Some periodic and solitary travelling-wave solutions of the short-pulse equation, Chaos Solitons Fractals 36

Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation are investigated. Attention is restricted to solutions that, for the appropriate choice of certain constant parameters, reduce to solutions of the reduced Ostrovsky equation. It is shown how the nature of the waves may be categorized in a simple way by considering the value of a certain single combination of constant parameters. The periodic waves may be smooth humps, cuspons, loops or parabolic corner waves. The latter are shown to be the maximum-amplitude limit of a one-parameter family of periodic smooth-hump waves. The solitary waves may be a smooth hump, a cuspon, a loop or a parabolic wave with compact support. All the solutions are expressed in parametric form. Only in one circumstance can the variable parameter be eliminated to give a solution in explicit form. In this case the resulting waves are either a solitary parabolic wave with compact support or the corresponding periodic corner waves.


Introduction
The equation where c 0 is the velocity of dispersionless linear waves, a is the nonlinear coefficient, and b and γ are dispersion coefficients, is a model for weakly nonlinear surface and internal waves in a rotating ocean. It was derived by Ostrovsky in 1978 [1] and is now known as the Ostrovsky equation. For long waves, for which high-frequency dispersion is negligible, b = 0 and (1.1) becomes the so called reduced Ostrovsky equation (ROE), namely The ROE has been studied by several authors (see [2] and references therein). By applying the transformation u → u/a, t → t/ |γ|, x → (x + c 0 t)/ |γ| (1. 3) to (1.2), we obtain the ROE in the neat form In [3] we found periodic and solitary travelling-wave solutions of (1.4). (By a travelling-wave solution we mean one in which the dependence on x and t is via a single variable χ := x−vt−x 0 , where v and x 0 are arbitrary constants.) As mentioned in [4] and references cited therein, equation (1.4), with δ = −1, is sometimes referred to as the Ostrovsky-Hunter equation (OHE). Vakhnenko derived equation (1.4), with δ = 1, in order to model the propagation of waves in a relaxing medium [5,6]. Parkes [7] dubbed this equation the Vakhnenko equation (VE).
In [3] we pointed out that equation (1.4) is invariant under the transformation so that the solutions of the OHE and VE are related in a simple way. For example, given a travelling-wave solution to one of the equations, the corresponding solution to the other equation is the inverted wave travelling in the opposite direction.
When v > 0 the travelling-wave solutions of the OHE are periodic smooth-hump waves [3, Fig. 1] which, in the limit of maximum amplitude, become 'corner waves' [3,Fig. 2]; the latter have discontinuous slope at each crest and are parabolic between crests. The corresponding solutions for the VE occur when v < 0, namely periodic smooth-hump waves and parabolic corner waves with discontinuous slope at the troughs [5, Fig. 2]. When v < 0 the travelling-wave solutions of the OHE are periodic inverted loops [3, Fig. 3] and a solitary inverted loop [3,Fig. 4]. The corresponding solutions for the VE occur when v > 0, namely periodic loops and a solitary loop [5, Fig. 1].
The corner wave solution for the OHE was discussed in detail by Boyd [4] and Parkes [3]. Clearly a similar discussion applies to the corner-wave solution of the VE.
In [8] we showed that the solitary-loop solution of the VE when v > 0 is a soliton, and that the VE has a multi-soliton solution in which each soliton is a loop that propagates in the positive x-direction. During interaction the loop-solitons combine in a rather remarkable way as is illustrated in Figs. 3-5 in [8] for example. Clearly, similar observations apply to the inverted solitary-loop solution of the OHE when v < 0.
In [9] we considered a Hirota-Satsuma-type 'shallow water wave' equation [10] of the form where p = 0, q = 0 and β are arbitrary constants. By using the transformation where x 0 is a constant, we obtained the following equation: With p = q and β = 0, (1.8) may be written Clearly solutions of the ROE are also solutions to equation (1.9) with p = ±1. Because of this, hereafter we shall refer to the more general form of equation (1.9), namely equation (1.8), as the extended ROE (exROE).
