Solutions Classification to the Extended Reduced Ostrovsky Equation
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2008), 073, 19 pages Solutions Classif icationto the Extended Reduced Ostrovsky Equation
Yury A. STEPANYANTSAustralian Nuclear Science and Technology,Organisation PMB 1, Menai (Sydney), NSW, 2234, Australia
E-mail: [email protected]
Received July 14, 2008, in final form October 13, 2008; Published online October 26, 2008Original article is available at
Abstract.
An alternative to the Parkes’ approach [
SIGMA (2008), 053, 17 pages] issuggested for the solutions categorization to the extended reduced Ostrovsky equation (theexROE in Parkes’ terminology). The approach is based on the application of the qualitativetheory of differential equations which includes a mechanical analogy with the point particlemotion in a potential field, the phase plane method, analysis of homoclinic trajectories andthe like. Such an approach is seemed more vivid and free of some restrictions contained in[ SIGMA (2008), 053, 17 pages]. Key words: reduced Ostrovsky equation; mechanical analogy; phase plane; periodic waves;solitary waves, compactons
In paper [1] E.J. Parkes presented a categorization of solutions of the equation dubbed theextended reduced Ostrovsky equation (exROE). The equation studied has the form ∂∂x (cid:18) D u + 12 pu + βu (cid:19) + q D u = 0 , where D = ∂∂t + u ∂∂x (1)with p , q , and β being constant coefficients. This equation was derived from the Hirota–Satsuma-type shallow water wave equation considered in [2] (for details see [1]).For stationary solutions, i.e. solutions in the form of travelling waves depending only on onevariable χ = x − V t − x , this equation reduces to the simple third-order ODE: ddχ (cid:20) w ddχ (cid:18) w dwdχ (cid:19) + 12 pw + ( pV + β ) w (cid:21) + qw dwdχ = 0 , (2)where V stands for the wave speed and w = u − V . (Note, that in many contemporary papersincluding [1] authors call such solutions simply “travelling-wave solutions”. Such terminologyseems not good as nonstationary propagating waves also are travelling waves. The term “sta-tionary waves” widely used earlier seems more adequate for the waves considered here.) Inpaper [1], equation (2) was reduced by means of a series of transformations of dependent andindependent variables to an auxiliary equation whose solutions were actually categorized subjectto some restrictions on the equation coefficients, viz.: p + q (cid:54) = 0 , qV − β (cid:54) = 0(one more restriction on the constant of integration for that auxiliary equation, B = 0, wasused in [1]). Under these restrictions, solutions to equation (2) were found in analytical form a r X i v : . [ n li n . S I] O c t Y.A. Stepanyantsand corresponding wave profiles were illustrated graphically. Among solutions obtained thereare both periodic and solitary type solutions including multivalued loop periodic waves andloop-solitons.Similar loop solutions to exROE and some other equations were earlier obtained by Ji-BinLi [3] who came to the conclusion that loop solutions actually are compound solutions whichconsist of three different independent branches. These branches may be used in various combi-nations representing several types of stationary propagating singular waves (waves with infinitegradients). This conclusion completely coincides with the conclusion of paper [4] where a com-plete classification of stationary solutions of ROE was presented. ROE derived by L.A. Ostrov-sky [5] in 1978 as a model for the description of long waves in a rotating ocean (see [4] andreferences therein) can be treated as a particular case of exROE with p = q and β = 0 (see [1]).Below an analysis of stationary solutions to equation (2) is presented by the direct methodavoiding any redundant transformations of variables. The method used is based on the phaseplane concept and analogy of the equation studied with the Newtonian equation for the pointparticle in a potential field. Such approach seems more vivid and free of aforementioned re-strictions. This work can be considered also as complementary to paper [1] as the analysispresented may be helpful in the understanding of basic properties of stationary solutions ofequation (2). Equation (2) can be integrated once resulting in w ddχ (cid:18) w dwdχ (cid:19) + 12 ( p + q ) w + ( pV + β ) w = C , (3)where C is a constant of integration. By multiplying this equation by dw/dχ and integratingonce again, the equation can be reduced to the form of energy conservation for a point particleof unit mass moving in the potential field P ( w ):12 (cid:18) dwdχ (cid:19) + P ( w ) = E, (4)where the effective “potential energy” as a function of “displacement” w is P ( w ) = p + q w − C w − C w , (5)and C is another constant of integration. The constant E = − ( pV + β ) / K = (1 / dw/dχ ) , andthe “potential energy”, P ( w ). As follows from equation (4), real solutions can exist only for E ≥ P ( w ). Various cases of the potential function (5) are considered below and correspondingbounded solutions are constructed. Unbounded solutions are not considered in this paper asthey are less interesting from the physical point of view; nevertheless, their qualitative behaviorbecomes clear from the general view of corresponding phase portraits. p + q = 0 Consider first a particular case when the coefficients in equation (1) are such that p + q = 0.Note, this is one of the cases which were omitted from the consideration in paper [1]. Theolutions Classification to the Extended Reduced Ostrovsky Equation 3 Figure 1. a) Potential function for the case p + q = 0, C = 0 and two values of C : C = 1 (solid lines),and C = − C = 1 and different values of E . Line 1: E = −
1; line 2: E = − .
