Superposition solutions to the extended KdV equation for water surface waves

Nonlinear Dynamics, Nov 2017

The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher-order effects. Although this equation has only one conservation law, exact periodic and solitonic solutions exist. Khare and Saxena (Phys Lett A 377:2761–2765, 2013; J Math Phys 55:032701, 2014; J Math Phys 56:032104, 2015) demonstrated the possibility of generating new exact solutions by combining known ones for several fundamental equations (e.g., Korteweg–de Vries, nonlinear Schrödinger). Here we find that this construction can be repeated for higher-order, non-integrable extensions of these equations. Contrary to many statements in the literature, there seems to be no correlation between integrability and the number of nonlinear one variable wave solutions.

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Superposition solutions to the extended KdV equation for water surface waves

Superposition solutions to the extended KdV equation for water surface waves Piotr Rozmej 0 1 Anna Karczewska 0 1 Eryk Infeld 0 1 0 E. Infeld National Centre for Nuclear Research , Hoz ̇a 69, 00-681 Warszawa , Poland 1 A. Karczewska Faculty of Mathematics , Computer Science and Econometrics , University of Zielona Góra , Szafrana 4a, 65-246 Zielona Góra , Poland The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher-order effects. Although this equation has only one conservation law, exact periodic and solitonic solutions exist. Khare and Saxena (Phys Lett A 377:27612765, 2013; J Math Phys 55:032701, 2014; J Math Phys 56:032104, 2015) demonstrated the possibility of generating new exact solutions by combining known ones for several fundamental equations (e.g., Korteweg-de Vries, nonlinear Schrödinger). Here we find that this construction can be repeated for higher-order, nonintegrable extensions of these equations. Contrary to many statements in the literature, there seems to be no correlation between integrability and the number of nonlinear one variable wave solutions. Shallow water waves; Extended KdV equation; Analytic solutions; Nonlinear equations 1 Introduction A long time ago, Stokes opened the field of nonlinear hydrodynamics by showing that waves described by nonlinear models can be periodic [ 1 ]. Although several related results followed, it took half a century before the Korteweg–de Vries equation became widely known [ 2 ]. A more accurate equation system, Boussinesq, was formulated in 1871. It is also the theme of several recent papers [ 3,4 ]. Another direction research has gone in is including perpendicular dynamics in KdV, e.g., [5]. The KdV equation is one of the most successful physical equations. It consists of the mathematically simplest possible terms representing the interplay of nonlinearity and dispersion. This simplicity may be one of the reasons for success. Here we investigate this equation, improved as derived from the Euler inviscid and irrotational water equations. Just as for conventional KdV, two small parameters are assumed: wave amplitude/depth (a/H ) and depth/wavelength squared (H/ l)2. These dimensionless expansion constants are called α and β. We take the expansion one-order higher. The new terms will then be of second order. This procedure limits considerations to waves for which the two parameters are comparable. Unfortunately some authors tend to be careless about this limitation. The next approximation to Euler’s equations for long waves over a shallow riverbed is (η is the elevation above a flat surface divided by H ) 3 1 3 2 2 ηt + ηx + 2 α ηηx + 6 β η3x − 8 α η ηx + αβ 23 5 24 ηx η2x + 12 ηη3x + 31690 β2η5x = 0. In (1) subsequently we use low indexes for derivatives n ηnx ≡ ∂∂xηn . This second-order equation was called by Marchant and Smyth [ 6,7 ] the extended KdV. It was also derived in a different way in [8] and [ 9,10 ]. We call it KdV2. It is not integrable. However, by keeping