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 \(\alpha \) and \(\beta \). 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 (\(\eta \) is the elevation above a flat surface divided by H)

$$\begin{aligned} \eta _t&+ \eta _x + \frac{3}{2} \alpha \,\eta \eta _x+ \frac{1}{6}\beta \, \eta _{3x} -\frac{3}{8}\alpha ^2\eta ^2\eta _x \nonumber \\&+ \alpha \beta \,\left( \frac{23}{24}\eta _x\eta _{2x}+\frac{5}{12}\eta \eta _{3x} \right) +\frac{19}{360}\beta ^2\eta _{5x} =0 . \end{aligned}$$
(1)

In (1) subsequently we use low indexes for derivatives \(\left( \eta _{nx}\equiv \frac{\partial ^n \eta }{\partial x^n} \right) \). 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 \(\frac{23}{24}\) by \(\frac{5}{6}\)) 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 [15,16,17] demonstrated that for several nonlinear equations which admit solutions in terms of elliptic functions \(\,\mathsf{cn}(x,m),\,\mathsf{dn}(x,m)\) there exist solutions in terms of superpositions \(\,\mathsf{cn}(x,m)\pm \sqrt{m}\,\mathsf{dn}(x,m)\). They also showed that KdV which admits solutions in terms of \(\,\mathsf{dn}^2(x,m)\) also admits solutions in terms of superpositions \(\,\mathsf{dn}^2(x,m)\pm \sqrt{m}\,\mathsf{cn}(x,m)\,\mathsf{dn}(x,m)\). Since then we found analytic solutions to KdV2 in terms of \(\,\mathsf{cn}^2(x,m)\) [18, 19], the results of Khare and Saxena [15,16,17] 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

$$\begin{aligned} \eta _t + \eta _x + \frac{3}{2} \alpha \,\eta \eta _x+ \frac{1}{6}\beta \, \eta _{3x} =0 . \end{aligned}$$
(2)

Assuming solution in the form

$$\begin{aligned} \eta (x,t) = A\,\mathsf{dn}^2 [B(x-vt),m] \end{aligned}$$
(3)

one finds

$$\begin{aligned} A= & {} \frac{4}{3}\,\frac{ B^2 \beta }{\alpha }\quad \text{ and } \quad v=1+\frac{2}{3}\beta B^2(2-m)\\&= 1+\frac{\alpha }{2}A(2-m). \end{aligned}$$

Next, the authors [15] showed that superpositions

$$\begin{aligned} \eta _{\pm }(x,t)&= \frac{1}{2} A \left( \,\mathsf{dn}^2 [B(x-vt),m] \right. \nonumber \\&\left. \quad \pm \, \sqrt{m} \,\mathsf{cn}[B(x-vt),m]\,\mathsf{dn}[B(x-vt),m] \right) \end{aligned}$$
(4)

are solutions to (2) with the same relation between A and B, but for a different velocity, \(v_{\pm } =1+\frac{1}{6}\beta B^2(5-m)\).

Now, we look for periodic nonlinear wave solutions of KdV2 (1). Introduce \(y:=x-v t\). Then \(\eta (x,t) =\eta (y)\), \(\eta _t=-\,v\eta _y\)  and Eq. (1) takes the form of an ODE

$$\begin{aligned}&(1-v) \eta _y + \frac{3}{2} \alpha \,\eta \eta _y+ \frac{1}{6}\beta \, \eta _{3y} -\frac{3}{8}\alpha ^2\eta ^2\eta _y \nonumber \\&\quad + \alpha \beta \,\left( \frac{23}{24}\eta _y\eta _{2y} +\frac{5}{12}\eta \eta _{3y} \right) \nonumber \\&\quad +\frac{19}{360}\beta ^2\eta _{5y} =0 . \end{aligned}$$
(5)

2.1 Single periodic function \(\,\mathsf{dn}^2\)

First, we recall some properties of the Jacobi elliptic functions (arguments are omitted)

$$\begin{aligned} \,\mathsf{sn}^2+\,\mathsf{cn}^2 =1, \quad \,\mathsf{dn}^2+m\,\mathsf{sn}^2 = 1. \end{aligned}$$
(6)

