Weakly-Nonlinear Solution of Coupled Boussinesq Equations and Radiating Solitary Waves

Abstract

Weakly-nonlinear waves in a layered waveguide with an imperfect interface (soft bonding between the layers) can be modelled using coupled Boussinesq equations. We assume that the materials of the layers have close mechanical properties, in which case the system can support radiating solitary waves. We construct a weakly-nonlinear d’Alembert-type solution of this system, considering the problem in the class of periodic functions on an interval of finite length. The solution is constructed using a novel multiple-scales procedure involving fast characteristic variables and two slow time variables. Asymptotic validity of the solution is carefully examined numerically. We also discuss the limiting case of an infinite interval for localised initial conditions. The solution is applied to study interactions of radiating solitary waves.

Appendix A: Numerical Methods

In the following methods we use the Discrete Fourier Transform (DFT) to calculate the Fourier transform of numerical data (e.g., [24]). Let us consider a function u(x, t) on a finite domain $x \in [-L, L]$ and we discretise the domain into N equally spaced points, so we have the spacing $\Delta x = 2L/N$. In all calculations we scale the domain from $x \in [-L, L]$ to $\tilde{x} \in [0, 2\pi ]$, which can be achieved by applying the transform $\tilde{x} = sx + \pi$, where $s = \pi /L$. Denoting $x_{j} = -L + j \Delta x$ for $j=0,\dots ,N$, we define the DFT for the function u(x, t) as

$$\begin{aligned} \hat{u} \left( k, t \right) = \frac{1}{\sqrt{N}} \sum _{j=1}^{N} u \left( x_{j}, t \right) e^{-i k x_{j}}, \ -\frac{N}{2} \le k \le \frac{N}{2} - 1, \end{aligned}$$

(A.1)

and similarly the IDFT is defined as

$$\begin{aligned} u \left( x, t \right) = \frac{1}{\sqrt{N}} \sum _{k=-N/2}^{N/2 - 1} \hat{u} \left( k, t \right) e^{i k x_{j}}, \quad j = 1, 2, \dots , N, \end{aligned}$$

(A.2)

where the discretised and scaled wavenumber is now $k \in \mathbb {Z}$. To perform these transforms we implement the FFTW3 algorithm in C [25].

For the coupled Boussinesq equations (1)–(2) we use a pseudospectral method similar to the one presented in [26], where this method was used to solve a single regularised Boussinesq equation in the context of microstructured solids. We introduce the change of variable

$$\begin{aligned} U = u - \epsilon u_{xx}, \quad W = w - \epsilon \beta w_{xx}, \end{aligned}$$

(A.3)

so that we have

$$\begin{aligned}&U_{tt} = u_{xx} + \epsilon \left[ \frac{1}{2} \left( u^2 \right) _{xx} - \delta \left( u - w \right) \right] , \nonumber \\&W_{tt} = c^2 w_{xx} + \epsilon \left[ \frac{\alpha }{2} \left( w^2 \right) _{xx} + \gamma \left( u - w \right) \right] . \end{aligned}$$

(A.4)

Taking the Fourier transform of (A.3) we obtain

$$\begin{aligned} \hat{u} = \frac{\hat{U}}{1 + \epsilon k^2}, \quad \hat{w} = \frac{\hat{W}}{1 + \epsilon \beta k^2}. \end{aligned}$$

(A.5)

We take the Fourier transform of (A.4) and substitute (A.5) into this expression to obtain an ODE in $\hat{u}$ and $\hat{w}$, taking the form

$$\begin{aligned} \hat{U}_{tt}&= -\frac{\epsilon \delta + s^2 k^2}{1 + \epsilon s^2 k^2} \hat{U} - \frac{\epsilon s^2 k^2}{2} \mathscr {F} \left\{ \mathscr {F}^{-1} \left[ \frac{\hat{U}}{1 + \epsilon s^2 k^2} \right] ^2 \right\} + \frac{\epsilon \delta }{1 + \epsilon s^2 \beta k^2} \hat{W}^2 = \hat{S}_1 \left( \hat{U}, \hat{W} \right) , \nonumber \\ \hat{W}_{tt}&= -\frac{\epsilon \gamma + c^2 s^2 k^2}{1 + \epsilon \beta s^2 k^2} \hat{W} - \frac{\epsilon \alpha s^2 k^2}{2} \mathscr {F} \left\{ \mathscr {F}^{-1} \left[ \frac{\hat{W}}{1 + \epsilon \beta s^2 k^2} \right] ^2 \right\} + \frac{\epsilon \delta }{1 + \epsilon s^2 k^2} \hat{U}^2 = \hat{S}_{2} \left( \hat{U}, \hat{W} \right) , \end{aligned}$$

