3.1. Solitonic modulation system

The analytical description of solitonic dispersive hydrodynamics is based on considering the soliton reduction of the Whitham modulation equations. Having the mKdV equation in mind, we present the general theory for the unidirectional, scalar case.

For a periodic travelling wave solution parametrised by three independent constants (as in the case of KdV or mKdV equations, third-order PDEs), the modulation equations can be written in terms of the physical wave parameters: the mean flow $\bar {u}$, the amplitude $a$ and the wavenumber $k$. Allowing $\bar {u}$, $a$, $k$ to be slow functions of $x,t$, the requirement for the modulated periodic wave to be an asymptotic solution to the dispersive hydrodynamic equation (2.1) results in the quasilinear modulation system,

(3.1)\begin{equation} \boldsymbol{u}_t + \mathrm{A} (\boldsymbol{u}) \boldsymbol{u}_x =0, \end{equation}

where $\boldsymbol {u}= ( \bar {u}, a, k)^\textrm {T}$ and $\mathrm {A}( \boldsymbol {u})$ is a $3 \times 3$ modulation matrix. We call the dispersive hydrodynamics convex if the associated Whitham modulation system (3.1) is strictly hyperbolic and genuinely nonlinear. If at least one of these conditions is violated, the system is non-convex. Strict hyperbolicity requires that the eigenvalues $v_i (\boldsymbol {u}),\ i=1,2,3$ of the matrix $\mathrm {A}( \boldsymbol {u})$ are real and distinct, $v_1 < v_2 < v_3$, for all ${\boldsymbol{u}}$ in the admissible set

(3.2)\begin{equation} \boldsymbol{u} \in\mathcal{A} = \left\{ (\bar{u},a,k)^\textrm{T} \,|\, \bar{u} \in \mathbb{R},\ a > 0,\ k > 0\right\}. \end{equation}

The genuine nonlinearity condition then reads $\boldsymbol {\nabla }_{\boldsymbol {u}} v_i \boldsymbol {\cdot } \boldsymbol {r}_i \ne 0,\ i=1,2,3$ for all $\boldsymbol {u} \in \mathcal {A}$, where $\boldsymbol {r}_i(\boldsymbol {u})$ is the right eigenvector corresponding to the eigenvalue $v_i$ (Lax Reference Lax1973). If the system is non-strictly hyperbolic (the eigenvectors $\boldsymbol {r}_i$ span $\mathbb {R}^{3}$ but multiple eigenvalues are admissible), then it is not genuinely nonlinear either (Dafermos Reference Dafermos2016). The converse is generally not true. Nevertheless, a non-convex system can exhibit convex properties in a restricted domain $\mathcal {D} \subset \mathcal {A}$.

The KdV–Whitham modulation system (3.1) is strictly hyperbolic and genuinely nonlinear for all admissible $\boldsymbol {u} \in \mathcal {A}$ (Levermore Reference Levermore1988), while for the mKdV equation, the properties of strict hyperbolicity and genuine nonlinearity depend on the sign of $\mu$ and on $\boldsymbol {u}$ (El et al. Reference El, Hoefer and Shearer2017).

An important ingredient for modulation theory is the equation for $k$ in (3.1)

(3.3)\begin{equation} k_t+ [\omega(\bar{u}, k, a)]_x=0, \end{equation}

known as the conservation of waves, where $\omega (\bar {u}, k, a)$ is the travelling wave frequency.

Soliton–mean interaction theory is based on the fundamental property of Whitham modulation systems that we postulate here in a general form and later explicitly justify for mKdV: in the $k \to 0$ soliton limit, the modulation system (3.1) admits the following exact reduction (Gurevich, Krylov & El Reference Gurevich, Krylov and El1990):

(3.4)\begin{equation} \begin{bmatrix} \bar{u}\\ a \end{bmatrix}_t+ \begin{bmatrix} f'(\bar{u}) & 0 \\ g(a,\bar{u}) & c(a,\bar{u}) \end{bmatrix} \begin{bmatrix} \bar{u}\\ a \end{bmatrix}_x = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \end{equation}

where $c(a,\bar {u})= \lim _{k \to 0} (\omega /k)$ is the soliton amplitude–speed relation for propagation on the background $\bar {u}$ and $g(a,\bar {u})$ is a coupling function that is system dependent. Equation (3.4) is called the solitonic modulation system.

The third modulation equation (3.3) is identically satisfied for $k=0$ while for $0 < k \ll 1$, it assumes at leading order the form

(3.5)\begin{equation} k_t + [c(a,\bar{u})k]_x = 0. \end{equation}

Equation (3.5) can be added to the solitonic modulation system (3.4) to give an approximate modulation system for a train of non-interacting solitons propagating on a variable mean flow. Equation (3.5) then signifies the conservation of the number of solitons in the train. We shall refer to the combined system (3.4) and (3.5) as the augmented solitonic modulation system. Note that a particular case of this system was derived in Grimshaw (Reference Grimshaw1979) for slowly varying soliton solutions of the variable coefficient KdV equation.

The soliton train interpretation of the modulation system (3.4) is instrumental for a solitonic dispersive hydrodynamics as it enables the description of a single modulated soliton by treating the soliton amplitude $a(x,t)$ as a spatio-temporal field, in contrast to standard soliton perturbation theory where the soliton's parameters evolve temporally along its trajectory in the $x,t$-plane; see, e.g. Kivshar & Malomed (Reference Kivshar and Malomed1989). Additionally, as we will show, the introduction of the fictitious wavenumber field $k(x,t)$ for a single soliton enables the determination of the soliton phase shift due to interaction with the mean flow.

The characteristic velocities of the system (3.4) are $f'(\bar {u})$ and $c(a,\bar {u})$. The right eigenvectors $\boldsymbol {r}_{1,2}$ of the Jacobian matrix in (3.4) for each characteristic velocity are

(3.6a,b)$$\begin{gather} v_1 = f'(\bar{u}),\quad \boldsymbol{r}_1 = \begin{bmatrix} f'- c \\ g \end{bmatrix}, \end{gather}$$

(3.7a,b)$$\begin{gather}v_2 = c(a,\bar{u}),\quad \boldsymbol{r}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. \end{gather}$$

Thus, the system (3.4) is strictly hyperbolic if $f' \ne c$ for all $(\bar {u},a)\in \mathcal {A}_0$, where

(3.8)\begin{equation} \mathcal{A}_0 = \mathbb{R} \times (0,\infty) \end{equation}

is the set of admissible states. The system (3.4) is genuinely nonlinear in the $j$th characteristic field if $\boldsymbol {\nabla } v_j \boldsymbol {\cdot } \boldsymbol {r}_j \ne 0$ for all $(\bar {u},a)\in \mathcal {A}_0$. For the first characteristic field,

(3.9)\begin{equation} \boldsymbol{\nabla} f'(\bar{u}) \boldsymbol{\cdot} \boldsymbol{r}_1 \ne 0 \implies f''(\bar{u})(f'(\bar{u})-c(a,\bar{u})) \ne 0, \end{equation}

which holds, provided the characteristic velocities are distinct (strict hyperbolicity) and the flux $f$ of the original scalar evolution equation (2.1) is convex. Thus, when two characteristic velocities merge (non-strict hyperbolicity), the corresponding characteristic field is not genuinely nonlinear.

The genuine nonlinearity of the second characteristic field requires

(3.10)\begin{equation} \boldsymbol{\nabla} c(a,\bar{u}) \boldsymbol{\cdot} \boldsymbol{r}_2 \ne 0 \implies c_{a}(a,\bar{u}) \ne 0. \end{equation}

To summarise, the quasi-linear system (3.4) is strictly hyperbolic when $f'(\bar {u}) \neq c(a,\bar {u})$ and is genuinely nonlinear when additionally $f''(\bar {u})\neq 0$ and $c_a(a,\bar {u})\neq 0$ for all $(\bar {u},a) \in \mathbb {R} \times (0,\infty )$. Negation of any of these three conditions gives rise to a non-convex solitonic dispersive hydrodynamics.