Their derivatives are

$$\begin{aligned}&\frac{\text {d}\,\,\mathsf{sn}}{\text {d}y} = \,\mathsf{cn}\,\,\mathsf{dn},\quad \frac{\text {d}\,\,\mathsf{cn}}{\text {d}y} = -\, \,\mathsf{sn}\,\,\mathsf{dn},\nonumber \\&\frac{\text {d}\,\,\mathsf{dn}}{\text {d}y} = -\,m \,\,\mathsf{sn}\,\,\mathsf{cn}. \end{aligned}$$
(7)

Assume a solution of (1) in the same form as KdV solution (3). Insertion of (3) into (5) yields

$$\begin{aligned} \frac{ABm}{180} \,\mathsf{cn}\,\mathsf{dn}\,\mathsf{sn}\left( F_0+F_2 \,\mathsf{cn}^2 + F_4 \,\mathsf{cn}^4 \right) =0. \end{aligned}$$
(8)

Equation (8) holds for arbitrary arguments when \(F_0,F_2,F_4\) vanish simultaneously. The explicit form of this set of equations is following

$$\begin{aligned} F_0 =\,&135 \alpha ^2 A^2 (m-1)^2+30 \alpha A (m-1)\nonumber \\&\times \left( \beta B^2 (63 m-20)+18\right) \nonumber \\&-8\left( 19 \beta ^2 B^4 \left( 17 m^2-17 m+2\right) \right. \nonumber \\&\left. +\,30 \beta B^2 (2 m-1)+45\right) \nonumber \\&+360 v = 0, \end{aligned}$$
(9)
$$\begin{aligned} F_2 =\,&-\,30 m \left[ 9 \alpha ^2 A^2 (m-1)+6 \alpha A \right. \nonumber \\&\left. \times \left( \beta B^2 (32 m-21)+3\right) -\,8 \beta B^2\right. \nonumber \\&\left. \times \left( 19 \beta B^2 (2 m-1)+3\right) \right] = 0, \end{aligned}$$
(10)
$$\begin{aligned} F_4 =\,&45 m^2 \left( 3 \alpha ^2 A^2+86 \alpha A \beta B^2-152 \beta ^2 B^4\right) = 0. \end{aligned}$$
(11)

Equation (11) is equivalent to the [10, Eq. (26)] obtained for solitonic solutions to KdV2. Denoting \(z:=\frac{B^2\beta }{A\,\alpha }\) one obtains from (11) two possible solutions

$$\begin{aligned} z_1=\frac{43-\sqrt{2305}}{152} <0 \quad \text{ and } \quad z_2=\frac{43+\sqrt{2305}}{152} >0 \,. \end{aligned}$$
(12)

The case \(z=z_1\) leads to \(B^2<0\) and has to be rejected as in previous papers [10, 18]. Then for \(z=z_2 \) the amplitude A is

$$\begin{aligned} A&= \frac{43+\sqrt{2305}}{3}\,\frac{B^2\beta }{\alpha } >0 \,. \end{aligned}$$
(13)

Inserting this into (10) yields

$$\begin{aligned} B^{2}&= \frac{3 \left( \sqrt{2305}-14\right) }{703 \beta (2-m)} \end{aligned}$$
(14)

and then using (13) one has

$$\begin{aligned} A&= \frac{3 \left( \sqrt{2305}-51\right) }{37 \alpha (m-2)} . \end{aligned}$$
(15)

Finally from (9) one obtains

$$\begin{aligned} v =\,&\left\{ 4 \left( 129877+314 \sqrt{2305}\right) m^2. \right. \nonumber \\&\left. +\left( 18409 \sqrt{2305}-3209623\right) (m-1)\right\} \nonumber \\&\Big / \left\{ 520220 (m-2)^2\right\} \end{aligned}$$
(16)

Despite the same form of solutions to KdV and KdV2, there is a fundamental difference. KdV only imposes two conditions on coefficients A, B, v for given m, so there is one parameter freedom. This is no longer the case for KdV2.

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, \(A_{\text {KdV2}}\). Such comparison is displayed in Fig. 1 for \(\alpha =\beta =\frac{1}{10}\).

