1 Introduction

The mKdV equation is, importantly, known to be connected to the canonical Korteweg–de Vries equation of soliton theory via a transformation due to Miura [1, 2]. It has physical application in the analysis of nonlinear Alfvén waves in a collisionless plasma [3] and of acoustic wave propagation in an anharmonic lattice [4]. The remarkable connection of [1] may be shown to constitute the spatial part of a Bäcklund transformation [5, 6]. The key property of invariance under Bäcklund transformations with their associated nonlinear superposition principles (permutability theorems) with genesis in classical geometry of pseudospherical surfaces is generic to solitonic systems. In geometric terms the mKdV equation can be associated with motion of an inextensible curve of zero torsion [6]. Moreover, it can be linked via a reciprocal transformation to a loop soliton equation. It is recalled that the canonical Ablowitz–Kaup–Newell (AKNS) and Wadati–Konno–Ichikawa (WKI) inverse scattering schemes are connected by a conjugation of reciprocal and gauge transformations [7].

Moving boundary problems of Stefan-type arise notably in the analysis of the melting of solids and freezing of liquids (see e.g., [8,9,10,11,12,13,14] and cited literature). Therein, the heat balance requirement on the moving boundary which separates the phases characteristically leads to a nonlinear boundary condition. Reciprocal-type transformations have previously been applied in this connection in [15] to derive novel analytic solution to moving boundary problems for a class of nonlinear heat equations as delimited by Storm [16] to model heat conduction in a range of metals with temperature-dependent specific heat and thermal conductivity. Reciprocal-type transformations may be applied in such Stefan problems to determine conditions for the onset of melting for this class of metals subjected to applied boundary flux. Threshold melting conditions as previously derived by Tarzia [17] and Solomon et al. [18] for analogous types of moving boundary problems for the classical heat equation can be thereby extended.

Reciprocal-type transformations as originally introduced by Bateman [19] were applied in connection with lift and drag phenomena in two-dimensional homentropic gasdynamics. Canonical reduction in subsonic régimes may thereby be obtained via Kármán–Tsien model pressure-density gas laws to the classical Cauchy–Riemann system of hydrodynamics. The gasdynamic reciprocal transformations of [19] which characteristically depend on admitted conservation laws were subsequently shown to constitute particular Bäcklund transformations in [20].

The study of moving boundary problems in continuum mechanics has been and continues to be a subject of significant research interest (see e.g., [21,22,23] and work cited therein). By constrast, the literature on moving boundary problems for solitonic equations is sparse indeed. However, an intriguing solitonic connection of physical importance occurs in the analysis in [24] of the classical Saffman–Taylor problem with surface tension [25]. Therein, a one-parameter class of solutions was isolated in a description of the motion of an interface between a viscous and non-viscous liquid. This class, remarkably, was shown to be associated with travelling wave solutions of the canonical Dym equation of modern soliton theory [26]. Importantly, the latter may be linked to the potential mKdV equation via a reciprocal transformation. Novel integrable extensions of the Dym equation were derived in [27] in the geometric context of the binormal motion of an inextensible curve of constant curvature. Connection was made, in particular, with a Camassa–Holm equation of shallow water hydrodynamics.

Reciprocal transformations were applied in [28] to obtain exact solution to moving boundary problems such as arise in the context of the percolation of liquids through a porous medium such as soil.

In [29], a Painlevé II symmetry reduction was used to derive exact solutions to a class of moving boundary problems for the Dym equation. The latter type of reduction had its origin in a study of the evolution of the interface in a Hele–Shaw problem in [30]. A Bäcklund transformation was applied in [29] to generate iteratively a novel sequence of analytically solvable moving boundary problems for the Dym equation with interface of the type \(x=\gamma t^n\). Exact solutions were obtained therein in terms of Yablonski–Vorob’ev polynomials [31, 32] with n corresponding to a sequence of values of the Painlevé II parameter. The procedure was subsequently applied in [33] to generate such exact solutions involving Yablonski–Vorob’ev polynomials for an extended Dym equation.