In [9] we explained why the exROE (with β = 0) is integrable in two special cases, namely when p = q and when p = 2q. In [11] we considered the exROE with p = q = 1 and β = 0. We referred to the resulting equation as the generalised VE (GVE) and went on to find its N -soliton solution. The ith soliton may be a hump, loop or cusp depending on the value of β/k 2 i , where 6k 2 i is the amplitude of the ith soliton. In [9] we considered the exROE with p = 2q and β = 0. We referred to the resulting equation as the modified generalised VE (mGVE) and went on to find its N -soliton solution. In [9] we assumed that q > 0 and then the ith soliton may be a hump, loop or cusp depending on the value of β/k 2 i , where 4k 2 i /q is the amplitude of the ith soliton. For q < 0, the three possible types of soliton are the inverted versions of those for q > 0.
In [9,11] we considered two-soliton interactions, i.e the case N = 2. We found that for both the GVE and the mGVE, 'hump-hump', 'loop-loop' and 'hump-loop' 2-soliton interactions are possible. In addition, 'cusp-loop' and 'hump-cusp' interactions are possible for the GVE, and a 'cusp-cusp' interaction is possible for the mGVE.
Liu et al [12] investigated periodic and solitary travelling-wave solutions of the exROE. Their first step was to follow the procedure described in [9], namely to introduce new independent variables X and T as defined in (1.7) and hence to transform the exROE into equation (1.6). Then they used the Jacobi elliptic-function expansion method (see [13], for example) to find a solution to equation (1.6) in terms of the elliptic sn function. A transformation back to the original independent variables leads to implicit periodic and solitary-wave solutions of the exROE.
The first aim of the present paper is to find implicit periodic and solitary travelling-wave solutions of the exROE that have the property that they reduce to the bounded solutions of the ROE for the appropriate choice of parameters, namely p = q = ±1 and β = 0. This consideration, together with the fact that bounded solutions of the ROE have v = 0, lead us to seek solutions of the exROE subject to the restrictions where B is a constant of integration that is defined in Section 2.
The solution procedure that we adopt is the one that we have used previously to find implicit periodic and solitary travelling-wave solutions of the ROE [3], the Degasperis-Procesi equation [14], the Camassa-Holm equation [15] and the short-pulse equation [16]. An important feature of the method is that it delivers solutions in which both the dependent variable and the independent variable χ are given in terms of a parameter. It may or may not be possible to eliminate the parameter in order to obtain an explicit solution in which the dependent variable is given explicitly in terms of χ. Qiao et al. have tackled similar problems but have restricted attention only to explicit solutions for solitary waves for which the dependent variable tends to a constant as |x| → ∞; see, for example, [17,18] for the Degasperis-Procesi equation and [19] for the Camassa-Holm equation. (Related aspects of the Camassa-Holm hierarchy are discussed in [20].) We claim that Qiao's method does not yield such a solution for the problem considered in this paper; the only explicit solitary-wave solution derived by our method is a wave with compact support.
Our solution procedure is quite different from the one used in [12]. After correcting some minor errors in [12], we will show that the solutions in [12] agree with our results. Liu et al [12] mention that their solutions may be of different types such as loops, humps or cusps, but they made no attempt to categorize the solutions according to appropriate parameter ranges. The second aim in the present paper is to provide such a categorization.
In Section 2 we find that the quest for travelling-wave solutions of the exROE leads to a simple integrated form of equation (1.8). As this is similar in form to the corresponding equation for the VE, in Section 3 we use known results for the VE to generate implicit periodic and solitarywave solutions to the exROE. In Section 4 we categorize these solutions according to the shape of the corresponding wave profile. In Section 5 we illustrate our results with two examples. Some conclusions are given in Section 6. In Appendix A we point out some errors in [12]. In Appendix B we indicate how single-valued composite solutions may be obtained from the results for multi-valued solutions derived in this paper.