Fig. 1
figure 1

Coefficients A, B, v, \((z=z_2)\) as functions of m for the periodic solution of KdV2 in the form of a single function (3). Lines represent KdV2 coefficients, points KdV coefficients. (\(A_{\text {KdV}}=A_{\text {KdV2}}\)). (Color figure online)

Full size image

It is clear that \(v_{\text {KdV2}}\) and \(v_{\text {KdV}}\) are very similar. We have the following relations: for KdV \(\frac{B^2}{A}=\frac{3\,\alpha }{4\,\beta }\), whereas for KdV2 \(\frac{B^2}{A}=\frac{\alpha }{\beta }z_2\). Since \(z_2\approx 0.6\), \(B_{\text {KdV}}/B_{\text {KdV}}=\sqrt{\frac{3~}{4 z_2}}\approx 1.12\). The same relations hold between KdV2 and KdV coefficients for superposition solutions shown in Fig. 3.

The above examples for the case \(\alpha =\beta =\frac{1}{10}\) show that for somewhat small values of \(\alpha \) the coefficients of KdV2 \(\,\mathsf{dn}^2\) solutions are not much different from those of KdV.

However, physically relevant exact solutions of \(\,\mathsf{dn}^2\) to KdV2 can be found for much larger values of the parameter \(\alpha \). In Fig. 2, the amplitude \(A(\alpha ,m)\) given by (15) is shown as a contour plot for the region \(\alpha \in [0.01,0.5], m\in [0,1]\). It is clear that reasonable amplitudes occur in wide regions of \(\alpha \) and m values. The amplitudes become too big only for \(\alpha \rightarrow 0\), but in such cases KdV works very well.

Fig. 2
figure 2

Amplitude \(A(\alpha ,m)\) given by (15) as function of \((\alpha ,m)\). (Color figure online)

Full size image

2.3 Superposition “\(\,\mathsf{dn}^2+\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)”

Now assume the periodic solution to be in the same form as the corresponding superposition solution of KdV [15] function

$$\begin{aligned} \eta _{+}(y)= & {} \frac{1}{2} A \left[ \,\mathsf{dn}^2(By,m) +\sqrt{m}\,\,\mathsf{cn}(By,m) \right. \nonumber \\&\quad \left. \,\mathsf{dn}(By,m) \right] , \end{aligned}$$
(17)

where A, B, v are yet unknown constants. (m is the elliptic parameter.) We will need

$$\begin{aligned} \eta _y~= & {} -\,\frac{1}{2}AB\,\sqrt{m} \left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2\,\mathsf{sn}, \end{aligned}$$
(18)
$$\begin{aligned} \eta _{2y}= & {} \frac{1}{2}AB^2\,\sqrt{m}\nonumber \\&\times \left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2\left( -\,\mathsf{cn}\,\mathsf{dn}+2\sqrt{m}\,\mathsf{sn}^2\right) , \end{aligned}$$
(19)
$$\begin{aligned} \eta _{3y}= & {} \frac{1}{2}AB^3\,\sqrt{m} \left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2 \nonumber \\&\times \,\mathsf{sn}\, \left( m\,\mathsf{cn}^2+6\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}+\,\mathsf{dn}^2-4m\,\mathsf{sn}^2\right) , \end{aligned}$$
(20)
$$\begin{aligned} \eta _{5y}= & {} -\,\frac{1}{2} AB^5 \,\sqrt{m}\left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2 \nonumber \\&\times \,\mathsf{sn}\left[ m^2\,\mathsf{cn}^4+30 m^{3/2}\,\mathsf{cn}^3\,\mathsf{dn}+\,\mathsf{dn}^4 -44\,\mathsf{dn}^2\,\mathsf{sn}^2\right. \nonumber \\&+\,16m^2\,\mathsf{sn}^4 -30\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\left( -\,\mathsf{dn}^2+4m\,\mathsf{sn}^2\right) \nonumber \\&\left. +\, \,\mathsf{cn}^2\left( 74m\,\mathsf{dn}^2-44m^2\,\mathsf{sn}^2\right) \right] . \end{aligned}$$
(21)