Since the exact soliton reduction (3.4) is a $2\times 2$ quasi-linear hyperbolic system, it can be reduced to Riemann invariant form. We refer to the mean flow $\bar {u}$ as the ‘hydrodynamic’ Riemann invariant and the other is found by integrating the differential form $g\,\mathrm {d}\bar {u} + (c - f')\,\mathrm {d}a$ provided $c \ne f'$ (Whitham Reference Whitham1974). Denoting the second, solitonic Riemann invariant as $q = q(a,\bar {u})$, the diagonalised system can be written as

(3.11)\begin{equation} \begin{bmatrix} \bar{u}\\ q \end{bmatrix}_t+ \begin{bmatrix} f'(\bar{u}) & 0 \\ 0 & C(q,\bar{u}) \end{bmatrix} \begin{bmatrix} \bar{u}\\ q \end{bmatrix}_x = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \end{equation}

where $C (q(a,\bar {u}),\bar {u} ) \equiv c(a,\bar {u})$. In terms of the diagonal system (3.11), the condition of strict hyperbolicity reads $f'(\bar {u}) \ne C(q,\bar {u})$ and the conditions of genuine nonlinearity of the first and second characteristic fields are written respectively as

(3.12a,b)\begin{equation} f''(\bar{u}) \ne 0, \quad \hbox{and} \quad C_q \ne 0. \end{equation}

It is important to stress that the existence of the solitonic Riemann invariant $q$ is not reliant on the diagonalisability of the full quasi-linear system (3.1) in Riemann invariants. In fact, as was shown in El (Reference El2005), this Riemann invariant can be obtained directly, as the integral $q=Q(\tilde k, \bar {u}) = const$ on $\mathrm {d}\kern0.06em x/\mathrm {d}t=C$, of the following characteristic ordinary differential equation (ODE)

(3.13)\begin{equation} \frac{\mathrm{d} \tilde k}{{\textrm{d}} \bar{u}} = \frac{\partial _{\bar{u}} \tilde \omega_0}{f'(\bar{u}) - \partial_{\tilde k} \tilde \omega_0}, \end{equation}

where $\tilde k$ and $\tilde \omega _0$ are called the conjugate wavenumber and conjugate frequency, respectively. They are defined in terms of the soliton amplitude–speed relation $c(a, \bar {u})$ and the linear dispersion relation (2.2) $\omega _0(k, \bar {u})$ by

(3.14)\begin{equation} \tilde \omega_0 (\tilde k, \bar{u}) ={-}\textrm{i}\omega_0 (\textrm{i} \tilde k, \bar{u} );\quad c(a, \bar{u}) = \frac{\tilde \omega_0}{\tilde k}. \end{equation}

We note that the Riemann invariant $q=Q(\tilde k, \bar {u})$ is not defined uniquely, as any smooth function of a Riemann invariant is also a Riemann invariant. In the case of convex systems, a convenient normalisation is suggested by the requirement to maintain strict hyperbolicity of the solitonic system in the limit of vanishing amplitude where the long-wave speed $f'(\bar {u})$ and soliton speed $c(a, \bar {u})$ must coincide. The variable $\tilde k$ can be identified as an amplitude-type variable (El Reference El2005), so that $\tilde k =0 \Longleftrightarrow a =0$, and requires that the hydrodynamic and solitonic Riemann invariants coincide when $\tilde k \to 0$, i.e. $Q(0, \bar {u})=\bar {u}$. As a result, the system (3.11) reduces to a single hyperbolic equation $\bar {u}_t + f'(\bar {u})\bar {u}_x=0$. The situation is different for non-convex systems, where two or more distinct Riemann invariants associated with the same characteristic speed may exist. For example, for cubic flux $f(\bar {u})=\bar {u}^{3}$, the mean flow equation $\bar {u}_t +3 \bar {u}^{2} \bar {u}_x=0$ is invariant with respect to the transformation $\bar {u} \to - \bar {u}$ so another possible normalisation is $Q(0, \bar {u})=-\bar {u}$. To avoid ambiguity, we will be using the normalisation

(3.15)\begin{equation} Q(0, \bar{u})=\bar{u}, \end{equation}

for the initial configuration. For the case of a general non-convex flux, we assumed, without loss of generality, that it satisfies $f''(0)=0$. Then, if the solution curve crosses $\bar {u}=0$, the normalisation of the Riemann invariant should be changed to $Q(0, \bar {u})=-\bar {u}$ across this point to maintain smoothness of $Q$.

The two Riemann invariants $\bar {u}$ and $q$ for the $2\times 2$ system (3.11) are also Riemann invariants for the $3 \times 3$ augmented solitonic modulation system (3.11) and (3.5). But the latter quasi-linear system is not hyperbolic because its corresponding Jacobian matrix is deficient, with just two eigenvalues and two linearly independent eigenvectors. Nevertheless, it has another hyperbolic subsystem, in addition to (3.11), which is obtained by setting $q \equiv q_0$ constant, as will be the case for the soliton–mean flow interaction problems we consider. Then the remaining simple wave equation $\bar u_t + f(\bar u)_x=0$, together with the approximate equation (3.5), where we replace $c(a, \bar u)$ with $C(q_0, \bar u)$, form a hyperbolic subsystem. Equation (3.5) is diagonalised by the quantity $kp(q_0,\bar {u})$, where

(3.16)\begin{equation} p(q_0,\bar{u}) = \exp\left(-\int_{\bar{u}_0}^{\bar{u}} \frac{C_u(q_0,u)}{f'(u)-C(q_0,u)}\mathrm{d}u\right),\quad \bar{u}_0\in \mathbb{R}. \end{equation}

In other words, if $q=q_0$ is constant, we can use

(3.17)\begin{equation} (kp)_t + C(q_0,\bar{u})(kp)_x = 0 \end{equation}

instead of (3.5). The quantities $q$ and $kp$ have been identified in Maiden et al. (Reference Maiden, Anderson, Franco, El and Hoefer2018) as adiabatic invariants of soliton–mean flow interaction.

3.2. Soliton–mean interaction

Solutions to the solitonic modulation system can now be sought subject to an initial mean flow $\bar {u}(x,0)=\bar {u}_0(x)$ and an initial soliton with amplitude $a_0$ located at $x=x_0$. However, we need an initial amplitude and wavenumber field $a(x,0)$, $k(x,0)$ defined for all $x$. This is obtained by invoking the soliton train description and asserting that the required solution of the augmented solitonic system (3.4), (3.5) is a simple wave (to be justified), meaning all but one Riemann invariant are constant. The non-constant Riemann invariant is $\bar {u}$, in order to satisfy the initial condition. Then $a(x,0)$ is selected to maintain constant $q$

(3.18)\begin{equation} q (a(x,0),\bar{u}_0(x)) = q_0 \equiv q (a_0,\bar{u}_0(x_0)) . \end{equation}

Since $q$ constant is a solution, this reduces the augmented solitonic modulation system to the hyperbolic subsystem consisting of two diagonalised equations for $\bar u$ and $kp(q_0, \bar u)$. In order to define the initial wavenumber field $k(x,0)$, we set the latter Riemann invariant to also be constant

(3.19)\begin{equation} k(x,0) = k_0 \frac{p_0}{p (q_0,\bar{u}_0(x))}, \end{equation}

where $p_0 \equiv p ( q_0 , \bar {u}_0(x_0) )$ and $k_0 \equiv k(x_0,0) \ll 1$ is a small, positive quantity whose particular value is not important for our consideration since we assume the limit $k_0 \to 0$ in the soliton number conservation equation (3.5), and, therefore in (3.17).