An integrated form of the exROE
As explained in Section 1, here, and subsequently, we assume that p + q = 0 and qv − β = 0. In order to seek travelling-wave solutions of equation (1.8), it is convenient to introduce a new dependent variable z defined by and to assume that z is an implicit or explicit function of η, where It is also convenient to introduce the variable ζ defined by the relation (Note that ζ is not a new spatial variable; it is the parameter in the parametric form of solution that we obtain eventually.) Then (1.8) becomes After one integration, (2.4) gives where B is a constant of integration. We impose the requirement that, for p = q = 1 and β = 0, the solutions that we seek reduce to the corresponding solutions of the VE. To do this we note that, with p = q = 1 and β = 0,  .7) give The corresponding relations for the VE are given in [3]; (2.6) agrees with (2.6) in [3], and (2.8) with B = 0 agrees with (2.7) in [3]. Accordingly we set B = 0 from here on. With B = 0, equation (2.5) can be integrated once more to give where A is a real constant. Equation (2.9) is one of the differential equations that arise in solving the VE; it is equivalent to equations (2.9) and (2.10) in [3] with δ = 1. We make use of this in Section 3. Also, as noted in [3], the cubic equation f (z) = 0 has three real roots provided that 0 ≤ A ≤ 1.

Travelling-wave solutions of the exROE
The bounded solutions of equation (2.9) that we seek are such that z 1 ≤ z 2 ≤ z ≤ z 3 , where z 1 , z 2 and z 3 are the three real roots of f (z) = 0. In [3,Appendix] we gave expressions for these roots and m := (z 3 − z 2 )/(z 3 − z 1 ) in terms of an angle θ. By eliminating θ, we obtain z 1 , z 2 and z 3 in terms of m, namely where 0 ≤ m ≤ 1. Following [3, Section 3], we may integrate equation (2.9) by using result 236.00 in [21] to obtain Result 310.02 in [21] leads to In (3.4) sn(w|m) is a Jacobian elliptic function and the notation is as used in [22,Chapter 16]; in (3.5) E(w|m) is the elliptic integral of the second kind and the notation is as used in [22,Section 17.2.8].
In view of the definition of c in (2.1), it is convenient to let where κ is a positive constant. It is also convenient to define the positive constant k by By using (3.1)-(3.7) in (2.1)-(2.3), we obtain The travelling-wave solution to the exROE is given in parametric form by (3.8) and (3.9) with w as the parameter, so that u is an implicit function of χ. This solution agrees with the corrected versions of (3.25) and (3.27) in [12]. (We discuss the corrections in Appendix A.) With respect to w, u in (3.8) is periodic with period 2K(m), where K(m) is the complete elliptic integral of the first kind. It follows from (3.9) that the wavelength λ of u regarded as an implicit function of χ is where E(m) is the complete elliptic integral of the second kind. When m = 1, the solution given by (3.8) and (3.9) becomes This solution agrees with the corrected versions of (3.26) and (3.28) in [12].

Categorization of solutions
In this section we categorize solutions according to the shape of the corresponding wave profile. We discuss the cases for which 0 < m < 1 and m = 1 separately. Firstly we present some preliminary results.

Preliminaries
In this sub-section we assume that 0 < m < 1. It is convenient to define a quantity ψ by ψ(m) := (p + q)β 4qκ 2 + pc q . (4.1) Note that by eliminating β between (4.1) and (3.6), we obtain and by eliminating κ 2 between (4.1) and (3.6), we obtain Observe that, for a given choice of p, q, β and v, ψ and κ are constants independent of m, but k depends on m; on the other hand, for a given choice of p, q, β and k, ψ, κ and v depend on m. From (3.9) we have which is a periodic function of w. From (4.4) and (3.7) we find that the maximum value of (p + q) dχ dw is zero when For ψ > ψ 1 (m), (p+q)χ is a strictly monotonic decreasing function of w. Similarly the minimum value of (p + q) dχ dw is zero when For ψ < ψ 3 (m), (p + q)χ is a strictly monotonic increasing function of w. In this case, χ may be written The range of χ is [−χ m , χ m ], where and w m is such that 0 < w m < K(m) and is given by (4.10) The curves ψ = ψ 1 (m), ψ = ψ 2 (m) and ψ = ψ 3 (m) are plotted in Fig. 1.