Denote (5) as

$$\begin{aligned} E_1+E_2+E_3+E_4+E_5+E_6+E_7=0, \end{aligned}$$
(22)

where

$$\begin{aligned} E_1&= (1-v) \eta _y =-\,\frac{1}{2}AB(1-v)\sqrt{m} \left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2\,\mathsf{sn}, \end{aligned}$$
(23)
$$\begin{aligned} E_2&= \frac{3}{2} \alpha \,\eta \eta _y =-\,\frac{3}{8} \alpha \, A^2B\,\sqrt{m} \left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^3\,\mathsf{sn}\,\mathsf{dn}, \end{aligned}$$
(24)
$$\begin{aligned} E_3&= \frac{1}{6}\beta \eta {3y}= \frac{1}{12}\beta \, A B^3 \,\sqrt{m}\left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2 \,\mathsf{sn}\nonumber \\&\quad \times \left( m\,\mathsf{cn}^2+6\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}+\,\mathsf{dn}^2-4m\,\mathsf{sn}^2 \right) , \end{aligned}$$
(25)
$$\begin{aligned} E_4&= -\,\frac{3}{8}\alpha ^2\eta ^2\eta _y = \frac{3}{64}\alpha ^2 A^3B \sqrt{m}\,\mathsf{dn}^2\left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^4\,\mathsf{sn}, \end{aligned}$$
(26)
$$\begin{aligned} E_5&= \frac{23}{24}\alpha \beta \,\eta _y\eta _{2y}=-\frac{23}{96}\alpha \beta \,A^2B^3 \,m\,\left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^4 \nonumber \\&\quad \times \, \,\mathsf{sn}\left( -\,\mathsf{cn}\,\mathsf{dn}+2 \sqrt{m}\,\mathsf{sn}^2 \right) , \end{aligned}$$
(27)
$$\begin{aligned} E_6&= \frac{5}{12}\alpha \beta \,\eta \eta _{3y} = \frac{5}{48}\alpha \beta \,A^2B^3\, \sqrt{m}\,\mathsf{dn}\left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^3 \nonumber \\&\quad \times \, \,\mathsf{sn}\left( m\,\mathsf{cn}^2+6 \sqrt{m}\,\mathsf{cn}\,\mathsf{dn}+\,\mathsf{dn}^2-4m\,\mathsf{sn}^2 \right) , \end{aligned}$$
(28)
$$\begin{aligned} E_7&=\frac{19}{360}\beta ^2\eta _{5y} = -\,\frac{19}{720}\beta ^2 A B^5 \,\sqrt{m}\left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2\nonumber \\&\quad \times \, \,\mathsf{sn}\left[ m^2\,\mathsf{cn}^4+30 m^{3/2}\,\mathsf{cn}^3\,\mathsf{dn}+\,\mathsf{dn}^4-44\,\mathsf{dn}^2\,\mathsf{sn}^2 \right. \nonumber \\&\quad +\,16m^2\,\mathsf{sn}^4 -30\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\left( -\,\mathsf{dn}^2+4m\,\mathsf{sn}^2\right) \nonumber \\&\quad \left. +\, \,\mathsf{cn}^2\left( 74m\,\mathsf{dn}^2-44m^2\,\mathsf{sn}^2\right) \right] . \end{aligned}$$
(29)

Then (22) becomes

$$\begin{aligned}&\frac{1}{2} AB\,\sqrt{m} \left( \sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2\,\mathsf{sn}\times \big ( F_0 +F_{cd} \,\mathsf{cn}\,\mathsf{dn}\nonumber \\&\quad + F_{c^2} \,\mathsf{cn}^2 + F_{c^3d}\,\mathsf{cn}^3\,\mathsf{dn}+ F_{c^4} \,\mathsf{cn}^4\big ) =0. \end{aligned}$$
(30)