(A.6)

where $\mathscr {F}$ denotes the Fourier transform. We solve this system of ODEs using a 4th-order Runge-Kutta method for time stepping, such as the one used in [16]. Let us rewrite the system as

(A.7)

where we defined $\hat{S}_{1,2}$ as the right-hand side of (A.6). We discretise the time domain and functions as $t = t_n$, $\hat{U}(k, t_{n}) = \hat{U}_n$, $\hat{W}(k, t_{n}) = \hat{W}_n$, $\hat{G}(k, t_{n}) = \hat{G}_{n}$, $\hat{H}(k, t_{n}) = \hat{H}_{n}$ for $n=0,1,2,\dots$, where $t_{n} = n \Delta t$, and k discretises the Fourier space. Taking the Fourier transform of the initial conditions as defined in (3) and (4), and making use of (A.5), we obtain initial conditions $\hat{U}_{0}$, $\hat{W}_{0}$, and $\hat{G}_0$, $\hat{H}_{0}$ of the form

(A.8)

Now we have initial conditions, we implement a 4th-order Runge-Kutta method, taking the form

$$\begin{aligned} \begin{array}{ll} \hat{U}_{n+1} = \hat{U}_n + \frac{1}{6} \left[ k_1 + 2k_2 + 2k_3 + k_4 \right] , &{} \hat{G}_{n+1} = \hat{G}_n + \frac{1}{6} \left[ l_1 + 2l_2 + 2l_3 + l_4 \right] , \\ \hat{W}_{n+1} = \hat{W}_n + \frac{1}{6} \left[ m_1 + 2m_2 + 2m_3 + m_4 \right] , &{} \hat{H}_{n+1} = \hat{H}_n + \frac{1}{6} \left[ n_1 + 2n_2 + 2n_3 + n_4 \right] ,\\ \end{array} \end{aligned}$$

where

(A.9)

To obtain the solution in the real domain, we transform $\hat{U}$ back to u, and similarly $\hat{W}$ back to w, through relation (A.5). Explicitly we have

$$\begin{aligned} u(x,t) = \mathscr {F}^{-1} \left\{ \frac{\hat{U}}{1 + \epsilon s^2 k^2} \right\} , \quad w(x,t) = \mathscr {F}^{-1} \left\{ \frac{\hat{W}}{1 + \epsilon s^2 \beta k^2} \right\} . \end{aligned}$$

(A.10)

We now consider the solution to the coupled Ostrovsky equations. The method is similar to that used in [27]. It is presented for the equations (75), (76), as this method can be reduced to solve the system (29), (31). We present the equations for the negative superscript i.e. for $\phi _{1}^{-}$ and $\phi _{2}^{-}$. We omit the superscript in the subsequent equations. Let us consider the system of coupled Ostrovsky equations defined as

$$\begin{aligned} \left( \phi _{1 t} + \omega _{1} \phi _{1 x} + \alpha _1 \left( f_{1} \phi _{1} \right) _{x} + \beta _1 \phi _{1 xxx} \right) _{x}&= \delta \left( \phi _{1} - \phi _{2} \right) + H_{1} \left( f_{1}(x), f_{2}(x) \right) , \nonumber \\ \left( \phi _{2 t} + \omega _{2} \phi _{2 x} + \alpha _2 \left( f_{2} \phi _{2} \right) _{x} + \beta _2 \phi _{2 xxx} \right) _{x}&= \gamma \left( \phi _{2} - \phi _{1} \right) + H_{2} \left( f_{1}(x), f_{2}(x) \right) , \end{aligned}$$