The soliton–mean interaction problem can now be formulated and solved. Given the initial mean flow profile $\bar {u}(x,0) = \bar {u}_0(x)$, the soliton amplitude $a_0$ and location $x_0$, $\bar {u}(x,t)$ is the simple wave solution

(3.20a,b)\begin{equation} x - f'(\bar{u}) t = H(\bar{u}), \quad H = u_0^{{-}1} \end{equation}

and the soliton amplitude and wavenumber fields satisfy

(3.21a,b)\begin{equation} q(a,\bar{u}) = q_0, \quad k = k_0 \frac{p_0}{p(q_0,\bar{u})} . \end{equation}

We will focus our analysis on a generalised GP problem, in which initial conditions for the mean flow are given as in the original Riemann problem (2.4)

(3.22) \begin{equation} \bar{u}(x,0) = \left\{ \begin{array}{@{}ll} u_-, & x<0, \\[2pt] u_+, & x> 0, \end{array}\right.\end{equation}

and the amplitude and wavenumber fields exhibit step variations

(3.23a,b)\begin{equation} a(x,0) = \begin{cases} a_- & x < 0 \\[2pt] a_+ & x > 0 \end{cases},\quad k(x,0) = \begin{cases} k_- & x < 0 \\[2pt] k_+ & x > 0. \end{cases} \end{equation}

A sketch illustrating the generalised GP problem is shown in figure 2.

Figure 2. Sketch of the generalised GP problem.

Depending on the initial location $x_0$ of the soliton relative to the mean flow discontinuity at $x=0$, either the left $(a_-, k_-)$ or right $(a_+, k_+)$ part of the initial wave field $a(x,0), k(x,0)$ is prescribed with the other part to be determined as described below.

Due to the scaling invariance of both the quasilinear solitonic modulation system (3.4), (3.5) and the step initial data (3.22), (3.23a,b), the soliton–mean interaction problem is solved by a simple wave solution of the Riemann problem, thus justifying the constant Riemann invariant assumption for $q$ and $kp$ expressed by (3.18), (3.19). Therefore, the amplitude and wavenumber fields in the soliton–mean flow interaction must satisfy (3.21a,b), yielding the relations between admissible values of $a_{\pm }$ and $k_{\pm }$ in (3.23a,b). These are formulated as transmission and phase conditions

(3.24)$$\begin{gather} q(a_-,u_-) = q(a_+,u_+), \end{gather}$$

(3.25)$$\begin{gather}k_- p(q_0,u_-)= k_+ p(q_0,u_+), \end{gather}$$

where $p(q, \bar {u})$ is defined by (3.16). It is important to stress that the existence of the simple wave solution leading to the conditions in (3.24) and (3.25) requires convexity (genuine nonlinearity) of the characteristic field (3.6a,b) along the integral curve so that the conditions (3.9) are not violated.

In the context of a single soliton interacting with a varying mean flow connecting two equilibrium states $\bar {u}=u_-$ and $\bar {u}=u_+$, the conditions (3.24) and (3.25) should be interpreted as follows. The initial discontinuity (3.22) initiates the varying mean flow that is generally confined to the bounded, expanding region $s_-t < x < s_+ t$. There is an exception to this for the undercompressive DSW mean flow, which is a non-expanding travelling wave and requires a separate treatment. Then two basic scenarios of soliton–mean interaction can be realised that we describe by assuming positive polarity of the propagating soliton. The generalisation to negative polarity (dark) solitons is straightforward.

(i) Forward (left to right) transmission/trapping.

Assuming that the soliton with amplitude $a_->0$ is initially placed at $x_0=x_-<0$ on the left, background mean flow state $\bar {u}=u_-$, then if the soliton velocity satisfies $c(a_-, u_-)> s_-$, soliton–mean flow interaction occurs for times $t>t_1=|x_-/(c(a_-, u_-)-s_-)|$. As a result, the soliton either (a) gets transmitted (tunnels) through the variable mean flow and emerges on the right state $\bar {u} = u_+$ with the new amplitude $a_+>0$ determined by the condition (3.24) or (b) gets trapped within the variable mean flow. Trapping occurs if the transmitted soliton amplitude defined by (3.24) is negative or zero, $a_+ \le 0$.

For this case of forward transmission, the trajectory of the soliton post interaction is given by $x = c(a_+, u_+) t + x_+$, where generally $x_+ \ne x_-$. This implies that soliton–mean flow transmission is accompanied by both an amplitude change and a soliton phase shift $\varDelta =x_+-x_-$, which can be determined from the condition (3.25). To relate the $x$-intercepts $x_\pm$ of the soliton characteristic pre- and post-mean flow interaction we note that the conservation of the number of solitons in the fictitious modulated train of non-interacting solitons implies

(3.26)\begin{equation} k_- x_-\,{=}\, k_+ x_+ . \end{equation}

Given $x_-$, only the ratio of $k_+/k_-$ is needed to determine $x_+$, so, by virtue of the linear relationship between $k_+$ and $k_-$, the particular value of $k_-$ in (3.21a,b) is irrelevant. The soliton phase shift $\varDelta = x_+ - x_-$ due to interaction with the mean flow is then given by

(3.27)\begin{equation} \frac{\varDelta}{x_-} = \left (\frac{k_-}{k_+} -1 \right ) = \left(\frac{p_+}{p_-} -1 \right ), \end{equation}

where we have used the shorthand notation $p_{\pm } \equiv p(q_0,u_\pm )$.

(ii) Backward (right to left) transmission/trapping.

If the soliton with amplitude $a_+$ is initially placed at $x_0=x_+>0$ on the right background $\bar {u}=u_+$ and $c(a_+, u_+) < s_+$, then soliton–mean flow interaction occurs for times $t > t_{2}=x_+/(s_+-c(a_+, u_+))$. If the soliton eventually emerges from mean flow interaction onto the opposite constant background $\bar {u}=u_-$, its amplitude $a_->0$ and phase shift $\varDelta = x_- - x_+ = x_+(p_-/p_+ -1)$ are determined by the same transmission and phase conditions (3.24), (3.25). Otherwise, if the transmitted amplitude $a_- \le 0$, the soliton remains trapped within the mean flow.

The generalisation to negative (dark) soliton interaction with mean flow is straightforward. For this, it is convenient to introduce a signed amplitude $a$, which enables the representation of both bright $a > 0$ and dark $a < 0$ solitons. Assuming negative initial amplitude $a_\pm <0$, forward/backward transmission requires that the transmitted amplitude $a_\mp$ maintains the same, negative, sign. Generally, the condition $a_+a_->0$ is the sufficient condition for transmission in both bright and dark soliton cases. Its negation implies trapping.

In all cases of forward/backward transmission/trapping, the soliton trajectory for $t>0$ is given by the characteristic,

(3.28a,b)\begin{equation} \frac{\mathrm{d}\kern0.06em x}{\mathrm{d}t} = c(a(x,t),\bar{u}(x,t)),\quad x(0) = x_0, \end{equation}

where $|x_0| \gg 1$ so that the soliton is initially well separated from the initial step in the mean flow at $x=0$.

In the present work, we consider the implications of a non-convex solitonic modulation system (3.4) on the above soliton transmission and trapping scenarios. As described in § 3.1, non-convexity enters when strict hyperbolicity and/or genuine nonlinearity is lost via one of the three conditions: $f''(\bar {u})=0$, $f'(\bar {u})=c(a, \bar {u})$ or $c_a(a, \bar {u})=0$ for any $(\bar {u},a) \in \mathcal {A}_0$.

In Maiden et al. (Reference Maiden, Anderson, Franco, El and Hoefer2018), positivity of the transmitted amplitude (one of $a_\pm$) was proposed as a necessary and sufficient condition for bright soliton tunnelling to occur through a mean flow for convex dispersive hydrodynamics. In fact, this condition coincides with a less restrictive definition of strict hyperbolicity for (3.4) where $a = 0$ is included in the set of admissible states $\mathcal {A}_0' = \mathbb {R}\times [0,\infty )$. Generally, the soliton speed coincides with the long-wave speed when its amplitude vanishes, $c(0,\bar {u}) = f'(\bar {u})$, which signifies the onset of soliton trapping. Within the context of Whitham modulation theory, states in which $a = 0$ or $k = 0$ are not considered admissible when assessing strict hyperbolicity and genuine nonlinearity of the modulation equations because they coincide with a degeneracy in which the number of modulation equations is reduced; see, e.g. Levermore (Reference Levermore1988) and Bikbaev (Reference Bikbaev1989). We will utilise the traditional definition in which $a = 0$ is not included in the set of admissible states (3.8).