5; lines 3: E = 0 .
5; lines 4: E = 1. potential function (5) simplifies in this case. However, a variety of subcases can be distinguishednevertheless even in this case depending on the coefficients C and C . All these subcases arestudied in detail below. If C = 0, the potential function represents a set of antisymmetric hyperbolas locatedeither in the first and third quadrants or in the second and fourth quadrants as shown in Fig. 1a.The corresponding phase plane ( w, w (cid:48) ), where w (cid:48) = dw/dχ , is shown in Fig. 1b for C = 1 (for C = − C phase portraits are qualitatively similar to that shown in Fig. 1 for C = 1.Analysis of the phase portrait shows that there are no bounded solutions for any positive E ;corresponding trajectories both in the left half and right half of the phase plane go to infinityon w (see, e.g., lines 3 and 4 in Fig. 1b). Meanwhile, solutions bounded on w do exist for negativevalues of E (i.e. for V > − β/p ), but they possess infinite derivatives when w = 0. Consider,for instance, motion of an affix along the line 2 in Fig. 1b ( C = 1) from w (cid:48) = ∞ towards theaxis w where w (cid:48) = 0. The qualitative character of the motion becomes clear if we interpret it interms of “particle coordinate” w and “particle velocity” w (cid:48) treating ξ as the time. The motionoriginates at some “time” ξ with infinite derivative and zero “particle coordinate” w = 0. Then,the “particle coordinate” w increases to some maximum value w max = − C /E ( E <
0) as the“particle velocity” is positive. Eventually it comes to the rest having zero derivative w (cid:48) = 0and w = w max . Another independent branch of solution for the same value of E correspondsto the affix motion along the line 1 from the previously described rest point at axis w towards w (cid:48) = −∞ and w = 0.All bounded analytical solutions for this case can be presented in the universal implicit form: ξ ( y ) − ξ = ± (cid:20) arctan (cid:18)(cid:114) y − y (cid:19) − (cid:112) y (1 − y ) (cid:21) , (6)where y = − Ew/C , ξ = −√ − E ) / χ/C and ξ is an arbitrary constant of integration. Thissolution consists of two independent branches which correspond to signs plus or minus in frontof the square brackets in equation (6). Each branch is defined only on a compact support ofaxis ξ : either on − π/ ≤ ξ − ξ ≤ ≤ ξ − ξ ≤ π/ (cid:48) in Fig. 2).With the appropriate choice of constants ξ one can create a variety of different solutions, e.g.,the V -shape wave (see lines 2 and 2 (cid:48) ), or a smooth-crest compacton, i.e. a compound solitarywave defined only for | ξ − ξ | ≤ π/ (cid:48) ). Using a translational invariance of Y.A. Stepanyants Figure 2.
Various particular solutions described by equation (6).
Figure 3.
Maximum of the compacton solution (6) against speed in the original variables, equation (7).Dashed vertical line corresponds to the limiting value of V = − β/p . The plot is generated for C = p = β = 1. solutions and their independency of each other, one can create periodic or even chaotic sequencesof compactons randomly located on axis ξ .The maximum of the function y ( ξ ), y max = 1, corresponds in terms of w to w max = − C /E .Using the relationship between w and the original variable u (see above), as well as the definitionof the constant E , one can deduce the relationship between the wave extreme value (wavemaximum) and its speed: u max = V − C E = V + 2 C pV + β . (7)Taking into account that we consider the case of C = 1, and negative values of E are possibleonly when V > − β/p , the plot of u max ( V ) is such as presented in Fig. 3.As follows from equation (7), a wave is entirely negative ( u max < V < p (cid:16)(cid:112) β − pC − β (cid:17) , provided that p < β / (8 C ). At greater values of V , the wave profile contains both positive andnegative pieces, and for certain value of V the total wave “mass” I = (cid:82) u ( χ ) dχ vanishes (theintegral here is taken over the entire domain where function u ( χ ) is defined).olutions Classification to the Extended Reduced Ostrovsky Equation 5 Figure 4. a) Potential function for the case p + q = 0, C = 0 and two values of C : C = 1 (solid lines),and C = − C = 1 and different values of E . Line 1: E = −
1; line 2: E = − .
5; lines 3: E = 0 .