the same terms but changing one numerical coefficient (specifically, replacing 2243 by 56 ) we can obtain an integrable equation [ 11,12 ]. Not only is KdV2 non-integrable, it only seems to have one conservation law (volume or mass) [ 13 ]. However, a simple derivation of adiabatically conserved quantities can be found in [ 14 ]. Recently, Khare and Saxena [ 15–17 ] demonstrated that for several nonlinear equations which admit solutions in terms of elliptic functions cn(x , m), dn(x , m) there exist solutions in terms of superpositions cn(x , m) ± √m dn(x , m). They also showed that KdV which admits solutions in terms of dn2(x , m) also admits solutions in terms of superpositions dn2(x , m) ± √m cn(x , m) dn(x , m). Since then we found analytic solutions to KdV2 in terms of cn2(x , m) [ 18,19 ], the results of Khare and Saxena [ 15–17 ] inspired us to look for solutions to KdV2 in similar form. 2 Exact periodic solutions for KdV2 First, we repeat shortly the results obtained by Khare and Saxena [ 15 ], but formulating them for KdV in a fixed frame, that is, for the equation 3 1 ηt + ηx + 2 α ηηx + 6 β η3x = 0. Assuming solution in the form η(x , t ) = A dn2[B(x − vt ), m] one finds 4 B2β A = 3 α 2 and v = 1 + 3 β B2(2 − m) (1) (2) (3) Next, the authors [ 15 ] showed that superpositions 1 η±(x , t ) = 2 A dn2[B(x − vt ), m] ± √m cn[B(x −vt ), m] dn[B(x −vt ), m] (4) are solutions to (2) with the same relation between A and B, but for a different velocity, v± = 1 + 61 β B2(5 − m). Now, we look for periodic nonlinear wave solutions of KdV2 (1). Introduce y := x − vt . Then η(x , t ) = η(y), ηt = − vηy and Eq. (1) takes the form of an ODE (1 − v)ηy + 23 α ηηy + 61 β η3y − 83 α2η2ηy + αβ + 31690 β2η5y = 0. 23 5 24 ηy η2y + 12 ηη3y 2.1 Single periodic function dn2 (5) (6) First, we recall some properties of the Jacobi elliptic functions (arguments are omitted) sn2 + cn2 = 1, dn2 + m sn2 = 1. Their derivatives are d sn d cn dy = cn dn, dy = − sn dn, dddyn = − m sn cn. (7) Assume a solution of (1) in the same form as KdV solution (3). Insertion of (3) into (5) yields A Bm 180 cn dn sn F0 + F2 cn2 + F4 cn4 = 0. (8) Equation (8) holds for arbitrary arguments when F0, F2, F4 vanish simultaneously. The explicit form of this set of equations is following F0 = 135α2 A2(m − 1)2 + 30α A(m − 1) × β B2(63m − 20) + 18 Equation (11) is equivalent to the [10, Eq. (26)] obtained for solitonic solutions to KdV2. Denoting z := BA2αβ one obtains from (11) two possible solutions z1 = 43 − √2305 152 < 0 and z2 = 43 + √2305 152 The case z = z1 leads to B2 < 0 and has to be rejected as in previous papers [ 10,18 ]. Then for z = z2 the amplitude A is A = 43 + √2305 B2β 3 α Despite the same form of solutions to KdV and KdV2, there is a fundamental difference. KdV only (9) (10) 2.2 Comparison to KdV solutions Is a solution of KdV2 much different from the KdV solution for the same m? In order to compare solutions of both equations, remember that the set of three Eqs. (9)–(11) fixes all A, B, v coefficients for KdV2 for given m. In the case of KdV, the equation analogous to (8) only imposes two conditions on three parameters. Therefore, one parameter, say amplitude A, can be chosen arbitrary. Then we compare coefficients of solutions to KdV2 and KdV choosing the same value of A, that is, AKdV2. Such comparison is displayed in Fig. 1 for α = β = 110 . It is clear that vKdV2 and vKdV are very similar. We have the following relations: for KdV B2 A = 43 βα , whereas for KdV2 B2 A = βα z2. Since z2 ≈ 0.6, BKdV/BKdV = 43z2 ≈ 1.12. The same relations hold between KdV2 and KdV coefficients for superposition solutions shown in Fig. 3. The above examples for the case α = β = 110 show that for somewhat small values of α the coefficients of KdV2 dn2 solutions are not much different from those of KdV. However, physically relevant exact solutions of dn2 to KdV2 can be found for much larger values of the 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 