and DMS References [1] T. B. Benjamin, J. L. Bona & J. J. Mahony, \Model equations for long waves in nonlinear dispersive systems," Phil. Trans. Royal Soc. London A 7 (197), 47{78. [] J. Bona & M. Chen, \A Boussinesq system for two-way propagation of nonlinear dispersive waves," Submitted to Physica D. (1996). [] J. L. Bona, J.-C. Saut & J. F. Toland, \Boussinesq equations for small-amplitude long wavelength water waves," preprint (1997). [4] J. L. Bona & R. Smith, \A model for the two-way propagation of water waves in a channel," Math. Proc. Cambridge Phil. Soc. 79 (1976), 167{18. [5] J. Boussinesq, \Theorie de l'intumescence liquide appelee onde solitaire ou de translation se propageant dans un canal rectangulaire," Comptes Rendus de l'acadmie de Sciences 7 (1871), 755{759. [6] W. Craig, \An existence theory for water waves, and Boussinesq and Kortewegde Vries scaling limits," Comm. in Partial Dierential Equations 10 (1985), 787{ 100. [7] J. D. Fenton, \A ninth-order solution for the solitary wave," J. Fluid Mech. 5 (197), 57{71. 0

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