Superposition solutions to the extended KdV equation for water surface waves
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Abstract
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 higherorder 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 higherorder, 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.
Keywords
Shallow water waves Extended KdV equation Analytic solutions Nonlinear equations1 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 oneorder 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.
Not only is KdV2 nonintegrable, 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
2.1 Single periodic function \(\,\mathsf{dn}^2\)
2.2 Comparison to KdV solutions
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.
2.3 Superposition “\(\,\mathsf{dn}^2+\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)”
2.4 Discussion of mathematical solutions

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$$\begin{aligned} A&= \frac{12 \left( \sqrt{2305}51\right) }{37 \alpha (m5)} >0, \end{aligned}$$(48)and \(v_2\) is given by (47). Since \(m\in [0,1],~(m5)<0\) then \(B_2\) is real. The solution in this case is$$\begin{aligned} B&= \sqrt{ \frac{12\left( \sqrt{2305}14\right) }{703(5m)\beta } } \end{aligned}$$(49)$$\begin{aligned}&\eta _2(x v_2t,m) = \frac{1}{2} A_2 \left[ \,\mathsf{dn}^2(B_2 (xv_2t),m) \right. \nonumber \\&\quad \left. +\sqrt{m}\,\,\mathsf{cn}(B_2 (xv_2t),m)\,\mathsf{dn}(B_2(xv_2t),m) \right] . \end{aligned}$$(50)
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 \(A_{\text {KdV}}=A_{\text {KdV2}}\).
2.5 Superposition 1 “\(\,\mathsf{dn}^2\sqrt{m}\,\mathsf{cn}\,\mathsf{dn}\)”
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.
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.
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