5; lines 4: E = 1. All lines are symmetrical with respect toaxis w (cid:48) and are labelled only in the left half of the phase plane. A similar analysis can be carried out for the case when C = 0, C (cid:54) = 0. The potentialfunction in this case represents a set of symmetric quadratic hyperbolas located either in the firstand second quadrants or in the third and fourth quadrants as shown in Fig. 4a for C = ±
1. Thecorresponding phase plane is shown in Fig. 4b for C = 1 only (there are no bounded solutionsfor C = −
1, therefore this case is not considered here). For other positive values of C phaseportraits are qualitatively similar to that shown in Fig. 4b.Analysis of the phase portrait shows that there are no bounded solutions for C = −
1, aswell as for C = 1 and any positive E (see, e.g., lines 3 and 4 in Fig. 4b); they exist however for C = 1 and negative values of E , but possess infinite derivatives at some values of χ . In nor-malized variables y = ( − E/C ) / w , ξ = − E (2 /C ) / χ all possible solutions can be presentedin terms of independently chosen function branches describing a unit circle in one of the fourquadrants, i.e.( ξ − ξ ) + y = 1 , (8)where ξ is an arbitrary constant of integration.Playing with the constant ξ one can create again a variety of compacton-type solutionsincluding multi-valued solutions. Some examples of solitary compacton solutions are shown inFig. 5a; they include N -shaped waves, multi-valued circle-shaped waves and semicircle positive-polarity pulses (due to symmetry, the polarity of the first and last waves can be inverted). Inaddition to those, various periodic and even chaotic compound waves can be easily constructed;one of the possible examples of a periodic solution is shown in Fig. 5b. Each positive or negativehalf-period of any wave consists of two independent branches originating at y = 0 and endingat y = ±
1. The same is true for the pulse-type solutions shown in Fig. 5a; they consist ofindependent symmetrical branches as shown, for example, for the semicircle pulse in Fig. 5awhere they are labelled by symbols 1 and 2.The maximum of the function y ( ξ ), y max = 1, corresponds in terms of w to the wave maxi-mum, w max = ( − C /E ) / . Using a relationship between w and the original variable u (seeabove), as well as definition of the constant E , one can deduce the relationship between the Y.A. Stepanyants Figure 5. a) Some examples of pulse-type waves described by equation (8): N -shaped wave; circle waveand semicircle compacton. b) One of the examples of a periodic wave with infinite derivatives at y = 0, ξ = 2 n + 1, where n is an entire number. wave maximum and its speed: u max = V + (cid:114) − C E = V + (cid:115) C pV + β . (9)The plot of u max ( V ) is presented in Fig. 6 for V > − β/p in accordance with the chosen constant C = 1 and E < U max ( − U min ), which occurs at some speed V , where U max = 1 p (cid:34)(cid:18) C p (cid:19) / − β (cid:35) + 2 (cid:18) C p (cid:19) / , V = 1 p (cid:34)(cid:18) C p (cid:19) / − β (cid:35) . For all possible values of wave maximum u max > U max , two values of wave speed are pos-sible, i.e. two waves of the very same “amplitude” can propagate with different speeds. This isillustrated by horizontal dashed line in Fig. 6 drawn for u max = 2 .
5. The same is true for wavesof negative polarity.
Consider now the case when both C and C are nonzero but p + q is still zero. Thereare in general four possible combinations of signs of the parameters C and C :i) C > , C >
0; ii) C < , C >
0; iii) C > , C <
0; iv) C < , C < . The shape of the potential function P ( w ) and corresponding solutions are different for allthese cases. However, among them there are only two qualitatively different and independentcases, whereas the two others can be obtained from those two cases using simple symmetryreasons. This statement is illustrated by Fig. 7, where the potential relief is shown for all fouraforementioned cases i)–iv).As one can see from Fig. 7, cases i) and ii), as well as iii) and iv), are mirror symmetricalcounterparts of each other with respect to the vertical axis. This implies that solutions for thecases i) and ii), and correspondingly, iii) and iv), are related by the simple sign interchangeolutions Classification to the Extended Reduced Ostrovsky Equation 7 Figure 6.
Dependence of the wave maximum on speed in original variables, equation (9), as follows fromsolution (8). Dashed vertical line corresponds to V = − β/p . The plot is generated for C = p = β = 1. Figure 7.
