Extended Reduced Ostrovsky Equation
Periodic and Solitary Travelling-Wave Solutions
of an Extended Reduced Ostrovsky EquationThis paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The full collection is available at http://www.emis.de/journals/SIGMA/symmetry2007.html
E. John PARKES
Received October 29, 2007, in final form June 16, 2008; Published online June 19, 2008
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.
Ostrovsky equation; Ostrovsky–Hunter equation; Vakhnenko equation; periodic waves; solitary waves; corner waves; cuspons; loops
35Q58; 35Q53: 35C05
where is the velocity of dispersionless linear waves, is the nonlinear coefficient, and and are dispersion coefficients, is a model for weakly nonlinear surface and internal waves in a rotating ocean. It was derived by Ostrovsky in 1978  and is now known as the Ostrovsky equation. For long waves, for which high-frequency dispersion is negligible, and (1.1) becomes the so called reduced Ostrovsky equation (ROE), namely
The ROE has been studied by several authors (see  and references therein).
By applying the transformation
to (1.2), we obtain the ROE in the neat form
In  we found periodic and solitary travelling-wave solutions of (1.4). (By a travelling-wave solution we mean one in which the dependence on and is via a single variable , where and are arbitrary constants.)
As mentioned in  and references cited therein, equation (1.4), with , is sometimes referred to as the Ostrovsky–Hunter equation (OHE). Vakhnenko derived equation (1.4), with , in order to model the propagation of waves in a relaxing medium [6, 7]. Parkes  dubbed this equation the Vakhnenko equation (VE).
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 the travelling-wave solutions of the OHE are periodic smooth-hump waves [4, Fig. 1] which, in the limit of maximum amplitude, become ‘corner waves’ [4, Fig. 2]; the latter have discontinuous slope at each crest and are parabolic between crests. The corresponding solutions for the VE occur when , namely periodic smooth-hump waves and parabolic corner waves with discontinuous slope at the troughs [6, Fig. 2]. When the travelling-wave solutions of the OHE are periodic inverted loops [4, Fig. 3] and a solitary inverted loop [4, Fig. 4]. The corresponding solutions for the VE occur when , namely periodic loops and a solitary loop [6, Fig. 1].
In  we showed that the solitary-loop solution of the VE when is a soliton, and that the VE has a multi-soliton solution in which each soliton is a loop that propagates in the positive -direction. During interaction the loop-solitons combine in a rather remarkable way as is illustrated in Figs. 3–5 in  for example. Clearly, similar observations apply to the inverted solitary-loop solution of the OHE when .
where , and are arbitrary constants. By using the transformation
where is a constant, we obtained the following equation:
With and , (1.8) may be written
Clearly solutions of the ROE are also solutions to equation (1.9) with . 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  we explained why the exROE (with ) is integrable in two special cases, namely when and when . In  we considered the exROE with and . We referred to the resulting equation as the generalised VE (GVE) and went on to find its -soliton solution. The th soliton may be a hump, loop or cusp depending on the value of , where is the amplitude of the th soliton. In  we considered the exROE with and . We referred to the resulting equation as the modified generalised VE (mGVE) and went on to find its -soliton solution. In  we assumed that and then the th soliton may be a hump, loop or cusp depending on the value of , where is the amplitude of the th soliton. For , the three possible types of soliton are the inverted versions of those for .
In [10, 12] we considered two-soliton interactions, i.e the case . 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  investigated periodic and solitary travelling-wave solutions of the exROE. Their first step was to follow the procedure described in , namely to introduce new independent variables and 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 , 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 and . This consideration, together with the fact that bounded solutions of the ROE have , lead us to seek solutions of the exROE subject to the restrictions
where 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 , the Degasperis–Procesi equation , the Camassa–Holm equation  and the short-pulse equation . An important feature of the method is that it delivers solutions in which both the dependent variable and the independent variable 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 . 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 ; see, for example, [18, 19] for the Degasperis–Procesi equation and  for the Camassa–Holm equation. (Related aspects of the Camassa–Holm hierarchy are discussed in .) 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 . After correcting some minor errors in , we will show that the solutions in  agree with our results. Liu et al  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 solitary-wave 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 . In Appendix B we indicate how single-valued composite solutions may be obtained from the results for multi-valued solutions derived in this paper.