Waves with 0 < m < 1
For 0 < m < 1, the travelling-wave solution of the exROE is given by (3.8) and (3.9). From (3.8) it can be seen that u as a function of w has a periodic smooth hump profile. The nature of the corresponding profile of u as a function of χ clearly depends on the behaviour of χ as a function of w as given by (3.9). For all values of ψ with 0 < m < 1, except for ψ = ψ 2 (m), the range of χ as a function of w is (−∞, ∞); the corresponding possible periodic-wave profiles for (p + q)u may be categorized as follows: inverted cuspons ψ < ψ 3 (m): smooth humps Furthermore, from (4.2), (p + q)v ≷ 0 according as ψ ≷ −c. When ψ = ψ 2 (m), χ is the periodic function of w given by (4.8), and has a finite range given by (4.9). In this case the parametric solution given by (3.8) and (3.9) is just a closed curve in the (χ, u) plane. (A similar scenario was discussed in [14, Section 3.3] for the Degasperis-Procesi equation.) This curve is symmetrical with respect to χ and has infinite slope at the two points where u = v. Periodic composite weak solutions may be constructed from the closed curve. (The notion of composite waves is discussed in [2,23], for example.) For example, a periodic bell solution with wavelength 4χ m is given in parametric form as follows: where u(w), χ(w), χ m and w m are given by (3.8), (3.9), (4.9) and (4.10), respectively, and j = 0, ±1, ±2, . . ..

Waves with m = 1
For m = 1, the travelling-wave solution of the exROE is given by (3.11) and (3.12). From ; (4.13) this can be obtained directly from(4.13) or from (4.9) with m = 1. By eliminating tanh w between (3.11) and (4.12), we obtain i.e. a solitary wave with compact support. Thus we can construct a composite weak solution for (p + q)u in the form of spatially parabolic waves, i.e. corner waves, with discontinuous slope at the troughs, where The corner waves given by (4.15) are the maximum-amplitude limit of a one-parameter family of periodic smooth-hump waves. An example of such a family may be identified as follows. When ψ = −1 and 0 < m < 1, (4.2) and (4.3) give the two possibilities c = +1 : v = 0, β < 0 with p and q arbitrary; c = −1 : Consider the one-parameter family of waves with parameter m and for which p, q, β and v are fixed and satisfy either of the conditions in (4.16). From Fig. 1 it can be seen that, for 0 < m < 1, ψ = −1 lies below the lower curve ψ = ψ 3 (m). Hence, for 0 < m < 1, the family are periodic smooth-hump waves. From (3.6)-(3.8), we deduce that the amplitude of these waves is In the maximum-amplitude limit, i.e. m = 1, these waves become the corner waves given by (4.15) and have amplitude 3|qv − β| |p + q| . (4.18)

Examples
We illustrate the results in Section 4 by considering two examples.

Example 1
Here we consider the simplest case, namely the VE (for which p = q = 1 and β = 0). In this case, ψ given by (4.1) reduces to ψ = c, and v ≷ 0 according as ψ ≷ −c. Hence, with c = 1, we have ψ = 1 and v > 0. From Fig. 1 we deduce that the solution comprises periodic upright loops for 0 < m < 1 and a solitary upright loop for m = 1. On the other hand, for c = −1, we have ψ = −1 and v < 0. From Fig. 1 we deduce that the solution comprises periodic smooth humps for 0 < m < 1 and a periodic corner-wave for m = 1. These are the results first given in [5,7].

Example 2
Here we consider the GVE (for which p = q = 1) with c = 1 and arbitrary β. (Note that the particular case for which β = 0 is just the VE with c = 1 as discussed in Example 1.) In this case, ψ given by (4.1) reduces to and v ≷ 0 according as ψ ≷ −1. From (3.7) and (5.1) with (4.5), (4.7) and (4.6), the curves ψ = ψ 1 (m), ψ = ψ 2 (m) and ψ = ψ 3 (m) correspond to  figure, χ is plotted as an explicit function of w, and u is plotted as an implicit function of χ. In Figs. 3-8, the wave profile is periodic. In Fig. 9, the solution is a closed curve in the (χ, u) plane. Fig. 10 illustrates the corresponding composite solution given by (4.11).