Potential relief for the four different cases, i)–iv), of various signs of constants C and C .The plot was generated for C = ± C = ± . operation, i.e. w i) = − w ii) , w iii) = − w iv) . Therefore, below only two qualitatively different casesare considered in detail, namely the cases i) and iii).Case i) is characterized by an infinite potential well at the origin, w = 0. This singularityin the potential function corresponds to the existence of a singular straight line w = 0 on thephase plane (see Fig. 8). On both sides from this singular line there are qualitatively similartrajectories which correspond to bounded solutions having infinite derivatives at the edges.Quantitative difference between the “left-hand side solutions” and “right-hand side solutions”,apart of their different polarity, is the former solutions (of negative polarity, w ≤
0) exist for E ≤ P max , whereas the latter ones (of positive polarity, w ≥
0) exist for E ≤
0. The potentialfunction has a maximum P max = C / (4 C ) at w = − C /C . There are no bounded solutionsfor E > P max .Consider first bounded solutions which correspond to trajectories shown in the left half-plane, w ≤
0, in Fig. 8. For a positive value of the parameter E in the range 0 ≤ E ≤ P max , theanalytical solution can be presented in the form ξ ( y ) = ± (cid:112) Q (cid:34)(cid:112) ( y + 2 Q ) − Q ( Q − − Q ln y + 2 Q + (cid:112) ( y + 2 Q ) − Q ( Q − (cid:112) Q ( Q − (cid:35) , (10)where ξ = χ √ C ( C /C ) , y = w ( C /C ), Q = C / (4 C E ). Y.A. Stepanyants Figure 8.
Phase portrait of equations (3), (4) for the case i) (only those trajectories are shown whichcorrespond to particle motion within the potential well in Fig. 7a). Line 1: E = 2 .
5; line 2: E = 1;lines 3: E = −
1; line 4: E = −
2; line 4: E = − The range of variability on ξ is: | ξ | ≤ Q (cid:110) − (cid:112) Q ln (cid:104)(cid:0)(cid:112) Q + 1 (cid:1) / (cid:112) Q − (cid:105)(cid:111) , whereas y varies in the range − (cid:104) Q − (cid:112) Q ( Q − (cid:105) ≤ y ≤ . The relationship between the wave minimum and its speed is: u min = V + C pV + β + (cid:115) C pV + β (cid:20) C C ( pV + β ) + 1 (cid:21) , (11)where pV + β < E >
E <
0, then the solution is ξ ( y ) = ± (cid:112) − Q (cid:34)(cid:112) Q ( Q − − ( y + 2 Q ) + 2 Q arctan (cid:32) y + 2 Q (cid:112) Q ( Q − − ( y + 2 Q ) (cid:33) + πQ (cid:35) . (12)The range of variability on ξ is: | ξ | ≤ − Q (cid:2) √− Q (cid:0) arctan √− Q − π/ (cid:1)(cid:3) , whereas y variesin the range − (cid:104) Q + (cid:112) Q ( Q − (cid:105) ≤ y ≤
0. The relationship between the wave minimum andits speed is also given by equation (11), but with pV + β > Q = 1 ( E = P max ), solution (10)with the appropriate choice of the integration constant reduces to ξ ( y ) = ± (cid:104) y − ln (cid:16) y (cid:17)(cid:105) . (13)olutions Classification to the Extended Reduced Ostrovsky Equation 9This solution is unbounded on ξ , i.e. it is defined in the range: | ξ | ≤ ∞ . However, thesolution is bounded on y : − ≤ y ≤
0. The relationship between the wave minimum and itsspeed is simple as both of them are constant values in this special case: V = − p (cid:18) β + C C (cid:19) , u min = V − C C = − p (cid:18) β + C C + 2 p C C (cid:19) . (14)Another special case corresponds to Q = ∞ ( E = 0); in this case equation (10) after appro-priate choice of integration constant reduces to: ξ ( y ) = ± (cid:112) y + 1( y − . (15)The range of variability on ξ is: | ξ | ≤ /
3, whereas y varies in the range: − ≤ y ≤
0. Therelationship between the wave minimum and its speed is also very simple as both of them areagain constants but different from those given by equation (14); in this case they are: V = − βp , u min = V − C C = − (cid:18) βp + C C (cid:19) . Bounded solutions corresponding to the trajectories shown in the right half-plane in Fig. 8with w ≥
0, exist only for negative E ; they are given by equation (12), but with the differentrange of variability of y : 0 ≤ y ≤ (cid:104) − Q + (cid:112) Q ( Q − (cid:105) . The relationship between the wavemaximum and its speed is given again by equation (11) where u max should be substituted insteadof u min and pV + β > E < (cid:48) represent an example when two branches are matched so thatthey form a semi-oval; lines 3 and 3 (cid:48) represent another example when two branches are matchedso that they form an inverted “seagull”. On the basis of these “elementary” solutions, variouscomplex compound solutions can be constructed including periodic or chaotic stationary waves.The dashed line 1 in the figure corresponds to E = 0 ( Q = 8). Another branch of the solutionwith the same value of E = 0 represents a solution of positive polarity which is unbounded bothon ξ and y . For positive values of E , solutions of negative polarity become wider and of greater“amplitude” (see line 2). When E further increases and approaches P max , the solution becomesinfinitely wide, but its minimum goes to −
2. In the limiting case E = P max ( Q = 1) twoindependent branches of the solution can be matched differently as shown by dashed-dottedlines 3 and 3 (cid:48) in Fig. 9. The solution vanishes in this case when ξ = 0 and goes to − ξ → ±∞ ; this situation is described by equation (13).For the negative E there are two families of solutions: negative one, corresponding to theleft-hand side trajectories in Fig. 8, and positive one, corresponding to the right-hand sidetrajectories. When E , being negative, increases in absolute value ( Q varies from −∞ to 0 − ),solutions depart from the line 1 in Fig. 9 and gradually squeeze to the origin (see line 4 forinstance). For the same values of negative E , positive solutions originated at infinity alsogradually shrink and collapse in the origin (lines 6 and 5 demonstrate this tendency).Consider now the case iii) shown in Fig. 7b. The potential function in this case has only onewell of a finite depth so that P min = C / (4 C ) at w = − C /C , where C is negative now. Thereare no bounded solutions for negative w; they exist however for positive w and E varying in therange P min ≤ E <
0. The finite value of the potential minimum corresponds to the equilibriumpoint of the centre type in the phase plane. There is also a family of closed trajectories forthe above indicated range of E variation (see Fig. 10); these trajectories correspond to periodicsolutions.0 Y.A. Stepanyants Figure 9.