Here, a class of Stefan-type moving boundary problems for the mKdV equation is solved by application of a similarity reduction to Painlevé II involving Airy’s equation. A novel reciprocal transformation is then introduced to isolate a linked class of solvable moving boundary problems for a canonical base Casimir member of a compacton hierarchy. In addition, a class of involutory transformations with origin in the autonomisation procedure for the Ermakov–Ray–Reid system is used to construct a wide class of solitonic Ermakov-modulated mKdV equations together with associated solvable moving boundary problems of Stefan-type.

2 A class of moving boundary problems for the mKdV equation

The canonical mKdV equation of modern soliton theory adopts the form

$$\begin{aligned} u_t-6u^2u_x+u_{xxx}=0 \end{aligned}$$
(1)

and constitutes the base member of the mKdV soliton hierarchy (see e.g., [6] and literature cited therein). Here, a class of Stefan-type moving boundary problems for the mKdV equation is considered, namely

$$\begin{aligned} \begin{array}{c} u_t-6u^2u_x+u_{xxx}=0, \quad 0<x<S(t), \quad t>0\\ \left. \begin{array}{c} u_{xx}-2u^3=L_m\ S^i\dot{S}, \ \quad \\ u=P_m\ S^j \end{array}\right\} \quad on \quad x=S(t), \quad t>0 \end{array} \end{aligned}$$
(2)

together with

$$\begin{aligned} (u_{xx}-2u^3)|_{x=0}=H_0(t+a)^k, \quad t>0 \end{aligned}$$
(3)
$$\begin{aligned} S(0)=S_0. \end{aligned}$$
(4)

In the preceding, \(L_m, P_m\) and \(H_0\ \epsilon \mathbb {R}\) while i, j, k are indices to be determined by admittance of a viable symmetry reduction. In the latter connection, an ansatz

$$\begin{aligned} u=(t+a)^m\varPhi (x/(t+a)^n) \end{aligned}$$
(5)

is introduced into the mKdV equation (1), so that

$$\begin{aligned} \begin{array}{c} m\varPhi -n\xi \varPhi '-6(t+a)^{2m-n+1}\varPhi ^2\varPhi '+(t+a)^{-3n+1}\varPhi ^{'''}=0 \\ (\xi =x/(t+a)^n) \end{array} \end{aligned}$$
(6)

whence \(n=1/3\). Accordingly,

$$\begin{aligned} m\varPhi -(1/3)\xi \varPhi '-6(t+a)^{2m+2/3}\varPhi ^2\varPhi '+\varPhi ^{'''}=0 \end{aligned}$$

so that \(m=-1/3\) and

$$\begin{aligned} \varPhi ^{'''}-6\varPhi ^2\varPhi '-(1/3)(\xi \varPhi )'=0. \end{aligned}$$
(7)

Integration of the latter yields

$$\begin{aligned} \varPhi ^{''}-2\varPhi ^3-(1/3)\ \xi \ \varPhi =\zeta , \quad \zeta \ \epsilon \ \mathbb {R} \end{aligned}$$
(8)

and, on introduction of the scalings \(\varPhi =\delta w, \ \xi =\epsilon \ z\) with

$$\begin{aligned} \delta ^2\epsilon ^2=1, \end{aligned}$$
(9)
$$\begin{aligned} \epsilon ^3/3=1 \end{aligned}$$
(10)

the classical Painlevé II equation

$$\begin{aligned} w_{zz}=2w^3+zw+\alpha \end{aligned}$$
(11)

results with Painlevé parameter \(\alpha =\zeta \ \epsilon ^2/\delta \).

3 Airy solution

Painlevé II admits an important class of exact solutions \(w=-\phi '(z)/\phi (z)\) corresponding to the parameter \(\alpha =1/2\) where \(\phi \) is governed by the classical Airy equation

$$\begin{aligned} \phi ^{''}+\frac{1}{2}z\phi =0 \end{aligned}$$
(12)

wherein \(':=d/dz\). Iterative application of a Bäcklund transformation due to Lukashevich [34] to this class generates a chain of exact solutions of \(P_{II}\) corresponding to a sequence of Painlevé parameters \(\alpha =k-1/2,\ k\ \epsilon \ \mathbb {N}^+\). Application of this result in the context of stationary boundary value problems for the Nernst–Planck electrolytic system has been detailed in [35].