2 An integrated form of the exROE
and to assume that is an implicit or explicit function of , where
It is also convenient to introduce the variable defined by the relation
(Note that 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 is a constant of integration.
and then (2.3) gives
With , equation (2.5) can be integrated once more to give
where 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  with . We make use of this in Section 3. Also, as noted in , the cubic equation has three real roots provided that .
3 Travelling-wave solutions of the exROE
The bounded solutions of equation (2.9) that we seek are such that , where , and are the three real roots of . In [4, Appendix] we gave expressions for these roots and in terms of an angle . By eliminating , we obtain , and in terms of , namely
Result 310.02 in  leads to
In view of the definition of in (2.1), it is convenient to let
where is a positive constant. It is also convenient to define the positive constant by
The travelling-wave solution to the exROE is given in parametric form by (3.8) and (3.9) with as the parameter, so that is an implicit function of . This solution agrees with the corrected versions of (3.25) and (3.27) in . (We discuss the corrections in Appendix A.) With respect to , in (3.8) is periodic with period , where is the complete elliptic integral of the first kind. It follows from (3.9) that the wavelength of regarded as an implicit function of is
where is the complete elliptic integral of the second kind.
4 Categorization of solutions
In this section we categorize solutions according to the shape of the corresponding wave profile. We discuss the cases for which and separately. Firstly we present some preliminary results.
In this sub-section we assume that .
It is convenient to define a quantity by
Observe that, for a given choice of , , and , and are constants independent of , but depends on ; on the other hand, for a given choice of , , and , , and depend on .
From (3.9) we have
For , is a strictly monotonic decreasing function of . Similarly the minimum value of is zero when
For , is a strictly monotonic increasing function of .
For , changes sign periodically. At the values of where , i.e. where as can be seen from (2.3), the profile of (regarded as an implicit function of ) has infinite slope.
Finally, we note that is periodic if in (3.10) is zero; this condition gives
In this case, may be written
The range of is , where
and is such that and is given by
The curves , and are plotted in Fig. 1.
4.2 Waves with
For , the travelling-wave solution of the exROE is given by (3.8) and (3.9). From (3.8) it can be seen that as a function of has a periodic smooth hump profile. The nature of the corresponding profile of as a function of clearly depends on the behaviour of as a function of as given by (3.9). For all values of with , except for , the range of as a function of is ; the corresponding possible periodic-wave profiles for may be categorized as follows:
Furthermore, from (4.2), according as .
When , is the periodic function of 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 plane. (A similar scenario was discussed in [15, Section 3.3] for the Degasperis–Procesi equation.) This curve is symmetrical with respect to and has infinite slope at the two points where . Periodic composite weak solutions may be constructed from the closed curve. (The notion of composite waves is discussed in [3, 24], for example.) For example, a periodic bell solution with wavelength is given in parametric form as follows:
4.3 Waves with
For , the travelling-wave solution of the exROE is given by (3.11) and (3.12). From (4.5)–(4.7), and . For all values of , except for , the range of as a function of is ; the corresponding possible solitary-wave profiles for may be categorized as follows:
Furthermore, from (4.2), according as .
i.e. a solitary wave with compact support. Thus we can construct a composite weak solution for in the form of spatially parabolic waves, i.e. corner waves, with discontinuous slope at the troughs, where
The corner waves given by (4.14) 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 and , (4.2) and (4.3) give the two possibilities
Consider the one-parameter family of waves with parameter and for which , , and are fixed and satisfy either of the conditions in (4.15). From Fig. 1 it can be seen that, for , lies below the lower curve . Hence, for , 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. , these waves become the corner waves given by (4.14) and have amplitude
We illustrate the results in Section 4 by considering two examples.
5.1 Example 1
Here we consider the simplest case, namely the VE (for which and ).