Various solutions described by equations (10), (12), (13) and (15). Compactons of negativepolarity: line 1: Q = ∞ ; lines 2 and 2 (cid:48) : Q = 2; lines 3 and 3 (cid:48) : Q = 1: line 4: Q = − .
1. Compactons ofpositive polarity: line 5: Q = − .
1; line 6: Q = − . Figure 10.
Phase portrait of equations (3), (4) for the case iii) (only those trajectories are shownwhich correspond to particle motion within the potential well in Fig. 7b). The dot at the center of closedlines indicates an equilibrium point corresponding to the potential minimum ( E = − . C = 1, C = − . E = −
2; line 2: E = − .
5; line 3: E = −
1; line 4: E = − . As usual, closed trajectories around the center ( E ≥ P min ) correspond to quasi-sinusoidalsolutions. Whereas other closed trajectories ( E > P min ) correspond to non-sinusoidal periodicwaves with smooth crests and sharp narrow troughs. The larger is the value of E , the longer isthe wave period. The period tends to infinity when E → − . The analytical form of this familyof solutions is described by the following equation: ξ ( y ) = ± (cid:112) Q (cid:34)(cid:112) Q ( Q − − ( y + 2 Q ) olutions Classification to the Extended Reduced Ostrovsky Equation 11 Figure 11.
Various solutions described by equation (16). Line 1 (quasi-sinusoidal wave): Q = 1 . Q = 1 .
5; line 3: Q = 2; line 4: Q = 2 .
5. Dashed lines shows the equilibrium state y = − + 2 Q arctan (cid:32) y + 2 Q (cid:112) Q ( Q − − ( y + 2 Q ) (cid:33) − πQ (cid:35) , (16)where ξ = χ √− C ( C /C ) , y = w ( C /C ), Q = C / (4 C E ). Solution (16) is shown in Fig. 11for different values of Q (note that the solution is negative in terms of y because C < y varies in the range: − (cid:104) Q + (cid:112) Q ( Q − (cid:105) ≤ y ≤ − (cid:104) Q − (cid:112) Q ( Q − (cid:105) , whereas the dependence of wave period Λ on Q is: Λ( Q ) = 8 πQ √ Q . The wave period variesfrom 8 π to infinity when Q increases from unity to infinity.From the extreme values of y (see above indicated range of its variability) one can deducethe dependences of wave maximum and minimum on speed in the original variables. The corre-sponding formulae are: u max , min ( V ) = V + C pV + β ∓ C (cid:115) C ( pV + β ) (cid:20) C C ( pV + β ) + 1 (cid:21) , (17)where the upper sign in front of the root corresponds to the wave maximum and lower sign –to the wave minimum. These dependences are plotted in Fig. 12 for V > − β/p in accordancewith the chosen values of constants C = 1, C = − . E <
0. The asymptote V = − β/p isshown in the figure by the vertical dashed line. As follows from equation (15), wave maximumcannot be less than the certain value, U max , which occurs at some speed V shown in Fig. 12.For all possible values of wave maximum u max > U max , two values of wave speed are pos-sible, i.e. two periodic waves of the same maximum (but not minimum!) can propagate withdifferent speeds. This is illustrated by the horizontal dashed line shown in Fig. 12 and drawnfor u max = 2 .
5. In original variables quasi-sinusoidal waves exist when the speed is close to itslimiting value V max = − p (cid:16) C C + β (cid:17) ; there are no waves with greater speed. When V ≤ V max ,the wave minimum and maximum are close to each other. Then, when the speed decreases, thegap between wave maximum and minimum gradually increases and goes to infinity when thespeed approaches its minimum value V min = − β/p .2 Y.A. Stepanyants Figure 12.