Here, the class of similarity solutions

$$\begin{aligned} u=-\delta (t+a)^{-1/3}\phi '(z)/\phi (z) \end{aligned}$$
(13)

with

$$\begin{aligned} \begin{array}{c} \phi (z)=\alpha Ai(-2^{-1/2}z)+\beta Bi(-2^{-1/2}z), \\ z=x/\epsilon (t+a)^{1/3} \end{array} \qquad \alpha ,\ \beta \ \epsilon \ \mathbb {R} \end{aligned}$$
(14)

is applied to obtain exact solution to a class of moving boundary problems (2) for the solitonic mKdV equation. In the above, Ai and Bi denote Airy functions of the first and second kind respectively.

3.1 Boundary conditions

With (13), namely in extenso

$$\begin{aligned} u=-\delta (t+a)^{-1/3}\phi '(x/\epsilon (t+a)^{1/3})/\phi (x/\epsilon (t+a)^{1/3}) \end{aligned}$$

it is seen that

$$\begin{aligned} \begin{array}{ll} u_x=-\delta (t+a)^{-1/3}\left[ \dfrac{1}{\epsilon (t+a)^{1/3}}\phi ^{''}/\phi -\dfrac{1}{\epsilon (t+a)^{1/3}}(\phi '/\phi )^2\right] \\ \quad =-(\delta /\epsilon )(t+a)^{-2/3}[-z/2-(\phi '/\phi )^2]\\ \quad =(\delta /\epsilon )(t+a)^{-2/3}\left[ \dfrac{x}{2\epsilon (t+a)^{1/3}}+(\phi '/\phi )^2\right] , \end{array} \end{aligned}$$

whence

$$\begin{aligned}{} & {} u_{xx}=(\delta /2\epsilon ^2)\frac{1}{t+a}\\{} & {} \quad +(\delta /\epsilon )(t+a)^{-2/3/}[2(\phi '/\phi )[\phi ^{''}/\phi -(\phi '/\phi )^2]]\left( \dfrac{1}{\epsilon (t+a)^{1/3}}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} \begin{array}{l} u_{xx}-2u^3=(\delta /2\epsilon ^2)\dfrac{1}{t+a} +(\delta /\epsilon )(t+a)^{-2/3}\\ \qquad \left[ 2\left( \dfrac{-u(t+a)^{1/3}}{\delta }\right) \left( -\dfrac{z}{2}\right) -2\left( \dfrac{\phi '}{\phi }\right) ^3\right] \dfrac{1}{\epsilon (t+a)^{1/3}} \\ \qquad +2\delta ^3(t+a)^{-1}(\phi '/\phi )^3 \\ \qquad = \dfrac{1}{\epsilon ^2(t+a)}[\ \delta /2+ux/\epsilon \ ] \end{array} \end{aligned}$$
(15)

on use of the relation \(\delta ^2\epsilon ^2=1\). Accordingly, with \(S(t)=\gamma (t+a)^{1/3}\),

$$\begin{aligned}{} & {} (u_{xx}-2u^3)|_{x=\gamma (t+a)^{1/3}}\nonumber \\{} & {} =\frac{1}{\epsilon ^2(t+a)}[\ \delta /2+(\gamma /\epsilon )(t+a)^{1/3}P_mS^j\ ]=L_mS^i\dot{S} \end{aligned}$$
(16)

while

$$\begin{aligned} u|_{x=\gamma (t+a)^{1/3}}=P_m\gamma ^j(t+a)^{j/3}. \end{aligned}$$
(17)

Thus, \(i=j=-1\) and

$$\begin{aligned} L_m=(3/\epsilon ^2)[\ \delta /2+P_m/\epsilon \ ] \end{aligned}$$
(18)

with

$$\begin{aligned} P_m=-\gamma \delta \phi '(\gamma /\epsilon )/\phi (\gamma /\epsilon ). \end{aligned}$$
(19)