In the more general non-convex case, we find that in order for the soliton to tunnel through the mean flow, we must require the additional condition that the modulation system (3.4) remain strictly hyperbolic along the entire soliton trajectory for all admissible states $(\bar {u},a) \in \mathcal {A}_0$. If the characteristic speeds $f'(\bar {u})$ and $c(a,\bar {u})$ coincide for non-zero $a$, then strict hyperbolicity is lost and the soliton is trapped inside the mean flow. If the speeds remain separated, the soliton amplitude on the transmitted side is non-zero and the phase shift is well defined according to (3.25). In summary, the necessary and sufficient conditions for tunnelling in a non-convex solitonic modulation system (3.11) with initial data (3.22), (3.23a,b) are

(3.29a–c)\begin{equation} q(a_-,u_-) = q(a_+,u_+),\quad a_+a_-{>} 0,\quad f'(\bar{u}(x,t)) \ne c(a(x,t),\bar{u}(x,t)), \end{equation}

where $x = x(t)$ is the characteristic (3.28a,b) and $t \ge 0$.

3.3. Hydrodynamic reciprocity

So far, we have assumed that the mean flow satisfies the simple wave equation $\bar {u}_t + f'(\bar {u})\bar {u}_x = 0$. For step initial data (3.22), the only candidate continuous solution is a RW

(3.30)\begin{equation} \bar{u}(x,t) =\begin{cases} u_- & x < f'(u_-)t, \\ (f')^{{-}1}(x/t) & f'(u_-) t < x < f'(u_+) t, \\ u_+ & f'(u_+) t < x, \end{cases} \end{equation}

so long as the admissibility criterion $f'(u_-) < f'(u_+)$ holds, corresponding to expansive initial data. As will be shown in the next section, there is a much richer variety of dispersive mean flows generated by the mKdV GP problem when the initial data are compressive. Thus, we need soliton–mean flow modulation theory to be flexible enough to accommodate a wide class of mean flows.

The solitonic modulation equations (3.4), (3.5) directly apply for expansive mean flow initial data, yielding a description of soliton–RW interaction. For compressive initial data (3.22), rather than form a discontinuous shock solution, a DSW is formed that occupies the space–time region $A \subset \mathbb {R} \times (0,\infty )$ where the solution is described by the full system of Whitham modulation equations for a slowly varying nonlinear periodic wave. As a result, the Riemann invariant $q$ and secondary invariant $kp$ of the augmented solitonic system (3.4), (3.5) are not conserved in $A$, and our arguments leading to the transmission and phase conditions (3.24), (3.25) do not apply to the soliton interaction with the DSW mean flow.

To address this, we invoke an important property of the dispersive conservation law (2.1): time reversibility. A consequence of time reversibility is the continuity of the modulation solution for all $(x,t) \in \mathbb {R}^{2}$. For compressive data, we consider the solution for $t < 0$ that consists of a simple wave described by (3.30), i.e. the expansive mean flow case. Then, since $q$ and $kp$ are constant for all $x \in \mathbb {R}$ and $t < 0$, they remain constant by continuity for $(x,t)$ in the complement of $A$, outside of the oscillatory region, where the augmented solitonic system (3.4), (3.5) remains valid. Note that for the Riemann data (3.22), (3.23a,b), the solution remains continuous outside $\mathbb {R}^{2} \setminus \{(0,0)\}$, which is justified by taking the limit of smooth solutions. This property was called hydrodynamic reciprocity in Maiden et al. (Reference Maiden, Anderson, Franco, El and Hoefer2018) and has been used previously in the characterisation of DSWs for a single or pair of dispersive hydrodynamic conservation laws (El Reference El2005). Since the transmission and phase conditions (3.24), (3.25) hold outside the oscillatory region, hydrodynamic reciprocity allows us to predict the transmitted amplitude and phase shift $\varDelta$ of a soliton interacting with DSW mean flows entirely within the framework of the augmented solitonic modulation system (3.4), (3.5).

The details of the modulation dynamics for the soliton within the interior of the oscillatory region $A$ can, in principle, be described by a degenerate two-phase solution (see Flaschka, Forest & McLaughlin (Reference Flaschka, Forest and McLaughlin1980) for multiphase modulation theory of the KdV equation). However, as we will show, this rather technical approach can be partially, approximately circumvented by replacing $f(\bar {u})$ in the characteristic equation (3.13) by an appropriate choice of the mean flow variation and effectively defining a new adiabatic invariant $q$ holding within $A$.

5.1. Convex mean flows

RW mean flows (regions II and VI)

RWs that emerge from the Riemann problems in regions II and VI of figure 3 are described to leading order by

(5.1) \begin{equation} \bar{u}(x,t) = \left\{\begin{array}{@{}ll} u_-, & x<3u_-^{2} t \\ [2pt] \text{sgn}(u_+{-}u_-) \sqrt{\dfrac{x}{3t}}, & 3u_-^{2}t < x < 3u_+^{2}t\\[2pt] u_+, & x>3u_+^{2} t. \end{array}\right. \end{equation}

This is the long-time approximation of the full mKdV solution that includes dispersive regularisation of weak discontinuities at $x = 3 u_\pm ^{2} t$.

DSW mean flows (regions I and V)

The GP modulation solution describing a DSW depends on the sign of the dispersion coefficient.

According to El et al. (Reference El, Hoefer and Shearer2017) for $\mu >0$ a DSW$^{+}$ is realised as the solution to the Riemann problem with $u_- < u_+ < 0$, see quadrant V in figure 3(a). The relevant GP solution to the mKdV modulation equations (4.3) is a centred simple wave given by

(5.2a–c)\begin{equation} \lambda_1 = u_-,\quad \lambda_3 = u_+,\quad W_2(u_-,\lambda_2,u_+) = \frac{x}{t}, \end{equation}

where the characteristic speed $W_2$ is given by (B2), (B3) so

(5.3a,b)\begin{equation} \left.\begin{gathered} W_2(u_-,\lambda_2,u_+) = u_-^{2} + u_+^{2} + \lambda_2^{2} + 2(u_-^{2}-\lambda_2^{2})\frac{(1-m)K(m)}{E(m) - (1-m)K(m)},\\ m=\frac{\lambda_2^{2} - u_-^{2} }{u_+^{2} -u_-^{2}} . \end{gathered}\right\} \end{equation}

To obtain the mean flow $\bar {u}$ through a DSW, we average the mKdV periodic solution for $u$ over a period. Integrating (A3) over the period $2K(m)$ and writing the solution in terms of the Riemann invariants gives

(5.4)\begin{equation} \bar{u} ={-}(\lambda_1+\lambda_2+\lambda_3)+ 2\frac{(\lambda_2+\lambda_3)}{K(m)}\varPi \left(\frac{\lambda_1 - \lambda_2}{\lambda_1+\lambda_3},m\right), \end{equation}

where $\varPi$ is the complete elliptic integral of the third kind. The dependence $\bar {u}(x,t)$ is obtained by inserting the modulation solution (5.2a–c), (5.3a,b) in (5.4).

For $\mu <0$, a similar averaging over a period of (A11) gives

(5.5)\begin{equation} \bar{u} = \lambda_1 + \lambda_2 - \lambda_3 - 2\frac{(\lambda_1+\lambda_2)}{K(m)}\varPi \left(\frac{\lambda_2+\lambda_3} {\lambda_3-\lambda_1},m \right). \end{equation}

For a DSW$^{+}$ with $u_->u_+>0$ (quadrant I in figure 3b), the GP solution to the modulation equations is

(5.6a–c)\begin{equation} \lambda_1 ={-}u_-, \quad \lambda_3 = u_+, \quad W_2({-}u_-,\lambda_2,u_+) = \frac{x}{t}, \end{equation}

where the characteristic speed is given by (B1), (B3) is

(5.7)\begin{equation} W_2 = u_-^{2} + u_+^{2} + \lambda_2^{2} + 2(u_+^{2} - \lambda_2^{2})\frac{(1-m)K(m)}{E(m) - (1-m)K(m)}. \end{equation}

Either (5.3a,b) or (5.7) gives a parameterisation of the DSW mean flow in terms of $\lambda _2 \in (u_-,u_+)$, yielding $W_2(\bar {u})$. This behaviour is not affected by the sign of the dispersion coefficient $\mu$.