Dependences of wave maximum (solid line) and minimum (dashed-dotted line) on speed inthe original variables, equation (17), as follows from the solution (16). Dashed vertical line correspondsto V = − β/p . The plot is generated for C = 1, C = − . p = β = 1. p + q (cid:54) = 0 Consider now a more general case when the coefficients in equation (1) are such that p + q (cid:54) = 0.The basic equation (4) can be presented in the new variables η = ( p + q ) χ/ v = ( p + q ) w/ E = − ( pV + β ) /
2, but with new effective potentialfunction P ( v ) = v − C v − C v . (18)The potential function is monotonic when C = C = 0, and there are no bounded solutionsin this case. Bounded solutions may exist if at least one of these constants is nonzero. Below wepresent possible forms of the potential function and corresponding phase portraits of boundedsolutions for various relationships between constants C and C . Qualitatively all these casesare similar to those which have been described already in the previous section, therefore weomit the detailed analysis and do not present analytical solutions as they can be obtainedstraightforwardly and expressed in terms of elliptic functions. . If C = 0, the potential function represents a set of antisymmetric hyperbolas locatedeither in the first and third quadrants when C = −
1, or in the second and fourth quadrantswhen C = 1; this is shown in Fig. 13.For the case of C = 1 only bounded solutions of a compacton type are possible for positive v .Such solutions correspond to the motion of particle c shown in the figure down to the potentialwell. This family of pulse-type solutions exist both for negative and positive E ; all of them arebounded from the top with the maximum values depending on E , have zero minimum valuesand infinite derivatives when v = 0. Corresponding phase plane is presented in Fig. 14a.For the case of C = − v ≤
0; they correspond to the motion of the particle b down to the potential well(particle motion to the left from the top of the “hill” corresponds to unbounded solutions). Pos-sible values of particle energy E vary for such motions from minus infinity to P max = − √− C ,where P max is the local maximum of the lower branch of the potential function (see Fig. 13).The phase portrait of such motions is shown in the left half of the phase plane in Fig. 14b.ii) Another possibility appears for the particle motion within the potential well shown inthe first quadrant of Fig. 13 (see the particle a ). Within this well all phase trajectories areolutions Classification to the Extended Reduced Ostrovsky Equation 13 Figure 13.
Potential function for the case p + q (cid:54) = 0, C = 0 and two values of C : C = 1 (solid line),and C = − a , b and c illustrate possible motion of a point particle in the potentialfield. Figure 14. a) Phase plane corresponding to the potential function with C = 1 and various values of E .Line 1: E = −
2; lines 2: E = −
1; lines 3: E = 0; lines 4: E = 1; lines 5: E = 2. All trajectoriesin the left half-plane correspond to unbounded solutions. b) Phase plane corresponding to the potentialfunction with C = − E . Line 1: E = −
4; lines 2: E = −
3; lines 3: E = − E = −
1; lines 5: E = 2 .
1; lines 6: E = 2 .
5; lines 7: E = 3. closed and corresponding solutions are bounded and periodical; they can be expressed in termsof elliptic functions. The phase portrait of such motions is shown in the right half of the phaseplane in Fig. 14b. . If C = 0, but C (cid:54) = 0, the potential function also represents a set of antisymmetrichyperbolas located either in the third quadrant and right half-plane in Fig. 15 when C = 1, orin the first quadrant and left half-plane of that figure when C = − C = 1 there are two possibilities: i) there is a family of compacton-typesolutions with v ≤
0; they correspond to the motion of the particle b down to the potential well4 Y.A. Stepanyants Figure 15.
Potential function for the case p + q (cid:54) = 0, C = 0 and two values of C : C = 1 (solidlines), and C = − a , b , c and d illustrate possible motion of a point particle in thepotential field. Figure 16.
Phase plane corresponding to the potential function (18) with C = 0. a) C = 1 and variousvalues of E . Lines 1: E = −
3; lines 2: E = −
2; lines 3: E = − .
89; lines 4: E = −
1; lines 5: E = 0;lines 6: E = 1; lines 7: E = 2. b) C = − E . Line 1: E = −
1; line 2: E = 0;line 3: E = 1; lines 4: E = 2; lines 5: E = 2 .
5; lines 6: E = 3. All trajectories in the left half-planecorrespond to unbounded solutions. (particle motion to the left from the top of the “hill” corresponds to unbounded solutions). Possi-ble values of particle energy E vary for such motions from minus infinity to P max = 3( − C / / ,where P max is the local maximum of the left branch of the potential function (see Fig. 15). Thephase portrait of such motions is shown in the left half-plane in Fig. 16a.ii) Another family of compacton-type solutions exist with v ≥
0; they correspond to themotion of the particle c down to the potential well. Possible values of particle energy E for suchmotions vary from minus to plus infinity. Corresponding phase plane is presented in the righthalf-plane in Fig. 16a.olutions Classification to the Extended Reduced Ostrovsky Equation 15 Figure 17.