Conclusion
We          This question may be pursued elsewhere. Here we record that additional bounded solutions do exist, but no elegant categorization procedure appears to be possible.
Recently Li [24] discussed the interpretation of multi-valued solutions of several nonlinear wave equations. In Appendix B, we discuss briefly how Li's interpretation may be applied to the multi-valued waves derived in this paper. It turns out that single-valued composite solutions may be constructed from our results.
in order to try to find an explicit solitary-wave solution for which z tends to a constant as |η| → ∞. The method appears to give a negative result. This is in agreement with our result in Section 4.3 where we showed that the only explicit solitary-wave solution that arises from our method is the solitary wave with compact support given by (4.14). The only explicit periodicwave solution is the corner wave given by (4.15).

Appendix A
In this appendix we point out some errors in [12]. Liu et al [12] claim that the exROE reduces to the VE when p = β = 0 and q = 1. This is not true because the operators D and ∂ ∂x do not commute; in fact they satisfy If these operators did commute, then the exROE with p = β = 0 and q = 0 would reduce to this equation with q = 1 is satisfied by solutions of the VE and so the claim in [12] would be correct. The correct argument is as given in Section 1, namely that when p = q and β = 0 the exROE reduces to equation (1.9), and equation (1.9) with p = 1 is satisfied by solutions of the VE. In [12], the qk 3 term in equation (3.18) should be qk 2 and consequently k 3 should be k 2 in the second term of the first and third equations in (3.21) and in the fifth term of the second equation in (3.21). Incidentally, in [12] in order to get the first equation in (3.21), C in (3.18) has to be set to zero. This is equivalent to setting B = 0 in our (2.5).
As a consequence of these corrections, the expressions for β in (3.22) in [12] should be In (A.3), c is not the c in the present paper; it is equivalent to our 1/v. Liu et al [12] have adopted the convention used in [21] for the parameter of an elliptic function whereas we have adopted the convention used in [22]; it follows that m 2 in (A.3) is equivalent to m in the present paper. By noting this difference in notation and combining our (3.7) with (A.3), we see that (A.3) is equivalent to our (3.6). Finally, ξ 1 in (3.25) in [12] should be and ξ 2 in (3.27) in [12] should be It follows that the kq in (3.26) and (3.28) in [12] should be q. The illustrative figures for periodic waves in [12] are clearly incorrect; they bear no resemblance to the periodic humps, cusps or loops mentioned in the corresponding captions.

Appendix B
The wave profiles for u in Figs. 5, 6 and 13 are multi-valued. Recently, Li [24] discussed the interpretation of similar solutions for other wave equations. Here we will apply Li's ideas to the wave illustrated in Fig. 13. The solution is given by Note that η is not a monotonic function of w. The phase portrait in the (z, z ζ )-plane is a single closed trajectory with a saddle point at (−1, 0). However, the phase portrait in the (z, z η )-plane consists of the stable and unstable manifolds through the saddle in the region −1 ≤ z < 0, and an open curve through (1/2, 0) for which 0 < z ≤ 1/2. Li's point of view is that each of these three trajectories corresponds to a different single-valued travelling-wave solution. The multivalued solution illustrated in Fig. 13 may be regarded as a composite solution of these three single-valued solutions but the three single-valued solutions may also be combined in different ways so as to give a variety of composite single-valued solutions. For example, with z given by (B.3) and η given by 2w − 3 tanh w − 2η 0 , w ∈ (−∞, −w 0 ) so η ∈ (−∞, −η 0 ) −2w + 3 tanh w, w ∈ [−w 0 , +w 0 ] so η ∈ [−η 0 , +η 0 ] 2w − 3 tanh w + 2η 0 , w ∈ (w 0 , ∞) so η ∈ (η 0 , ∞), where w 0 = tanh −1 (1/3) and η 0 = −2w 0 + √ 3, we have the wave illustrated in Fig. 17. Note that, in this solution, η is a monotonic increasing function of the parameter w. The corresponding periodic single-valued composite solution may be constructed from the single-valued solutions making up the multi-valued waves in Figs. 5 and 6.