Equation (30) is valid for arbitrary arguments when all coefficients \(F_0,F_{cd},F_{c^2},F_{c^3d}, F_{c^4}\) vanish simultaneously. This gives us a set of equations for the coefficients A, B, v

$$\begin{aligned} F_0&=-\,1440 v-135 \alpha ^2 A^2 (m-1)^2 \nonumber \\&\quad -\,60 \alpha A (m-1) \left[ \beta B^2 (48 m-5) +18\right] \nonumber \\&\quad +\,4 \left[ 19 \beta ^2 B^4 \left( 61 m^2-46 m+1\right) \right. \nonumber \\&\quad \left. +\,60 \beta B^2 (5 m-1)+360\right] =0, \end{aligned}$$
(31)
$$\begin{aligned} F_{cd}&= 30 \sqrt{m} \left[ 9 \alpha ^2 A^2 (m-1) \right. \nonumber \\&\quad \left. +\,3 \alpha A\left( \beta B^2 (75 m-31)+12\right) \right. \nonumber \\&\quad \left. -\,4 \beta B^2 \left( 19 \beta B^2 (5 m-1)+12\right) \right] =0, \end{aligned}$$
(32)
$$\begin{aligned} F_{c^2}&= 15 m \left( 27 \alpha ^2 A^2 (m-1)\right. \nonumber \\&\quad \left. +12 \alpha A\left( \beta B^2 (59 m-37)+6\right) \right. \nonumber \\&\quad \left. -\,32 \beta B^2 \left( 19 \beta B^2 (2 m-1)+3\right) \right) =0 , \end{aligned}$$
(33)
$$\begin{aligned} F_{c^3d}&= -\,90 m^{3/2} \left( 3 \alpha ^2 A^2+86 \alpha A \beta B^2 \right. \nonumber \\&\quad \left. -152 \beta ^2 B^4\right) =0 , \end{aligned}$$
(34)
$$\begin{aligned} F_{c^4}&= -\,90 m^2 \left( 3 \alpha ^2 A^2+86 \alpha A \beta B^2\right. \nonumber \\&\quad \left. -\,152 \beta ^2 B^4\right) =0 . \end{aligned}$$
(35)

Equations (34) and (35) are equivalent and give the same condition as (11). Solving (34) with respect to \(B^2\), we obtain the same relations as in [10, Eq. (28)]

$$\begin{aligned} (B_{1/2})^2= \frac{A\alpha }{ \beta }\left( \frac{43\mp \sqrt{2305}}{152}\right) \end{aligned}$$
(36)

Denote

$$\begin{aligned} z_1=\frac{43-\sqrt{2305}}{152} \qquad \text{ and } \qquad z_2=\frac{43+\sqrt{2305}}{152} \,. \end{aligned}$$
(37)

It is clear that \( z_1<0\) and \( z_2>0\). B has to be real-valued. This is possible for the case \(z=z_1\) if \(A<0\), and for \(z=z_2\) if \(A>0\). The value of \(z_2\) is the same as that found for the exact soliton solution in [10, Eq. (28)]. In general

$$\begin{aligned} B^2= \frac{A\alpha }{ \beta }z . \end{aligned}$$
(38)

Now, we insert (38) into (31),(32) and (33). Besides a trivial solution with \(A=0\), we obtain

$$\begin{aligned}&1440(1-v) +A\alpha (1-m)\left[ 1080-135 A\alpha (1-m) \right] \nonumber \\&\quad -240A\alpha (1-5m)-30(A\alpha )^2(10-109m+96m^2)\,z \nonumber \\&\quad + 4(A\alpha )^2 (19-847m+1159m^2)\, z^2 = 0, \end{aligned}$$
(39)
$$\begin{aligned}&9 [A\alpha (m-1)+4]+3[A\alpha (75 m-31)-16]\,z\nonumber \\&\quad -76 A\alpha (5 m-1)\,z^2 = 0, \end{aligned}$$
(40)
$$\begin{aligned}&9 (3 A\alpha (m-1)+8) +12 (A\alpha (59 m-37)-8)\,z \nonumber \\&\quad -608 A\alpha (2 m-1)\,z^2 = 0. \end{aligned}$$
(41)

From (40) we find

$$\begin{aligned} A= -\,\frac{12 (4 z-3)}{\alpha \left[ 76z^2(5m-1)-z(225m-93)-9(m-1)\right] } \end{aligned}$$
(42)

but from (41) it follows that

$$\begin{aligned} A= -\,\frac{24 (4 z-3)}{\alpha \left[ 608z^2(2m-1)-4z(177m-111)-27(m-1)\right] }. \end{aligned}$$
(43)