Potential function for the case p + q (cid:54) = 0. a) Supercritical case: C = − C : C = 1 (solid lines), and C = − C = − C ; c) subcritical case: C = − C (in the last case the horizontaland vertical scales are doubled). For the case of C = − P min = 3( − C / / , where P min is the local minimumof the right branch of the potential function (see Fig. 15) to infinity. Analytical solution for suchwaves can be also expressed in terms of cumbersome elliptic functions. Corresponding phaseplane is presented in Fig. 16b. . Consider now the case when both C (cid:54) = 0 and C (cid:54) = 0. The shape of the potentialfunction is more complex in this case in general and depends on the relationship between theconstants C and C . The number and values of the potential extrema are determined by thenumber of real roots of the equation P (cid:48) ( v ) = 0, where prime denotes the derivative on v . Thiscondition yields (see equation (18)): v + C v + 2 C = 0 . For real constants C and C this equation always has at least one real root. The real rootis single when C ≥ C cr1 ≡ − C / ; its value is given by the expression v = (cid:20)(cid:113) ( C / + C − C (cid:21) / − ( C / (cid:20)(cid:113) ( C / + C − C (cid:21) − / . For the case C > C cr1 , possible qualitative configurations of the potential function are shownin Fig. 17a for the particular choices of constants: C = − C = ±
1. There is only one6 Y.A. Stepanyants
Figure 18.
Phase plane corresponding to the marginal case, C = C cr1 . a) C = − C = −
1. Line 1: E = − .
05; line 2: E = −
3; line 3: E = − .
9; line 4: E = −
2; line 5: E = 3 .
8; line 6: E = 4; line 7: E = 4 .
5. All trajectories in the left half-plane correspond to unbounded solutions. b) C = − C = 1.Line 1: E = −
5; lines 2: E = −
4; lines 3: E = − .
75; lines 4: E = − .
5; line 5: E = 2 .
75; line 6: E = 3;line 7: E = 3 .
25; line 8: E = 3 . local minimum at the right branch of the potential function for C = − C = 1. Almost the same configuration of thepotential function occurs for the marginal case C = C cr1 , as shown in Fig. 17b, however onemore local extremum appears – on the left branch when C = − C = 1. In the case C < C cr1 the potential function is shown in Fig. 17c; there are three localextrema of the potential function for any value of C = ± C > C cr1 qualitatively is similar to thecase shown in Fig. 15, therefore the corresponding phase portraits are similar to those shown inFig. 16. In the marginal case, C = C cr1 , the potential configuration is also similar to those twocases mentioned above, however there are some peculiarities in the phase planes reflecting theappearance of embryos of new equilibrium points. Corresponding phase portraits are shown inFig. 18. The embryos appear in the vicinity of E = − E = 3in Fig. 18b.In the subcritical case C < C cr1 the situation is different from the previous ones and shouldbe considered separately. In the case of C = −
1, there are two potential wells, one of a finitedepth on the left branch of function P ( v ) and another infinitely deep and wide well but boundedfrom the bottom on the right branch of function P ( v ) (see Fig. 17c).For the first potential well there is a family of closed trajectories in the phase plane corre-sponding to periodic solutions with the parameter E varying between the local minimum andmaximum of the potential function; these solutions are described by elliptic functions. All closedtrajectories are bounded by the loop of separatrix designated by symbol 3 in Fig. 19a. Trajecto-ries inside the separatrix loop next to center correspond to quasi-sinusoidal waves, and the loopof the separatrix corresponds to the solitary wave (soliton) which can be treated as the limitingcase of periodic waves. The soliton shape is described by the following implicit formula: η = ±√ (cid:20) v √ v − v ln (cid:18) √ v − v + √ v − v √ v − v − √ v − v (cid:19) + √ v − v (cid:21) , v ≤ v ≤ v , (19)olutions Classification to the Extended Reduced Ostrovsky Equation 17 Figure 19.
Phase plane corresponding to the subcritical case C < C cr1 . a) C = − C = −
1. Line 1: E = −
6; lines 2: E = −
5; line 3: E = − .
25; lines 4: E = −
4; line 5: E = 4 .
7; line 6: E = 5; line 7: E = 6; line 8: E = 8. All trajectories in the left half-plane outside of the closed loop of separatrixcorrespond to unbounded solutions. b) C = − C = 1. Line 1: E = −
7; line 2: E = −
5; lines 3: E = − . E = −
4; lines 5: E = 4 .