On the stationary boundary \(x=0\) with \(t>0\)

$$\begin{aligned} (u_{xx}-2u^3)|_{x=0}=(\delta /2\epsilon ^2)\frac{1}{t+a}=H_0(t+a)^k \end{aligned}$$
(20)

whence \(k=-1\) together with

$$\begin{aligned} H_0=\delta /2\epsilon ^2. \end{aligned}$$
(21)

The initial condition on the moving boundary \(x=S(t)\) yields

$$\begin{aligned} S|_{t=0}=\gamma (t+a)^{1/3}|_{t=0}=\gamma a^{1/3}=S_0. \end{aligned}$$
(22)

4 Application of a reciprocal transformation

4.1 A Casimir connection

Under the reciprocal transformation

$$\begin{aligned} \begin{array}{c} dx^*=u\ dx+[\ -u_{xx}+2u^3\ ]dt, \quad dt^*=dt \\ \psi =u^{-1} \end{array} \end{aligned}$$
(23)

applied to the mKdV equation (1), it is seen that

$$\begin{aligned} dx=(1/u)dx^*-(1/u)[\ -u_{xx}+2u^3\ ]dt^* \end{aligned}$$
(24)

whence

$$\begin{aligned} \partial \psi /\partial t^*+(2\partial _{x^*}-1/2\ \partial ^3_{x^*x^*x^*})\psi ^{-2}=0. \end{aligned}$$
(25)

The latter constitutes the base third order member of the compacton hierarchy of [36], namely the Casimir equation

$$\begin{aligned} \partial \psi ^\dagger /\partial t^\dagger =(\partial _{x^\dagger }-\partial ^3_{x^\dagger x^\dagger x^\dagger })\psi ^{\dagger -2} \end{aligned}$$
(26)

on appropriate scalings \(\psi =\lambda \psi ^\dagger , \ x^*=\mu x^\dagger \) together with \(t^*=t^\dagger \).

4.2 A class of reciprocal moving boundary problems

Here, the reciprocal of the mKdV moving boundary problems determined by (1)–(3) becomes

$$\begin{aligned} \begin{array}{c} \partial \psi /\partial t^*+(2\partial _{x^*}-1/2\ \partial ^3_{x^*x^*x^*})\psi ^{-2}=0, \ x^*|_{x=0}<x^*<S^*(t^*), \ t^*>0\\ \left. \begin{array}{c} {[}\ 1/2\ \partial ^2/\partial x^{*2}-2\ ]\psi ^{-2}=L_m\ S^i\dot{S}\psi \ \\ \psi ^{-1}=P_mS^j \end{array}\right\} \ on \ x^*=S^*(t^*), \ t^*>0 \\ {[}\ 1/2\ \partial ^2/\partial x^{*2}-2\ ]\psi ^{-2}|_{x^*|_{x=0}}=H_0(t^*+a)^k, \\ S^*(0)=S^*_0 \end{array} \end{aligned}$$
(27)

with \(i=j=k=-1\).

The reciprocal relations yield

$$\begin{aligned} x^*=-\delta \epsilon \ln \phi (x/\epsilon (t+a)^{1/3})+T(t) \end{aligned}$$
(28)

whence

$$\begin{aligned} \begin{array}{l} \partial x^*/\partial t=\delta /3\ (t+a)^{-4/3}x \phi '(x/\epsilon (t+a)^{1/3})/\phi (x/\epsilon (t+a)^{1/3})+\dot{T}(t) \\ \ =-u_{xx}+2u^3=-\dfrac{1}{t+a}[\ \delta /2\epsilon ^2+x u/\epsilon ^3\ ] \\ \ =-\dfrac{1}{t+a}\left[ \dfrac{\delta }{2\epsilon ^2}-\dfrac{x\delta }{\epsilon ^3(t+a)^{1/3}}\phi '\left( \dfrac{x}{\epsilon (t+a)^{1/3}}\right) /\phi \left( \dfrac{x}{\epsilon (t+a)^{1/3}}\right) \right] \end{array} \end{aligned}$$
(29)

so that

$$\begin{aligned} \dot{T}=-\frac{1}{t+a}\delta /2\epsilon ^2 \end{aligned}$$
(30)