This looks like a contradiction, but substitution  \(z=z_1=(43 - \sqrt{2305})/152\) in both (42) and (43) gives the same result

$$\begin{aligned} A_1= \frac{24 \,(71+\sqrt{2305})}{(-\,329+5\sqrt{2305})\, \alpha \,(m-5)}. \end{aligned}$$
(44)

For  \(z=z_2=(43 + \sqrt{2305})/152\), the common result is

$$\begin{aligned} A_2= \frac{24 (-\,71+\sqrt{2305})}{(329+5\sqrt{2305})\, \alpha \,(m-5)}. \end{aligned}$$
(45)

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 \(z=z_1\) and \(A_1\) given by (44) we obtain from (39)

$$\begin{aligned} v_1 = \frac{\text {vnum}_{-}(m) }{\text {vden}_{-}(m)} \end{aligned}$$
(46)

and with \(z=z_2\) and \(A_2\) given by (45)

$$\begin{aligned} v_2 =\frac{\text {vnum}_{+}(m) }{\text {vden}_{+}(m)}, \end{aligned}$$
(47)

where

$$\begin{aligned} \text {vnum}_{\mp }(m)= & {} ~6 \left\{ \left( 2912513\mp 58361 \sqrt{2305}\right) m^2 \right. \\&-\,54 \left( 584397\mp 10069 \sqrt{2305}\right) m \\&\left. +\,75245133\mp 1419141 \sqrt{2305}\right\} , \end{aligned}$$

and

$$\begin{aligned} \text {vden}_{\mp }(m) = 95 \left( 329\mp 5 \sqrt{2305}\right) ^2 (m-5)^2. \end{aligned}$$

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.

  • Case 1 \(\displaystyle z=z_1=\frac{43-\sqrt{2305}}{152}\approx -\,0.0329633<0\). This case leads to \(B^2<0\) and has to be rejected as in previous papers [10, 18].

  • Case 2 \(\displaystyle z=z_2=\frac{43+\sqrt{2305}}{152}\approx 0.598753 >0\). Then

    $$\begin{aligned} A&= \frac{12 \left( \sqrt{2305}-51\right) }{37 \alpha (m-5)} >0, \end{aligned}$$
    (48)
    $$\begin{aligned} B&= \sqrt{ \frac{12\left( \sqrt{2305}-14\right) }{703(5-m)\beta } } \end{aligned}$$
    (49)

    and \(v_2\) is given by (47). Since \(m\in [0,1],~(m-5)<0\)  then  \(B_2\)  is real. The solution in this case is

    $$\begin{aligned}&\eta _2(x -v_2t,m) = \frac{1}{2} A_2 \left[ \,\mathsf{dn}^2(B_2 (x-v_2t),m) \right. \nonumber \\&\quad \left. +\sqrt{m}\,\,\mathsf{cn}(B_2 (x-v_2t),m)\,\mathsf{dn}(B_2(x-v_2t),m) \right] . \end{aligned}$$
    (50)
Fig. 3
figure 3

Same as in Fig. 1 but for superposition solutions \(\eta _+\) (17) and \(\eta _-\) (51). (Color figure online)

Full size image

Coefficients \(A_2, B_2, v_2\) of superposition solutions (17) to KdV2 as functions of m are presented in Fig. 3 for \(\alpha =\beta =\frac{1}{10}\) and compared to corresponding solutions to KdV. Here, similarly as in Fig. 1, we assume that \(A_{\text {KdV}}=A_{\text {KdV2}}\).

Physically relevant exact superposition solutions to KdV2 can be found for greater values of the parameter \(\alpha \) than \(\frac{1}{10}\). In Fig. 4, the amplitude \(A(\alpha ,m)\) given by (48) is shown as a contour plot for the region \(\alpha \in [0.01,0.5], m\in [0,1]\). It is clear that reasonable amplitudes occur in wide regions of \(\alpha \) and m values, similarly like in the case \(\,\mathsf{dn}^2\). The amplitudes become too big only for \(\alpha \rightarrow 0\), but in such cases KdV works very well.