3; line 6: E = 5; lines 7: E = 6 . E = 7. Figure 20.
Soliton solutions on pedestals as described by equations (19). where v , = − (cid:0) C ∓ (cid:112) C − EC (cid:1) /E , ( v < v ) and E = P max ( C , C ), where P max ( C , C ) isthe value of the potential local maximum shown in the left half-plane of Fig. 17c. Solution (19)is shown in Fig. 20a.In original variables function u describing soliton varies in the range V + 6 v p + q ≤ u ≤ V + 6 v p + q ;thus, the soliton amplitude amounts A = 6 v − v p + q = 12 p + q (cid:112) C − C P max ( C , C ) P max ( C , C ) ; (20)The soliton velocity is V = − p [ β + 2 P max ( C , C )] . (21)8 Y.A. StepanyantsEquations (20) and (21) allow one to obtain a direct relationship between the soliton’s velocityand amplitude: A = − p + q (cid:113) C + C ( pV + β ) pV + β . For the second potential well located in the right half-plane of Fig. 17c, there is anotherfamily of closed trajectories in the phase plane corresponding to periodic solutions with theparameter E varying between the local minimum of the potential function and infinity; thesetrajectories are shown in the right half-plane of Fig. 19a.In the case of C = 1, there is a shallow potential well on the right branch of function P ( v ) andone infinitely deep well at the origin where the potential function is singular. For the shallow wellthere is a family of closed trajectories in the phase plane corresponding to periodic solutions withthe parameter E varying between the local minimum and maximum of the potential function.All such trajectories are also bounded by the loop of separatrix designated by symbol 7 inFig. 19b. The loop of separatrix corresponds to the solitary wave whose shape is described by thesame implicit formula (19), but with different values of constants C , C , E and P max ( C , C ),where P max ( C , C ) is the value of the potential local maximum shown in the right half-plane ofFig. 17c. This solution is shown in Fig. 20b. All above relationships between soliton amplitudeand velocity, as well as between soliton amplitude or velocity and constants C and C remainthe same as above.For the infinitely deep well at the origin there are two families of compactons with nonpositiveand nonnegative values; the phase plane for them is similar to that shown in Fig. 8 and solutionsare similar to those shown in Fig. 9. The entire phase portrait of the system in the case of C = 1is shown in Fig. 19b. Phase trajectories corresponding to positive compactons are not shown indetail in that figure because they are too close to each other and are in the narrow gap betweenthe axis v (cid:48) and external two unclosed branches of the separatix 7 (only one such trajectory,line 5, is shown in Fig. 19b; all other trajectories are similar). As was shown in the paper, the extended reduced Ostrovsky equation (1) possesses periodicand solitary type solutions in general. There is a variety of solitary-wave solutions includingcompactons with infinite derivatives at the edges, smooth solitons, and periodic waves. Allcompactons, however, actually are of the compound-type solutions, i.e., they consist of twoor more non-smooth branches. Among periodic waves depending on the equation parameters,there are also both smooth solutions and compound-type solutions which consist of periodicsequences of non-smooth branches (see, e.g., Fig. 5b). Moreover, using compacton solutions asthe elementary blocks, one can construct very complex compound solutions including stochasticstationary waves.The approach used in this paper and based on the qualitative theory of dynamical systemsis free from the limitations of paper [1] and allows us to present a complete classification of allpossible solutions of stationary exROE. In particular, solutions were obtained and analyzed indetails for the case p + q = 0 that was out of consideration in paper [1]. Another “prohibi-ted” combination of parameters, qV − β (cid:54) = 0, that was also out of consideration in paper [1],does not even appear in our study. The approach exploited in the present paper is based ona vivid mechanical analogy between a particle moving in a special potential field and consideredstationary exROE. This approach allows one to observe qualitatively an entire family of allpossible solutions even without construction of particular exact solution. A similar approach hasbeen exploited recently in application to the reduced Ostrovsky equation [4, 3] and exROE [3],although in the last case, the complete solution classification was not considered.olutions Classification to the Extended Reduced Ostrovsky Equation 19 References [1] Parkes E.J., Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation,
SIGMA (2008), 053, 17 pages, arXiv:0806.3155.[2] Morrison A.J., Parkes E.J., The N -soliton solution of the modified generalised Vakhnenko equation (a newnonlinear evolution equation), Chaos Solitons Fractals (2003), 13–26.[3] Li J.-B., Dynamical understanding of loop soliton solution for several nonlinear wave equations, Sci. ChinaSer. A (2007), 773–785.[4] Stepanyants Y.A., On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactonsand compound solitons, Chaos Solitons Fractals (2006), 193–204.[5] Ostrovsky L.A., Nonlinear internal waves in a rotating ocean, Okeanologiya (1978), 181–191 (Engl. transl: Oceanology18