In this case, given by (4.1) reduces to , and according as . Hence, with , we have and . From Fig. 1 we deduce that the solution comprises periodic upright loops for and a solitary upright loop for . On the other hand, for , we have and . From Fig. 1 we deduce that the solution comprises periodic smooth humps for and a periodic corner-wave for . These are the results first given in [6, 8].
5.2 Example 2
Here we consider the GVE (for which ) with and arbitrary . (Note that the particular case for which is just the VE with as discussed in Example 1.)
In this case, given by (4.1) reduces to
and according as .
respectively. Note also that corresponds to
so that according as .
The curves , , and are plotted in Fig. 2.
To illustrate the results in Section 4, we let and consider the cases and separately.
With and , (5.2)–(5.5) give , , and , respectively. Figs. 3–9 correspond to seven choices of . In each figure, is plotted as an explicit function of , and is plotted as an implicit function of . In Figs. 3–8, the wave profile is periodic. In Fig. 9, the solution is a closed curve in the plane. Fig. 10 illustrates the corresponding composite solution given by (4.2).
With and , (5.2)–(5.5) give and as may be seen in Fig. 2. Figs. 11–15 correspond to five choices of . In Figs. 11–14, the wave profile is a solitary wave. In Fig. 15, the solution is the solitary wave with compact support given by (4.13). Fig. 16 illustrates the corresponding composite solution comprising periodic corner waves given by (4.14). The solitary waves for with , i.e. , coincide with the single soliton solutions given in  for the GVE as derived via Hirota’s method and illustrated in Fig. 4 in . In  it was shown that Hirota’s method fails when . Now we know that, in this case, the solution is a solitary wave with compact support or a periodic corner wave.
We have found periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation. These solutions were derived under the restrictions in (1.10). We imposed these restrictions so that our solutions reduce to solutions of the reduced Ostrovsky equation when and . We categorized the solutions by considering the values of and .
The definitions and derivations in Sections 2, 3 and 4 clearly require the restrictions in (1.10) to hold. Of course, it is of interest to ask what happens when these restrictions do not hold. 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  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 tends to a constant as . 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 (4.13). The only explicit periodic-wave solution is the corner wave given by (4.14).
In this appendix we point out some errors in .
Liu et al  claim that the exROE reduces to the VE when and . This is not true because the operators and do not commute; in fact they satisfy
If these operators did commute, then the exROE with and would reduce to
this equation with is satisfied by solutions of the VE and so the claim in  would be correct. The correct argument is as given in Section 1, namely that when and the exROE reduces to equation (1.9), and equation (1.9) with is satisfied by solutions of the VE.
In , the term in equation (3.18) should be and consequently should be 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  in order to get the first equation in (3.21), in (3.18) has to be set to zero. This is equivalent to setting in our (2.5).
As a consequence of these corrections, the expressions for in (3.22) in  should be
In (A.3), is not the in the present paper; it is equivalent to our . Liu et al  have adopted the convention used in  for the parameter of an elliptic function whereas we have adopted the convention used in ; it follows that in (A.3) is equivalent to 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, in (3.25) in  should be
and in (3.27) in  should be
It follows that the in (3.26) and (3.28) in  should be .
The illustrative figures for periodic waves in  are clearly incorrect; they bear no resemblance to the periodic humps, cusps or loops mentioned in the corresponding captions.
The wave profiles for in Figs. 5, 6 and 13 are multi-valued. Recently, Li  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
where and are determined from
Note that is not a monotonic function of . The phase portrait in the -plane is a single closed trajectory with a saddle point at . However, the phase portrait in the -plane consists of the stable and unstable manifolds through the saddle in the region , and an open curve through for which . Li’s point of view is that each of these three trajectories corresponds to a different single-valued travelling-wave solution. The multi-valued 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 given by (B.3) and given by
where and , we have the wave illustrated in Fig. 17. Note that, in this solution, is a monotonic increasing function of the parameter . 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.
The author thanks the referees for some perceptive comments and for recommending some additional references.
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