(A.11)

where $\alpha _1$, $\alpha _2$, $\beta _1$, $\beta _2$, $\omega _{1}$, $\omega _{2}$, $\delta$ and $\gamma$ are constants, and the functions $f_{1}$, $f_{2}$ are known. We consider the equation on domains $t \in [0, T]$ and $x \in [-L, L]$. We calculate the nonlinear terms in the real domain and then transform them to the Fourier space. The spatial domain is discretised by N equidistant points with spacing $\Delta x = 2 \pi / N$, and we have the DFT and IDFT as defined in (A.1) and (A.2) respectively, with an appropriately similar transform for w. The discrete Fourier transform of equations (A.11) with respect to x gives

$$\begin{aligned} \hat{\phi }_{1t} + \left( i s k \omega _{1} - i s^3 k^3 \beta _1 \right) \hat{\phi }_{1} + i s k \alpha _1 \mathscr {F} \left\{ f_{1} \phi _{1} \right\}&= -\frac{i \delta }{sk} \left( \hat{\phi }_{1} - \hat{\phi }_{2} \right) - \frac{i}{sk} \hat{H}_{1}, \nonumber \\ \hat{\phi }_{2t} + \left( i s k \omega _{2} - i s^3 k^3 \beta _{2} \right) \hat{\phi }_{2} + i s k \alpha _2 \mathscr {F} \left\{ f_{2} \phi _{2} \right\}&= -\frac{i \gamma }{sk} \left( \hat{\phi }_{2} - \hat{\phi }_{1} \right) - \frac{i}{sk} \hat{H}_{2}. \end{aligned}$$

(A.12)

This system is solved numerically using a 4th-order Runge-Kutta method for time stepping as for the coupled Boussinesq equations. Assume that the solution at t is given by $\hat{\phi }_{1,j} = \hat{\phi }_{1}(k, j\kappa )$ and $\hat{\phi }_{2,j} = \hat{\phi }_{2}(k, j \kappa )$, where $\kappa = \Delta t$. Then the solution at $t = (j+1) \Delta t$ is given by

$$\begin{aligned} \hat{\phi }_{1, \left( j+1 \right) \kappa }&= \hat{\phi }_{1, j \kappa } + \frac{1}{6} \left( a_{1} + 2 b_{1} + 2 c_{1} + d_{1} \right) , \nonumber \\ \hat{\phi }_{2, \left( j+1 \right) \kappa }&= \hat{\phi }_{2, j \kappa } + \frac{1}{6} \left( a_{2} + 2 b_{2} + 2 c_{2} + d_{2} \right) , \end{aligned}$$

(A.13)

where $a_i$, $b_i$, $c_i$, $d_i$ are functions of k at a given moment in time, t, and are defined as

for $i=1, 2$. The functions $F_{i}$ are found as a rearrangement of (A.12) to contain all non-time derivatives. Explicitly we have

$$\begin{aligned} F_{1} \left( \hat{\phi }_{1, j}, \hat{\phi }_{2, j} \right) =&- i k s \alpha _1 \mathscr {F} \left\{ f_{1} \phi _{1} \right\} + \left( i k^3 s^3 \beta _1 - i s k \omega _{1} \right) \hat{\phi }_{1, j} - \frac{i \delta }{sk} \left( \hat{\phi }_{1, j} - \hat{\phi }_{2, j} \right) - \frac{i}{sk} \hat{H}_{1, j}, \nonumber \\ F_{2} \left( \hat{\phi }_{1, j}, \hat{\phi }_{2, j} \right) =&- i k s \alpha _2 \left\{ f_{2} \phi _{2} \right\} + \left( i k^3 s^3 \beta _2 - i s k \omega \right) \hat{\phi }_{2, j} - \frac{i \gamma }{sk} \left( \hat{\phi }_{2, j} - \hat{\phi }_{1, j} \right) - \frac{i}{sk} \hat{H}_{2, j}. \end{aligned}$$

To obtain a solution at the next step, the functions $a_{i}, b_{i}, c_{i}, d_{i}$, for $i=1,2$, must be calculated in pairs, that is we calculate $a_{1}$ followed by $a_{2}$, then $b_{1}$ followed by $b_{2}$, and so on.