on use of the relation \(\epsilon ^3=3\). Accordingly,

$$\begin{aligned} x^*=-\delta \epsilon \ln \phi (x/\epsilon (t+a)^{1/3})-(\delta /2\epsilon ^2) \ln |t+a|, \quad t^*=t \end{aligned}$$
(31)

and the reciprocal moving boundary is determined by

$$\begin{aligned} x^*|_{x=S(t)}=-(\delta /2\epsilon ^2) \ln |t^*+a|-\delta \epsilon \ln \phi (\gamma /\epsilon )=S^*(t^*) \end{aligned}$$

5 Modulation

Systems which incorporate spatial or temporal modulation arise naturally in both nonlinear physics and continuum mechanics. In physics, they occur notably in nonlinear optics and the theory of Bose–Einstein condensates [33, 37,38,39,40]. In continuum mechanics they have been shown to have importance in the analysis of initial and boundary value problems in elastodynamics, visco-elastodynamics and elastostatics of both inhomogeneous and initially stressed media.

In recent work [41], modulated coupled systems of sine-Gordon, Demoulin and Manakov-type have been systematically reduced to their solitonic unmodulated counterparts via a class of involutory transformations. Here, such transformations of the type

$$\begin{aligned} \left. \begin{array}{c} dx^*=\rho ^{-2}(x)dx, \quad t^*=t\, \quad \\ u^*=\rho ^{-1}(x)u \end{array}\right\} \quad \text {on} \quad \mathbb {R}^* \end{aligned}$$
(32)

are applied to the class of moving boundary problems for the mKdV equation (1) determined by the conditions (2)–(4) to construct an associated class of solvable modulated boundary value problems. It is seen that if the relations in \(\mathbb {R}^*\) are augmented by \(\rho ^*=\rho ^{-1}\) then the involutory property \(\mathbb {R}^{**}=\textrm{I}\) results. This kind of transformation was originally applied in [42] in the autonomisation of the Ermakov–Ray–Reid system [43, 44]. The latter arises notably in nonlinear optics [45,46,47,48,49,50,51], and in diverse other areas of physical application [52,53,54,55,56,57,58].

Here, under \(\mathbb {R}^*\) the mKdV equation (1) becomes

$$\begin{aligned}{} & {} \partial (\rho ^{*-1}u^*)/\partial t^*+\rho ^{*2}\frac{\partial }{\partial x^*}\left[ \rho ^{*2}\frac{\partial }{\partial x^*}\left[ \rho ^{*2}\frac{\partial }{\partial x^*}\left( \frac{u^*}{\rho ^*}\right) \right] \right] \\{} & {} -2\rho ^{*2}\frac{\partial }{\partial x^*}[\ \rho ^{*-3}u^{*3}\ ]=0 \end{aligned}$$

that is

$$\begin{aligned}{} & {} \partial (\rho ^{*-1}u^*)/\partial t^*+\rho ^{*2}\frac{\partial }{\partial x^*}[\ \rho ^{*2}(\rho ^*u^*_{x^*x^*}-u^*\rho ^*_{x^*x^*})\nonumber \\{} & {} -2\rho ^{*-3}u^{*3}\ ]=0 \end{aligned}$$
(33)

while the boundary conditions (2) and (3) respectively yield

$$\begin{aligned} \left. \begin{array}{c} \rho ^{*2}(\rho ^*u^*_{x^*x^*}-u^*\rho ^*_{x^*x^*})-2\rho ^{*-3}u^{*3}=L_mS^i\dot{S} \ \\ \rho ^{*-1}u^*=P_mS^j \end{array}\right\} \ \text {on} \ x^*=S^*(t^*), \ t^*>0 \ \end{aligned}$$
(34)

and

$$\begin{aligned}{} & {} [\ \rho ^{*2}(\rho ^*u^*_{x^*x^*}-u^*\rho ^*_{x^*x^*}-2\rho ^{*-3}u^{*3}\ ]|_{x^*|_{x=0}}\nonumber \\{} & {} =H_0(t^*+a)^k, \ t^*>0. \end{aligned}$$
(35)