For solutions between the roots $u_1$ and $u_2$, the DSW$^{-}$ mean flow can be found by applying the transformation (A16).

For application to soliton–DSW mean flow interaction, it is instructive to write down the evolution equation for the DSW mean flow $\bar {u}(x,t)$, the simple wave equation

(5.8)\begin{equation} \bar{u}_t + W_2(\bar{u}) \bar{u}_x=0. \end{equation}

As a matter of fact, $\bar {u}(x,t)$ given by (5.4), (5.2a–c) (or (5.5), (5.6a–c)) satisfies (5.8). The advantage of using the PDE (5.8), instead of the explicitly prescribed mean flow $\bar {u}(x,t)$ will become clear later, in § 7, where we shall use it instead of the original mean flow equation $\bar {u}_t + f'(\bar {u})\bar {u}_x=0$ as part of the solitonic modulation system (3.11).

5.2. Non-convex mean flows

As we have mentioned, non-convex mean flows are generated if the Riemann data (2.4) satisfy $u_-u_+<0$. In contrast to two-parameter convex mean flows, non-convex mean flows are constrained, one-parameter families of mKdV solutions and, because of this, generally occur in combination with a convex mean flow – either a RW or DSW, see regions III, IV, VII, VIII in figure 3. The two classes of ‘pure’ non-convex mKdV mean flows are kinks described by kinks satisfying (A10) for $\mu >0$ and CDSWs when $\mu <0$ described by modulated trigonometric solutions (A14) that exhibit an algebraic soliton (A15) at one of its edges.

Kink mean flows ($\mu >0$, $u_+=-u_-$).

Unlike other mean flows that solve the mKdV GP problem, kinks are localised, steady transitions between antisymmetric means $\bar {u}(-\infty )=u_-$ and $\bar {u}(+\infty )=u_+=-u_-$ described by (A10). It has been shown in Leach (Reference Leach2012) that kinks dominate the long-time asymptotic solution of defocusing mKdV Riemann problems with antisymmetric data. Kinks are special in the sense that, in addition to considering them as mean flows, we can also treat them as localised soliton solutions that interact with convex mean flows such as RWs and DSWs.

In the limit $\mu \to 0^{+}$, kinks are the weak discontinuous solutions

(5.9) \begin{equation} \bar{u}(x,t) = \left\{\begin{array}{@{}ll} u_-, & x < u_-^{2}t, \\[2pt] -u_-, & x> u_-^{2} t, \end{array}\right.\end{equation}

of the hydrodynamic modulation equation $\bar {u}_t + (\bar {u}^{3})_x=0$ for the solitonic modulation system. The weak solution (5.9) represents an undercompressive shock (Hayes & Shearer Reference Hayes and Shearer1999; LeFloch Reference LeFloch2002) since the hydrodynamic characteristic velocity $c=3u_-^{2}=3u_+^{2}$ is the same on both sides of the shock.

CDSW mean flows ($\mu <0$, $u_+ = - u_-$)

A CDSW is a modulated trigonometric solution (A14) connecting antisymmetric states $u_-$ and $-u_+$, the negative dispersion counterpart of the kink solution. The CDSW mean flow is given by (5.5) with $\lambda _2=\lambda _3$ for CDSW$^{+}$ and $\lambda _3=-\lambda _2$ for CDSW$^{-}$.

For CDSW$^{+}$, realised when $u_->0$, we have

(5.10)\begin{equation} \bar{u}= \lambda_1 + 2 \sqrt{\lambda_1^{2} - \lambda_3^{2}}, \end{equation}

where the modulations of $\lambda _1$ and $\lambda _3$ are given by (El et al. Reference El, Hoefer and Shearer2017)

(5.11a,b)\begin{equation} \lambda_1 ={-}u_-,\quad W_2=W_3={-}3u_-^{2} + 6 \lambda_3^{2} =\frac{x}{t}. \end{equation}

As earlier, the mean flow variations satisfy (5.8).

7.1. Soliton tunnelling through RWs: regions II and VI

RWs that emerge from the GP problems in regions II and VI of figure 3 are described to leading order by (5.1). The RW is confined to the interval $3u_-^{2} t < x < 3u_+^{2} t$.

To determine the admissible directionality for soliton–mean interaction, an admissible soliton's velocity must be compared with the edge velocity of the RW. For $\mu > 0$, it is only possible for solitons to travel from right to left, implying backward soliton–mean interaction while for $\mu < 0$, solitons can only go from left to right. Soliton tunnelling occurs in either case if the system maintains strict hyperbolicity and $a \ne 0$. The tunnelling parameters are determined by the transmission conditions (6.1a,b).

For $\mu > 0$, we focus on RWs in region VI of figure 3 because bright solitons are admissible. The case of region II with dark soliton-RW interaction can be obtained by the transformation (A16). Initialising $x_0 = x_+$, $u_+ < 0$, $0 < a(x_+,0) = a_+ < -2u_+$, we only need to check that $0 < a_- < -4u_-$ for $u_+ < u_- < 0$ to prove that the characteristic speeds did not cross because the RW transitions continuously and monotonically from $u_-$ to $u_+$. We use the transmission condition (6.1a,b) to express this inequality in terms of the initial soliton amplitude $a_+$ for a bright soliton starting on the right,

(7.1)\begin{equation} 2(u_-{-}u_+) < a_+{<} -2(u_+{+} u_-). \end{equation}

The second inequality is automatically satisfied by any valid initial soliton. Hence, the first is a sufficient condition for tunnelling,

(7.2)\begin{equation} a_+{>} a_{cr} = 2(u_-{-} u_+). \end{equation}

Since $u_+ < u_- < 0$ in region VI, (7.2) gives a positive critical amplitude for transmission to occur. A numerical example of a soliton tunnelling through a RW in this case is shown in figure 4(a–c).

Figure 4. Soliton–RW interaction with $\mu = 1$, $a_+ = 1.5$, $x_+ = 100$, $u_- = -1$, $u_+ =-1.5$. The soliton trajectory (light grey curve) is accurately predicted by soliton–mean theory (dashed curve). (a) Initial condition. (b) Space–time contour plot of solution with soliton characteristic (dashed). (c) Configuration at $t = 300$. The predicted amplitude $a_-$ and phase shift $\varDelta$ are 0.5 and 96.40, respectively. The numerical solution results in $a_- = 0.4981$ and $\varDelta = 95.02$ at $t = 300$.

The soliton trajectory is specified by $\mathrm {d}\kern0.06em x/\mathrm {d}t = C(q,\bar {u})$, where $C(q, \bar {u})$ is given by (4.13). Integrating this equation we obtain

(7.3) \begin{equation} x(t) = \left\{\begin{array}{@{}ll} (u_-^{2} + 2q^{2})t + E, & x<3u_-^{2} t \\[2pt] 3q^{2}t + Dt^{1/3}, & 3u_-^{2} t < x < 3u_+^{2} t \\[2pt] (u_+^{2} + 2q^{2})t + x_+, & x>3u_+^{2} t \end{array}\right. \end{equation}

where $D$ and $E$ are obtained by continuity of $x(t)$

(7.4)$$\begin{gather} D = \tfrac{3}{2}x_+^{2/3}(2u_+^{2}-2q^{2})^{1/3} \end{gather}$$

(7.5)$$\begin{gather}E = x_+ \sqrt{\frac{u_+^{2}-q^{2}}{u_-^{2}-q^{2}}}. \end{gather}$$

The phase shift is $\varDelta = E - x_+$, which matches the condition given by (6.1a,b).