parameter α. In Fig. 2, the amplitude A(α, m) given by (15) is shown as a contour plot for the region α ∈ [0.01, 0.5], m ∈ [ 0, 1 ]. It is clear that reasonable amplitudes occur in wide regions of α and m values. The amplitudes become too big only for α → 0, but in such cases KdV works very well. 2.3 Superposition “ dn2 + √m cn dn” Now assume the periodic solution to be in the same form as the corresponding superposition solution of KdV [ 15 ] function η+(y) = 21 A dn2(B y, m) + √m cn(B y, m) dn(B y, m) , (17) where A, B, v are yet unknown constants. (m is the elliptic parameter.) We will need ηy = − 21 AB √m √m cn + dn 2 sn, η2y = 21 AB2 √m √m cn + dn 2 × η3y = 21 AB3 √m √m cn + dn 2 η5y = − 21 AB5 √m √m cn + dn 2 × sn m cn2 + 6√m cn dn + dn2 − 4m sn2 , (20) (21) (22) (23) (24) (25) (26) (27) (29) Equations (34) and (35) are equivalent and give the same condition as (11). Solving (34) with respect to B2, we obtain the same relations as in [10, Eq. (28)] and z2 = It is clear that z1 < 0 and z2 > 0. B has to be realvalued. This is possible for the case z = z1 if A < 0, and for z = z2 if A > 0. The value of z2 is the same as that found for the exact soliton solution in [10, Eq. (28)]. In general 1440(1 − v) + Aα(1 − m) [1080 − 135 Aα(1 − m)] This looks like a contradiction, but substitution z = z1 = (43 − √2305)/152 in both (42) and (43) gives the same result A1 = . For z = z2 = (43 + is A2 = This means that not only are Eqs. (34) and (35) equivalent, but also (32) and (33), as well. Therefore, Eqs. (32)–(35) supply only three independent conditions for the coefficients of KdV2 solutions in the form (17). Now, using z = z1 and A1 given by (44) we obtain from (39) vnum−(m) v1 = vden−(m) vnum+(m) v2 = vden+(m) , and with z = z2 and A2 given by (45) (39) (40) (41) (44) (45) (46) (47) where vnum∓(m) = 6 2912513 ∓ 58361√2305 m2 2.4 Discussion of mathematical solutions From a strictly mathematical point of view, we found two families of solutions determined by coefficients A, B, v as functions of the elliptic parameter m. There are two cases. 43 − √2305 Case 2 z = z2 = Then Case 1 z = z1 = 152 This case leads to B2 < 0 and has to be rejected as in previous papers [ 10,18 ]. 43 + √2305 As Bs Vs B1 V1 0 0.2 0.8 1 0.4 0.6 m and v2 is given by (47). Since m ∈ [ 0, 1 ], (m−5) < 0 then B2 is real. The solution in this case is 1 η2(x − v2t, m) = 2 A2 dn2(B2(x − v2t), m) +√m cn(B2(x − v2t), m) dn(B2(x − v2t), m) . (50) Coefficients A2, B2, v2 of superposition solutions (17) to KdV2 as functions of m are presented in Fig. 3 for α = β = 110 and compared to corresponding solutions to KdV. Here, similarly as in Fig. 1, we assume that AKdV = AKdV2. Physically relevant exact superposition solutions to KdV2 can be found for greater values of the parameter α than 110 . In Fig. 4, the amplitude A(α, m) given by (48) is shown as a contour plot for the region 0.0 0.1 0.2 0.3 0.4 0.5 α ∈ [0.01, 0.5], m ∈ [ 0, 1 ]. It is clear that reasonable amplitudes occur in wide regions of α and m values, similarly like in the case dn2. The amplitudes become too big only for α → 0, but in such cases KdV works very well. 2.5 Superposition 1 “ dn2 − √m cn dn” Now we check the alternative superposition “ dn2 − √m cn dn” (51) η−(y) = 21 A dn2(By, m) − √m cn(By, m) dn(By, m) . In this case, the derivatives are given by formulas similar to (18)–(21) with some signs altered. Analogous changes occur in formulas (23)–(29). Then (22) has a similar form like (30) 21 A B √m −√m cn + dn 2 sn × (F0 + Fcd cn dn +Fc2 cn2 + Fc3d cn3 dn + Fc4 cn4 = 0. (52) Equation (52) is valid for arbitrary arguments when all coefficients F0, Fcd , Fc2 , Fc3d , Fc4 vanish simultaneously. This gives us a set of equations for the coefficients v, A, B. Despite some changes in signs on the way to (52), this set is the same as for “ dn2 + √m cn dn” superposition (31)–(35). Then the coefficients A, B, v for superposition “ dn2 √m cn dn” are the same as for superposition “ dn2 −+ √m cn dn” given above. This property for KdV2 is the same as for KdV, see [ 15 ]. It follows from periodicity of the Jacobi elliptic functions. From cn(y + 2K (m), m) = − cn(y, m), dn(y + 2K (m), m) = dn(y, m) it follows that dn2(y + 2K (m), m) + = dn2(x , m) − (m) cn(x + 2K (m), m) dn(x + 2K (m), m) (m) cn(x , m) dn(x , m). (53) So both superpositions η+ (17) and η− (51) represent the same solution, but shifted by the period of the Jacobi elliptic functions. This property is well seen in Figs. 5, 6 and 7. 