Fig. 4
figure 4

Same as in Fig. 2 but for superposition solutions \(\eta _+\). (Color figure online)

Full size image

2.5 Superposition 1 “\(\,\mathsf{dn}^2-\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)”

Now we check the alternative superposition “\(\,\mathsf{dn}^2-\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)”

$$\begin{aligned} \eta _{-}(y) = \frac{1}{2} A \left[ \,\mathsf{dn}^2(By,m) -\sqrt{m}\,\,\mathsf{cn}(By,m)\,\mathsf{dn}(By,m) \right] .\nonumber \\ \end{aligned}$$
(51)

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)

$$\begin{aligned}&\frac{1}{2} AB\,\sqrt{m} \left( -\sqrt{m}\,\mathsf{cn}+\,\mathsf{dn}\right) ^2\,\mathsf{sn}\, \times \left( F_0 +F_{cd} \,\mathsf{cn}\,\mathsf{dn}\right. \nonumber \\&\quad \left. + F_{c^2} \,\mathsf{cn}^2 + F_{c^3d}\,\mathsf{cn}^3\,\mathsf{dn}+ F_{c^4} \,\mathsf{cn}^4\right) =0 . \end{aligned}$$
(52)

Equation (52) is valid for arbitrary arguments when all coefficients \(F_0,F_{cd},F_{c^2},F_{c^3d}, F_{c^4}\) 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 “\(\,\mathsf{dn}^2+\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)” superposition (31)–(35). Then the coefficients A, B, v for superposition “\(\,\mathsf{dn}^2-\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)” are the same as for superposition “\(\,\mathsf{dn}^2+\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)” given above. This property for KdV2 is the same as for KdV, see [15]. It follows from periodicity of the Jacobi elliptic functions. From

$$\begin{aligned} \,\mathsf{cn}(y+2K(m),m)= & {} -\,\mathsf{cn}(y,m), \\ \,\mathsf{dn}(y+2K(m),m)= & {} \,\mathsf{dn}(y,m) \end{aligned}$$

it follows that

$$\begin{aligned}&\,\mathsf{dn}^2(y+2K(m),m)\nonumber \\&\qquad +\,\sqrt{(}m)\,\mathsf{cn}(x+2K(m),m) \,\mathsf{dn}(x+2K(m),m)\nonumber \\&\quad = \,\mathsf{dn}^2(x,m)-\sqrt{(}m)\,\mathsf{cn}(x,m)\,\mathsf{dn}(x,m). \end{aligned}$$
(53)

So both superpositions \(\eta _{+}\) (17) and \(\eta _{-}\) (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.

Fig. 5
figure 5

Profiles of KdV and KdV2 waves for \(m=0.1\). (Color figure online)

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Fig. 6
figure 6

Profiles of KdV and KdV2 waves for \(m=0.9\). (Color figure online)

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Fig. 7
figure 7

Profiles of KdV and KdV2 waves for \(m=0.99\). (Color figure online)

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Fig. 8
figure 8

Time evolution of single KdV2 solution (red lines) and \(\eta _+\) KdV2 solution for \(m=0.1\). (Color figure online)

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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 \(A_{\text {KdV}} = A_{\text {KdV2}}\). In Figs. 5, 6 and 7 below, KdV solutions of the forms (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 \(\alpha =\beta =0.1\) 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 \(m\rightarrow 1\) the distance between crests of \(\eta _{+}\) and \(\eta _{-}\) 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 [9,10,11, 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 \(\,\mathsf{dn}^2\) (3) and superposition \(\eta _+\) and \(\eta _-\) (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 \( T=\lambda /v\) 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).

Fig. 9
figure 9

Same as in Fig. 8 but for \(m=0.9\). (Color figure online)

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Fig. 10
figure 10

Same as in Fig. 8 but for \(m=0.99\). (Color figure online)

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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. This statement is true for our single solitonic solutions [10], periodic solutions in the form of single Jacobi elliptic functions \(\,\mathsf{cn}^2\) [18] or \(\,\mathsf{dn}^2\), and for periodic solutions in the form of superpositions \(\,\mathsf{dn}^2\pm \sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\) (this paper). Coefficients A, B, v of these solutions to KdV2 are fixed by coefficients of the equation, that is by values of \(\alpha ,\beta \) parameters. This is in contradiction to the KdV case where one coefficient (usually A) is arbitrary.