A similar analysis can be carried out for each region in figure 3 to determine the tunnelling criterion. We summarise the remaining results without detailing the analysis for each case in table 1 for either sign of $\mu$ in regions II and VI. Note that for $\mu < 0$ in region VI, the tunnelling criterion is different than the condition that $a_+ > 0$. This is because there are cases for valid initial soliton amplitudes $a_-$ where the amplitude crosses $-4\bar {u}$ during interaction with the RW, causing the soliton to become trapped. In the limit $t \to \infty$, the trapped soliton limits to an algebraic soliton moving with the mean flow velocity $3\bar {u}^{2}$.

Table 1. Results for the bright soliton tunnelling problem through RWs; $R\to L$ means that $x_0 = x_+$ and the soliton propagates from right ($R$) to left ($L$), otherwise $x_0 = x_-$ ($L \to R$).

Figure 5 illustrates the loss of strict hyperbolicity when $\mu < 0$ for non-zero amplitudes by depicting the wave curves $a(\bar {u})$ corresponding to constant $q(a,\bar {u})$ and the corresponding soliton speed $c(a(\bar {u}),\bar {u})$. For interaction to occur in region VI, solitons are initialised on the left at $x_0 = x_-$ with mean flow $u_- < 0$ (we take $u_- = -1$) and amplitudes satisfying (A7). For initial amplitudes $-4 u_- < a_- < a_{cr} = -2(u_+ + u_-)$, solitons pass through the mean flow from $u_-$ to $u_+ < u_-$ ($u_+ = -2$) maintaining positive amplitude. But these wave curves eventually intersect the critical line $-4\bar {u}$ (shown in red). In figure 7(b), the corresponding soliton speeds are plotted. Intersection of $a(\bar {u})$ with $-4\bar {u}$ corresponds to the intersection of the soliton velocity $c(a,\bar {u})$ with the characteristic velocity $3\bar {u}^{2}$, also shown in red. As the two coincide, the soliton asymptotically limits to a trapped algebraic soliton propagating with characteristic velocity.

Figure 5. Simple wave curves of constant $q(a,\bar {u})$ (solid) and curves of non-strict hyperbolicity (dashed) illustrating interaction with a RW with $u_- = -1$, $u_+ = -2$ and $\mu = -1$ for various initial amplitudes $a_-$. (a) Soliton amplitudes as a function of $\bar {u}$; (b) soliton speeds. When travelling from $u_-$ to $u_+$ with $a_-< a_{cr}$, the solid and dashed curves intersect, corresponding to loss of strict hyperbolicity.

7.2. Soliton tunnelling through DSWs: regions I and V

For $\mu > 0$, the DSW leading and trailing edge velocities for both regions I and V of figure 3 are given by $s_+ = 6u_-^{2} - 3u_+^{2}$ and $s_- = u_-^{2} + 2u_+^{2}$ (El et al. Reference El, Hoefer and Shearer2017). Comparing the soliton's initial and edge velocities, we see that interaction occurs in region V ($u_-< u_+<0$) for both directions. In region I ($0 < u_+ < u_-$), bright solitons do not exist.

First, we consider the backward interaction in region V in which $x_0 = x_+$. For the interaction to occur, the soliton speed $c(a_+,u_+)$ must be smaller than $s_+$, implying

(7.6)\begin{equation} 2(u_-{-} u_+) < a_+{<} -2u_+. \end{equation}

This condition is satisfied for any admissible, initial bright soliton $a_+$. It and the transmission condition (6.1a,b) imply that $0 < a_- < -4u_-$ so, invoking monotonicity of the DSW mean flow, we can see that strict hyperbolicity and amplitude positivity is maintained along the soliton trajectory. The soliton will always tunnel.

Next, we consider forward interaction where $x_0 = x_-$. In order for the soliton to overtake the DSW, we require $c(a_-,u_-) > s_-$, which implies that

(7.7)\begin{equation} q^{2} > u_+^{2}. \end{equation}

Then one of

(7.8)\begin{equation} a_-{>} -2(u_+{+}u_-), \quad \text{or} \quad a_-{<} 2(u_+{-} u_-) \end{equation}

holds. The first inequality in (7.8) cannot be satisfied for an admissible, initial bright soliton constrained by (A7). The second inequality in (7.8) can be satisfied by an initial bright soliton, but (6.1a,b) implies that $a_+ < 0$ so the soliton is trapped. We can also see this by comparing the characteristic velocities $c(a_-,u_-)$ and $3u_-^{2}$ where valid initial soliton amplitudes $a_-$ satisfying (A7) result in $q^{2} < u_-^{2}$. Since $q^{2} > u_+^{2}$ is necessary for the interaction to occur (see (7.7)), $u_+^{2} > u_-^{2}$, and $\bar {u}$ is continuous, the velocities must intersect and therefore the soliton is trapped by the DSW.

For $\mu < 0$, the DSW leading and trailing edge speeds are given by $s_+ = u_+^{2} + 2u_-^{2}$ and $s_- = 6u_+^{2} - 3u_-^{2}$ (El et al. Reference El, Hoefer and Shearer2017). Initial solitons with either $x_0 = x_+$ or $x_0 = x_-$ can exist in both regions I and V and the tunnelling criterion can again be determined by comparing velocities and then looking at the admissibility criterion for tunnelling. Table 2 summarises the results for bright soliton–DSW interaction.

Table 2. Results for the bright soliton–DSW interaction problem.

Numerical experiments on soliton tunnelling through DSWs were conducted for both signs of $\mu$ with results given in tables 3 and 4. Good agreement was found between the predicted amplitude and phase shift (6.1a,b), (6.3) and the numerical results, confirming that hydrodynamic reciprocity is maintained. Figure 6 shows one sample numerical solution.

Table 3. Numerical tests of backward ($R \to L$) bright soliton–DSW interaction for $\mu = 1$ and $x_+ = 100$ in region V.

Table 4. Numerical tests of forward ($L \to R$) bright soliton–DSW interaction, $\mu = -1$ and $x_- = 100$ in regions I and V.

Figure 6. Soliton–DSW interaction with $\mu = 1$, $a_+ = 1$, $x_+ = 100$, $u_- = -1.5$ and $u_+ = -1$. (a) Initial condition. (b) Space–time contour plot of solution with soliton characteristic (dashed). (c) Configuration at $t = 100$. The predicted $a_-$ is 2 and the predicted $\varDelta$ is $-38.76$. At $t=100$, the numerical solution gives $a_- = 1.9960$ and $\varDelta = -38.33$.

For a RW, the soliton trajectory $x(t)$ and amplitude $a(\bar {u})$ throughout the soliton–RW interaction is known. In contrast, the mean flow $\bar {u}(x,t)$ for a DSW is given by the modulation of a periodic travelling wave and is more complicated. Although the soliton amplitude and phase shift on either side of the DSW can be predicted without knowing the space–time variation of the mean flow in the DSW's interior by invoking hydrodynamic reciprocity (see § 3.3), the Riemann invariants $q$ and $kp$ of the augmented solitonic system (4.8), (4.9) are not held constant throughout the DSW. What is desired is some way to estimate the wave curve $a(\bar {u})$ within the DSW. Since $\bar {u}$ is known and now described by (5.8) rather than $\bar {u}_t + f'(\bar {u})\bar {u}_x=0$, we require an alternative approach to approximating the wave curve $a(\bar {u})$.

