Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation

Abstract

Two types of analytical undular bore solutions, of the initial value problem for the modified Korteweg-de Vries (mKdV), are found. The first, an undular bore composed of cnoidal waves, is qualitatively similar to the bore found for the KdV equation, with solitons occurring at the leading edge and small amplitude linear waves occurring at the trailing edge. The second, a newly identified type of undular bore, consists of finite amplitude sinusiodal waves, which have a rational form. At the leading edge is the mKdV algebraic soliton, while, again, small amplitude linear waves occur at the trailing edge. The initial-boundary value (IBV) problem for the mKdV equation is also examined. The solutions of the initial value problem are used to construct approximate analytical solutions of the IBV problem. An alternative analytical solution for the IBV problem, based on the assumption of an uniform train of solitons, is also developed. The parameter regimes, in which the different types of solution occur, for both the initial value and IBV problem are identified and excellent comparisons are obtained between the numerical and approximate solutions, for both problems.

Introduction

The modified Korteweg-de Vries (mKdV) equation

$$u_t+12u^2{u}_x+u_{\mathit{xxx}}=0\text{,}$$

describes weakly nonlinear long waves in physical systems for which the usual leading order nonlinearity, ${\mathit{uu}}_x$ is absent. The mKdV equation has many physical applications. These include electrodynamics, electromagnetic waves in size-quantised films, internal waves for certain special density stratifications, elastic media and traffic flow.

The mKdV equation is integrable and occurs in two versions; the cubic nonlinearity term can have either positive (the focusing case) or negative sign (the defocusing case). Hirota [1] found the N-soliton solution for the focusing mKdV equation, while both Perelman et al. [2] and Ono [3] developed the N-soliton solution for the defocusing case. Of particular interest in this case is the elastic interaction of a soliton with a dissipationless shock wave.

Whitham [4] developed modulation theory to study slowly varying wavetrains. In the case of the KdV equation these modulation equations form a system of three first-order hyperbolic partial differential equations (pdes) for the properties of the modulated wavetrain. An analytical solution of these modulation equations is a centred simple wave, see Gurevich and Pitaevskii [5]. Physically the simple wave solution represents an undular bore which describes the evolution of an initial step in mean height. The undular bore is composed of cnoidal waves, with solitons at the front of the bore and linear waves at the rear. It was found to be in good agreement with numerical solutions of the KdV equation by Fornberg Whitham [6].

The KdV undular bore solution has been widely used in applications, such as the resonant flow of a stratified fluid over topography, see Grimshaw and Smyth [7] or Smyth [8]. Other examples of integrable, or nearly-integrable, equations for which the undular bore solution has been derived include the Benjamin-Ono equation, see Jorge et al. [9], the Camassa-Holm equation, see Marchant and Smyth [10], the higher-order KdV equation, see Marchant and Smyth [11] and the Su-Gardner system, see El et al. [12].

Driscoll and O’Neil [13] derived the modulation equations for the mKdV equation, using Whitham’s theory. They found that the modulation equations were qualitatively similar to the KdV modulation equations as the Riemann invariants are composed of the sum of two of the roots (in a cyclic permutation) of the polynomial governing the periodic wave. For the defocusing mKdV equation it was found that the modulation equations are always hyperbolic, and hence the periodic solutions are always stable, while for the focusing case, the modulation equations can be either hyperbolic or elliptic, hence the modulated waves are unstable for the elliptic case. The stability of small amplitude cnoidal waves was also examined in further detail, using mode coupling theory.

Kamchatnov et al. [14] considered the small amplitude limit of a discrete nonlinear Schrödinger equation. The KdV, mKdV, KdV(2) equations were all obtained as special cases for certain parameter choices. For the defocusing mKdV equation an undular bore solution was obtained, using modulation theory, for a cubic initial condition. El [15], [16] considered the resolution of an initial-step for both integrable and non-integrable model equations and found relations for the width of the bore and the amplitude of the lead soliton in the bore. Examples considered include the KdV equation, defocusing mKdV equation and a non-integrable system describing ion-acoustic waves.