Herein, \(x^*=S^*(t^*)\) and hence the initial condition \(S^*(0)\) are determined via integration of the relation

$$\begin{aligned} dx^*=\rho ^{-2}(x)dx|_{x=\gamma (t+a)^{1/3}}. \end{aligned}$$
(36)

5.1 Ermakov modulation

If the modulation term \(\rho (x)\) in the class of involutory transformations \(\mathbb {R}^*\) is determined by the classical Ermakov equation in \(\rho =\rho ^{*-1}\), namely [59]

$$\begin{aligned} \rho _{xx}+\omega (x)\rho =c/\rho ^3 \end{aligned}$$
(37)

then the nonlinear superposition principle

$$\begin{aligned} \rho =\sqrt{c_1\Omega ^2_1+2c_2\Omega _1\Omega _2+c_3\Omega ^2_2} \end{aligned}$$
(38)

is admitted in which \(\Omega _1,\ \Omega _2\) constitute a pair of linearly independent solutions of the auxiliary equation

$$\begin{aligned} \Omega _{xx}+\omega (x)\Omega =0 \end{aligned}$$
(39)

and with constants c together with \(c_i, i=1,2\) such that

$$\begin{aligned} c_1c_3-c^2_2=c/\mathcal {W}^2 \end{aligned}$$
(40)

where \(\mathcal {W}=\Omega _1\Omega _{2x}-\Omega _{1x}\Omega _2\) is the constant Wronskian of \(\Omega _1, \Omega _2\). This result and generalisations may be obtained via Lie group methods [60]. Physical applications of the classical Ermakov equation (37) arise, in the elastodynamics of boundary-loaded neo-Hookean hyperelastic tubes of Mooney-Rivlin material as detailed in [61].

The class of Ermakov-modulated moving boundary problems with conditions (34)–(35) is connected parametrically to its unmodulated counterpart by the relations

$$\begin{aligned}{} & {} u^*=u/\sqrt{c_1\Omega ^2_1+2c_2\Omega _1\Omega _2+c_3\Omega ^2_2}, \end{aligned}$$
(41)
$$\begin{aligned}{} & {} x^*(x)=\int \frac{dx}{c_1\Omega ^2_1+2c_2\Omega _1\Omega _2+c_3\Omega ^2_2}\nonumber \\{} & {} \quad =\frac{1}{\mathcal {W}}\int ^{z=\Omega _2/\Omega _1}\frac{1}{c_1+2c_2z+c_3z^2}dz \end{aligned}$$
(42)

while

$$\begin{aligned} S^*(t^*)=x^*|_{x=\gamma (t^*+a)^{1/3}}. \end{aligned}$$
(43)

In this manner, such nonlinear Ermakov-modulated moving boundary systems may be reduced to their exactly solvable unmodulated counterparts. Modulations determined by specialisations of (39) include, inter alia, those which involve Lamé, Mathieu or Airy terms.

6 Conclusion

Application of Airy-type solutions of the classical Painlevé II equation has previously been made to time-independent boundary value problems associated with the Nernst–Planck electrolytic system [35]. It was established therein that a chain of exact solutions may be derived by iterated action of a Bäcklund transformation admitted by Painlevé II on the seed Airy class. Here, Airy-type solution of a Painlevé II reduction of a canonical mKdV equation is applied to solve classes of Stefan-type moving boundary problems. The application of iteration of the Painlevé II Bäcklund transformation in this context as conducted in the exact solution of moving boundary problems for the solitonic Dym equation in [29] remains to be investigated.

A moving boundary problem in plasma physics has been previously investigated in [62]. Diverse physical applications of the mKdV equation have been documented in [63]. The type of application of Airy-type representations to moving boundary problems as presented here for the mKdV equation may, in principle, be extended to the solitonic Gardner which represents a composition of the mKdV and KdV equations. This is under current investigation. The Gardner equation has important physical applications, notably in plasma physics, optical lattice theory and nonlinear wave propagation in hydrodynamics. It has recently been shown to arise in nonlinear elastodynamics [64].