To obtain $a(\bar {u})$ along the soliton trajectory $\mathrm {d}\kern0.06em x/\mathrm {d}t = c(a, \bar {u})$, we make an assumption that soliton–DSW interaction can be approximated by the interaction of a soliton with the DSW mean flow, and take advantage of the characteristic ODE (3.13) in which we replace the characteristic velocity $f'(\bar {u})$ in the RW solution with the characteristic velocity $W_2(\bar {u})$ of the simple wave equation (5.8) for the DSW modulation solution. The velocity $W_2(\bar {u})$ is specified parametrically by (5.3a,b), (5.4) (or equivalently, (5.7), (5.5)). Thus, we obtain the ODE

(7.9a,b)\begin{equation} \frac{\mathrm{d}\tilde{k}}{\mathrm{d}\bar{u}} = \frac{\partial _{\bar{u}}\tilde{\omega}_0}{W_2(\bar{u})- \partial_{\tilde{k}} \tilde{\omega}_0},\quad \tilde{k}(u_-) = \tilde{k}_- , \end{equation}

where, as earlier, $\tilde {\omega }_0 = -\textrm {i}\omega _0(\textrm {i}\tilde {k},\bar {u})$ and $\omega _0(k,\bar {u})$ is the mKdV linear dispersion relation (4.1). The relation between the conjugate wavenumber $\tilde k$ and the soliton amplitude $a$ is given by (3.14), which is

(7.10)\begin{equation} \tilde{k}^{2} ={-}\frac{1}{\mu}\left(\frac{1}{2}a^{2}+2a\bar{u}\right) , \end{equation}

so that $\tilde {k}_-$ is (7.10) evaluated at $a = a_-$, $\bar {u} = u_-$. Substituting the expression for $W_2(\bar {u})$ given by (5.3a,b) or (5.7) and the dispersion relation (4.1) into (7.9a,b), we can numerically integrate for $\tilde {k}(\bar {u})$ and invert (7.10) to solve for the approximate wave curve $a(\bar {u})$. Since (7.10) is double valued, we use the existence conditions (A7) for $\mu > 0$ and (A13) for $\mu < 0$ in order to determine the correct value for $a$.

We now consider the implications of this mean flow approach toward understanding soliton–DSW interaction for each sign of $\mu$ separately.

First, for $\mu > 0$ and a backward soliton ($x_0 = x_+ > 0$) tunnelling through a DSW$^{+}$ (region V), figure 7(a) shows the wave curves $a(\bar {u})$ representing the soliton amplitude as it passes through the DSW mean flow with corresponding local trajectory velocity in figure 7(b). As expected from table 2, the soliton always tunnels through the DSW from right to left ($x_0 = x_+ >0$). For any valid initial positive amplitude satisfying $0 < a_+ < -2 u_+$ (A7), the soliton's amplitude neither crosses the critical line $-2\bar {u}$ nor decays to zero during propagation. Correspondingly, the velocities of solitons starting at $u_+ = -1$ and moving to $u_- = -1.5$ mostly remain below the DSW velocity $W_2(\bar {u})$, save for very small $a_+$. However, examining solitons travelling from left to right ($x_0 = x_- <0$), if the initial amplitude is below the critical value $a_- < a_{cr} = 2(u_+-u_-)$, the soliton is trapped and the amplitude decays to zero. The initial amplitudes below $a_{cr}$ in figure 7(a) correspond to the velocities in figure 7(b) that lie between $3\bar {u}^{2}$ and the characteristic $W_2(\bar {u})$, indicating that the soliton is trapped. Initial amplitudes satisfying $a_- > a_{cr}$ correspond to soliton velocities that never catch up to the DSW, $c(a_-,u_-) < W_2(u_-)$, so soliton–DSW interaction does not occur.

Figure 7. Solition amplitudes (a) and corresponding local velocities (b) of a soliton interacting with a DSW$^{+}$ for $u_- = -1.5$, $u_+ = -1$ and $\mu = 1$. When traversing curves from $u_+$ to $u_-$ with $a_+ < a_{cr} =$, the curves intersect the characteristics corresponding to loss of strict hyperbolicity. In this case, these amplitudes decay to zero.

We make two consistency checks in figure 8. The soliton amplitude $a(u_-)$ computed from the wave curve $a(\bar {u})$ that includes the point $a(u_+) = a_+$ remains within 8 % of $a_- = a_+ + 2(u_+-u_-)$ predicted by the transmission condition (6.1a,b) when $u_- = -1.5$, $u_+ = -1$. This result holds for all initial amplitudes satisfying $0 < a_+ \lessapprox 1.7134$. The upper bound on admissible initial amplitudes is below the critical value $a_{cr} = 2$. When $1.7134 < a_+ < 2$, the wave curve terminates at $a(\bar {u}) = -2\bar {u}$ for $\bar {u} > u_- = -1.5$ as shown in figure 7(a). The error is shown in figure 8 as a function of initial soliton amplitude.

Figure 8. Relative error in the predicted transmitted soliton amplitude $a(u_-)$ and soliton phase shift $\bar {\varDelta }$ for soliton–DSW$^{+}$ interaction using the mean flow description from right to left. Parameter values are $\mu = 1$, $x_0 = x_+ = 100$, $u_- = -1.5$ and $u_+ = -1$.

Another consistency check is the predicted phase shift

(7.11a–c)\begin{equation} \bar{\varDelta} \equiv x_+{-} (x(t_-) - c(a(u_-),u_-) t_-),\quad \frac{\mathrm{d}\kern0.06em x}{\mathrm{d}t} = \frac{\tilde{\omega}(\tilde{k}(x,t),\bar{u}(x,t))}{\tilde{k}(x,t)},\quad x(t_+) = s_+ t_+, \end{equation}

where $t_\pm$ are the times that the soliton's trajectory crosses the DSW's leading ($s_+t$) and trailing ($s_- t$) edges. This is compared with the phase shift $\varDelta$ determined by the transmission condition (6.3) in figure 8.

An interesting case where convexity is lost, but the trapped soliton amplitude does not decay to zero is for $\mu < 0$ and a soliton interacting with a DSW$^{-}$ (region V). Figure 9(a) shows the soliton amplitude as it passes through the DSW and figure 9(b) shows the corresponding soliton velocity. When travelling from left to right ($x_0 = x_- < 0$) with $u_- = -1.5$, admissible solitons satisfying (A13) always have velocities faster than $W_2(\bar {u})$, so interaction occurs but the velocities never cross $W_2(\bar {u})$ so strict hyperbolicity is maintained. This can also be seen in the smooth amplitude curves from $u_-$ to $u_+$, which never intersect $-4\bar {u}$. However, when the soliton is slow enough to interact from right to left through the DSW from $u_+ = -1$ to $u_- = -1.5$, then the soliton becomes trapped and the velocities lie between $3\bar {u}^{2}$ and $W_2(\bar {u})$. This corresponds to amplitudes below the critical amplitude $a_{cr} = 2(u_+-u_-)$. For a soliton with amplitudes below this critical amplitude, the velocity crosses $W_2$ and limits to $3\bar {u}^{2}$, while the amplitude limits to $-4\bar {u}$, indicating that it behaves like an algebraic soliton.

Figure 9. Soliton interaction with a DSW$^{-}$ with $u_- = -1.5$, $u_+ = -1$ and $\mu = -1$ for various amplitudes. When initialised on the right at $u_+$ with $a_+< a_{cr}$, the wave curves do not reach $u_-$ so the soliton is trapped. (a) Soliton amplitudes and (b) soliton velocities.

When tunnelling occurs, the transmitted amplitude from numerical integration of the ODE (7.9a,b) is compared with the amplitude predicted by transmission conditions and the error is shown in figure 10. For initial amplitudes that satisfy $a_- > 6.1$ with $a_{cr} = 6$ for $u_- = -1.5$ and $u_+ = -1$, the computed soliton amplitude $a(u_+)$ is within 1 % of the predicted amplitude. The predicted phase shift based on (7.11a–c) with subscripts $-$ replaced with $+$ is also compared with the phase shift as determined from the transmission condition (6.3) in figure 10.

Figure 10. Relative error in the predicted transmitted soliton amplitude $a(u_+)$ and soliton phase shift $\bar {\varDelta }$ for soliton–DSW$^{+}$ interaction using the mean flow description from left to right. Parameter values are $\mu = -1$, $x_0 = x_- = -100$, $u_- = -1.5$ and $u_+ = -1$.