Ercolani et al. [17] examined the focusing Zakharov-Shabat hierarchy of integrable equations. They considered the zero-dispersion limit and found that, in general, the conservation equations of hierarchy members are elliptic and initial value problems are ill-posed. However, for the special case of the odd members of the hierarchy (which includes the focusing mKdV equation) and real initial conditions, the conservation equations are hyperbolic in the zero-dispersion limit, and hence initial-value problems are well posed. They also established a strong theoretical connection between the zero-dispersion limits of the odd hierarchy members and the zero-dispersion limits of the KdV hierarchy.

The initial-boundary value (IBV) problem for the KdV equation has been considered by a number of authors. It can be written as

$$\begin{array}{cc}\hfill & u_t+6{\mathit{uu}}_x+u_{\mathit{xxx}}=0\text{,}x>0\text{,}\hfill \\ \hfill & u(x\text{,}0)=u_i(x)\text{,}x>0\text{,}u(0\text{,}t)=u_b(t)\text{,}t>0\text{,}\hfill \end{array}$$

and is also referred to as the quarter-plane problem since $x>0$ and $t>0$. A number of physical applications exist for (1.2), such as the generation of waves in a shallow channel by a wave-making device or the critical withdrawal of a stratified fluid from a reservoir, see Clarke and Imberger [18].

Chu et al. [19] considered the IBV problem (1.2) numerically. For their examples, a train of solitons was generated at the boundary and it was found that the soliton amplitudes were related to the boundary condition by a simple formula. It was also shown that the addition of a damping term in the KdV equation led to a steady collisionless shock being generated. Camassa and Wu [20] found approximate solutions to (1.2) by using inverse scattering. Both trapezoidal and exponentially decaying functions were used for the time-varying boundary condition $u_b$. Their method gave a reasonable estimate for the soliton amplitudes (within 20%) for large amplitudes. However it was unable to accurately estimate the amplitudes of small solitons. A drawback to their method is that the values of $u_x$ and $u_{\mathit{xx}}$ at $x=0$ are needed, but these are not known, a priori.

Marchant and Smyth [21] also considered the KdV IBV problem (1.2) and found various types of approximate and exact solutions. For the case of constant boundary and initial conditions, various types of steady and transient solutions were derived, the particular form of the solution depending on the relationship between $u_b$ and $u_i$. In addition, a case was considered of a time-dependent boundary condition, for which an approximate solution was derived using KdV modulation theory and the fact that the solution consisted of a train of solitons. For all the cases considered, excellent comparisons were obtained between the numerical and approximate solutions of (1.2).

Marchant and Smyth [22] considered an IBV problem describing magma flow, where the governing pde

$$f_t+[f^3(1-f_{\mathit{tx}}){]}_x=0\text{,}$$

included the effects of compaction and distortion of the matrix, due to the magma. The modulation equations were derived, but due to the non-integrable nature of (1.3) undular bore solutions were obtained numerically. Analytical predictions were obtained for the amplitude of the lead solitary wave, with a good comparison obtained with numerical solutions. Depending on the parameter regime, it was found that the modulation equations could be hyperbolic (stable) or elliptic (unstable).

In Section 2 modulation theory for the mKdV equation and the relevant periodic solutions are presented. In Section 3 the two analytical undular bore solutions of the initial-value problem, for the focusing mKdV equation, are derived and compared with numerical solutions of (1.1). An excellent agreement between numerical and approximate solutions is found. In Section 4 the IBV problem for the mKdV equation is examined. Analytical solutions are constructed using the undular bore solutions of Section 3, and by assuming that an uniform train of solitons is generated at the boundary. Again, for the IBV problem, excellent comparisons are obtained between numerical and approximate solutions. For both problems, the parameter regimes in which the different types of solutions occur, are identified. Appendix A gives the details of the numerical scheme used to solve the IBV problem, for the mKdV equation.