3 Examples Below, some examples of wave profiles for both KdV and KdV2 are presented. We know from Sect. 2 that for a given m, the coefficients A, B, v of KdV2 solutions are fixed. As we have already written, this is not the case for A, B, v of KdV solutions. So, there is one free parameter. In order to compare KdV2 solutions to those of KdV for identical m, we set AKdV = AKdV2. In Figs. 5, 6 and 7 below, KdV solutions of the forms (3), 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 2.5 2 1 1.5 0.5 0 0 2.5 2 1 1.5 0 0 0 2 4 8 10 12 6 x sup+ KdV sup+ KdV2 single KdV single KdV2 sup- KdV sup- KdV2 Fig. 5 Profiles of KdV and KdV2 waves for m = 0.1. (Color figure online) 8 x sup+ KdV sup+ KdV2 single KdV single KdV2 sup- KdV sup- KdV2 2 4 6 10 12 14 16 18 Fig. 6 Profiles of KdV and KdV2 waves for m = 0.9. (Color figure online) 15 x sup+ KdV sup+ KdV2 single KdV single KdV2 sup- KdV sup- KdV2 5 10 20 25 30 35 Fig. 7 Profiles of KdV and KdV2 waves for m = 0.99. (Color figure online) 1.4 1.3 (17) and (51) are drawn with solid red, green and blue lines, respectively. For KdV2 solutions, the same color convention is used, but with dashed lines. In all the presented cases, the parameters α = β = 0.1 were used. Comparison of wave profiles for different m suggests several observations. For small m, solutions given by the single formula (3) differ substantially from those given by superpositions (17) and (51). Note that (3) is equal to the sum of both superpositions and when m → 1 the distance between crests of η+ and η− increases to infinity (in the m = 1 limit). All three solutions converge to the same soliton. In order to check whether the obtained analytic solutions are really true solutions to KdV2 several numerical simulations were performed. In each of them, the numerical FDM code used with success in previous studies [ 9–11,14,18,19 ] was applied. Since the calculations concerned periodic solutions, the periodic boundary conditions were used with an x interval equal to the particular wavelength. In Figs. 5, 6 and 7 dashed lines display profiles of single dn2 (3) and superposition η+ and η− (4) solutions for three values of m = 0.1, 0.9 and 0.99. Below in Figs. 6, 7 and 8 six examples of time evolution for these solutions obtained in numerics are presented. Profiles of solutions at time instants t = 0, T /4, T /2, 3T /4 and T , where T = λ/v are displayed. Open symbols represent the profiles at t = T which overlap with those at t = 0 with numerical deviations less than 10−11. In all the presented examples, as well as all others not shown here, numerics confirmed a uniform motion and a fixed shape for the considered solutions (Figs. 8, 9, 10). 1.5 0.5 2.5 2 1 1.5 0.5 0 0 0 0 1 2 3 single KdV2 4 5 6 sup+ KdV2 7 8 9 Fig. 9 Same as in Fig. 8 but for m = 0.9. (Color figure online) 2 4 single KdV2 6 8 10 sup+ KdV2 12 14 Fig. 10 Same as in Fig. 8 but for m = 0.99. (Color figure online) 4 Conclusions The most important results of the paper can be summarized as follows. It is shown that several kinds of analytic solutions of KdV2 have the same forms as corresponding solutions to KdV but with different coefficients. 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Piotr Rozmej, Anna Karczewska, Eryk Infeld. Superposition solutions to the extended KdV equation for water surface waves, Nonlinear Dynamics, 2017, 1-9, DOI: 10.1007/s11071-017-3931-1