8.1. Soliton–kink interaction

A kink solution to the GP problem when $\mu > 0$ is realised when $u_+ = -u_-$. To be definite, we assume that $u_-<0$. The kink velocity $u_-^{2}=u_+^{2}$ is slower than the soliton velocity for any amplitude so interaction happens from left to right with $x_0 = x_- < 0$. By the soliton existence conditions (A7), when $u_- < 0$, we must initialise with a bright soliton $(a_- > 0)$ on the left side. Since bright solitons cannot exist on the right side of the kink where $u_+ > 0$, we expect that the soliton polarity undergoes a switch as a result of kink interaction in order for the soliton to be a valid solution. To determine the transmitted soliton amplitude, we observe that, under the quadratic transformation (B3), the mKdV soliton–kink interaction problem in the limit $\mu \to 0$ is mapped onto the problem of KdV soliton train propagation on a background $-3\bar {u}^{2}$ described by the modulation system (cf. (4.5))

(8.1)\begin{equation} \left.\begin{gathered} \frac{\partial r_1}{\partial t} - r_1 \frac{\partial r_1}{\partial x} = 0, \\ \frac{\partial r_3}{\partial t} - \frac 13(r_1+2r_3) \frac{r_3}{\partial x} = 0, \end{gathered}\right\} \end{equation}

where $r_1=-3\lambda _1^{2}$ and $r_3=-3\lambda _3^{2}$. Since $\lambda _1=\bar {u}$, the quadratic transformation maps the discontinuous solution (5.9) for $\bar {u}$ to the constant solution $r_1 = -3 \bar {u}^{2}= -3 u_-^{2}$ of the first equation in (8.1). The constant solution $r_3=const$, mapped to $\lambda _3 = const$, solves the second equation, implying $|a|=2(\lambda _3-\lambda _1)= const$ for the mKdV equation.

The transformation of the soliton amplitude in soliton–kink tunnelling can be obtained from the transmission condition (6.1a,b) by assuming that the normalisation (3.15) of the Riemann invariant $q=Q(a, \bar {u})$ changes to $Q(0, \bar {u})=-\bar {u}$ when crossing the zero convexity point $\bar {u}=0$ at $x=0$, yielding the transmission condition $u_++\frac 12 a_+ = -(u_-+\frac 12 a_-)$. Since $u_-=-u_+$, we obtain $a_+ = - a_-$. Inserting this result into the soliton phase shift formula (6.3), we obtain $\varDelta =0$. To be clear, the predicted zero phase shift is an approximate result within the context of modulation theory in the limit $\mu \to 0$. For non-zero but small $\mu$, a small phase shift due to soliton–kink interaction is expected. Such an interaction for the mKdV equation can be investigated using the inverse scattering transform (IST) with non-zero boundary conditions developed for both signs of $\mu$ in Zhang & Yan (Reference Zhang and Yan2020). Within the IST formalism, the conservation of the absolute value of the soliton amplitude pre and post interaction is a consequence of the discrete spectrum's conservation.

Numerical experiments with results in table 5 confirm that, as the soliton propagates through the kink, it switches polarity while preserving the absolute value of the amplitude and that the phase shift is very small, as seen in figure 11. The observed small phase shift from numerical experiments is due to the non-zero value of $\mu$ used in numerical simulations. Note that the kink itself undergoes a phase shift in the direction opposite to the soliton.

Table 5. Numerical tests of bright soliton–kink interaction from left to right for $\mu = 1$ with $x_- = 50$. Values for $a_+$ and ${\rm \Delta} x$ according to the transmission conditions are compared with numerical (num) results.

Figure 11. Soliton–kink interaction for $\mu = 1$, $a_- = 1$, $x_- = -50$, $u_- = -1$ and $u_+ = 1$. (a) Initial condition. (b) Space–time contour plot of solution with soliton characteristic (dashed). (c) Configuration at $t = 200$. The predicted $a_+$ is –1, after flipping polarity in the interaction, and the predicted phase shift $\varDelta$ is 0. The numerical solution gives $a_+ = -1.0000$ and $\varDelta = 2.1484$ at $t = 200$.

Figure 12. (a–c) a soliton and CDSW$^{+}$ interaction where $\mu = -1$, $a_- = 1$, $x_- = -50$, $u_- = 0.5$ and $u_+ = -0.5$. (a) Initial condition. (b) Space–time contour plot of solution with soliton characteristic (dashed). (c) Configuration at $t = 120$. (d–f) soliton and CDSW$^{-}$ where $\mu = -1$, $a_- = 2.5$, $x_- = -50$, $u_- = -0.5$ and $u_+ = 0.5$. (d) Initial condition. (e) Space–time contour plot of solution with soliton characteristic (dashed). (f) Configuration at $t = 120$.

8.2. Soliton–CDSW interaction

When $\mu < 0$ and $u_- = -u_+$, the resulting solution that emerges from the GP problem is a CDSW. The leading and trailing edge travel at characteristic velocity $s_+ = 3u_-^{2}$ and $s_- = -3u_-^{2}$, so that solitons can only interact with the CDSW from left to right, $x_0 = x_- < 0$. Tunnelling always occurs in this interaction and the transmitted amplitude is obtained from (6.1a,b) as

(8.2)\begin{equation} a_+{=}a_-{+}4u_-. \end{equation}

By the existence conditions (A13), an initial soliton satisfies $a_- > -4u_-$. To maintain strict hyperbolicity, $a_+ > -4u_+$ as well. Using the transmission condition, we observe

(8.3)$$\begin{gather} a_+{>} -4u_+ \end{gather}$$

(8.4)$$\begin{gather}\implies a_-{>} 2u_+{-} 2u_-{=} -4u_-, \end{gather}$$

and this relation is always satisfied by $a_-$ so strict hyperbolicity is maintained.

The predicted phase shift is found from (6.3) with $u_+ = -u_-$ and $a_+$ given by (8.2). As with the pure kink interaction, $\varDelta = 0$. Numerical experiments shown in table 6 verify the conservation of $q$ and $kp$, although we do see a small phase shift, likely due to higher order effects. See figure 12 for depictions of soliton interaction with CDSWs of both polarities at the boundaries of regions VII and VIII (positive polarity) and III and IV (negative polarity). The predicted soliton trajectory (dashed) in figure 12 was generated by assuming that it was unchanged by the mean flow, rather than considering the CDSW mean flow given by (5.10) and the ODE (7.9a,b). This approximation is justified by the predicted zero phase shift and unchanged soliton velocity post CDSW interaction.

Table 6. Numerical tests of bright soliton–CDSW interaction from left to right for $\mu = -1$ and $x_- = -50$.

In contrast to the soliton–kink interaction, the soliton–CDSW interaction does not result in the soliton's polarity change. This is because strict hyperbolicity is maintained throughout and the existence condition (A13) for solitons with $\mu < 0$ allow for bright solitons on either side of the mean flow.

8.3. Hybrid mean flows

Regions III, IV, VII and VIII for the Riemann problem result in hybrid mean flow dynamics involving a CDSW or kink coexisting with a RW or DSW. The analysis for RWs, DSWs and pure kinks and CDSWs serve as the building blocks for determining the soliton tunnelling criterion through these combination flows. We summarise these results for $\mu > 0$ in table 7 and for $\mu < 0$ in table 8.

Table 7. Results for $\mu > 0$ with bright solitons interacting with hybrid mean flows.

Table 8. Results for $\mu < 0$ with bright solitons interacting with hybrid mean flows.

Figure 13. Backward kink–RW interaction with $\mu = 1$, $a_+ = 2$, $x_+ = 200$, $u_- = -0.5$ and $u_+ = -1$. (a) Initial condition. (b) Configuration at intermediate time $t = 300$. (c) Configuration at $t = 1000$. (d) Space–time contour plot of solution with kink characteristic (dashed). The predicted $a_-$ is 1 and the predicted $\varDelta$ is 200. The numerical solution gives $a_- = 0.9984$ and $\varDelta = 199.02$ at $t = 1000$.

Figure 14. Kink–DSW interaction with $\mu = 1$, $a_+ = 1$; $x_+ = 150$, $u_- = -1$ and $u_+ = -0.5$. (a) Initial condition. (b) Configuration at intermediate time $t = 50$. (c) Configuration at $t = 250$. (d) Space–time contour plot of solution with kink characteristic (dashed). The predicted $a_-$ is 2 and the predicted $\varDelta$ is -75. The numerical solution gives $a_- = 2.0022$ and $\varDelta = -74.56$ at $t = 250$.