Effects of vorticity on the travelling waves of some shallow water two-component systems
DDenys
Dutykh
CNRS, Université Savoie Mont Blanc, France
Delia
Ionescu-Kruse
Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Effects of vorticity on thetravelling waves of some shallowwater two-component systems arXiv.org / hal a r X i v : . [ phy s i c s . c l a ss - ph ] A p r ast modified: April 15, 2019 ffects of vorticity on the travelling waves of someshallow water two-component systems
Denys Dutykh ˚ and Delia Ionescu-Kruse Abstract.
In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves ona constant shear current. Namely, we consider the two-component Camassa–Holm equations, the Zakharov–It¯o system and the Kaup–Boussinesq equationsall including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase spaceanalysis. We get the pulse-type solitary wave solutions and the front solitarywave solutions. For the Zakharov–It¯o system we underline the occurrence ofthe pulse and anti-pulse solutions. The front wave solutions decay algebraicallyin the far field. For the Kaup–Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found.
Key words and phrases: waves on shear flow; solitary waves; cnoidal waves;Zakharov–Ito system; Camassa–Holm equations; Kaup–Boussinesq equations;phase-plane analysis; analytical solutions; multi-pulsed solutions
MSC: [2010] 74J30 (primary), 76F10, 35C07, 76B25, 70K05, 35Q35 (sec-ondary)
PACS: [2010] 47.35.Bb (primary), 47.11.Kb, 47.10.Df (secondary)
Key words and phrases. waves on shear flow; solitary waves; cnoidal waves; Zakharov–Ito system;Camassa–Holm equations; Kaup–Boussinesq equations; phase-plane analysis; analytical solutions;multi-pulsed solutions. ˚ Corresponding author. ontents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 . Dutykh and D. Ionescu-Kruse
1. Introduction
Water wave theory has been traditionally based on the irrotational (and thus po-tential) flow assumption [18]. Obviously, it allows to determine all components of thevelocity vector by taking partial derivatives of just one scalar function — the so-calledvelocity potential. This approximation was shown to be very good in many practical sit-uations. But in order to incorporate the ubiquitous effects of currents and wave-currentinteractions, the vorticity is very important. The rotational effects are significant inmany circumstances, for instance, for wind-driven waves, waves riding upon a shearedcurrent, or waves near a ship or pier. The modern and large-amplitude theory forperiodic surface water waves with a general vorticity distribution was established byConstantin and Strauss in [16], an investigation which initiated an intense study ofwaves with vorticity — see, for example, [13, 17, 43] and the references therein. Addition-ally, there has been significant progress in the construction of numerical solutions formany of these problems — see, for example, [39]. An intermediate situation between theirrotational and fully rotational flows is to consider a prescribed vorticity distribution.Among all possible vorticity distributions the simplest one is the constant vorticity. Itis precisely the case we consider in the present study. The choice of constant vorticityis not just a mathematical simplification since for waves propagating at the surface ofwater over a nearly flat bed, which are long compared to the mean water depth, theexistence of a non-zero mean vorticity is more important than its specific distribution(see the discussion in [19]). We also point out that the surface wave flows of constantvorticity are inherently two-dimensional (see [12, 44]), with the vorticity correlatedwith the direction of wave-propagation, and that the presence of an underlying shearedcurrent (signalled by constant non-zero vorticity) is, somewhat surprisingly, known notto affect the symmetry of the surface travelling waves, at least in the absence of flowreversal (see [14, 27]). Tidal flow is a well-known example when constant vorticity flowis an appropriate model [19]. Teles da Silva and Peregrine [19] were the first to showthat the strong background vorticity may produce travelling waves of unusual shapein the high amplitude region.In the approximate theories of long waves on flows with an arbitrary vorticity distri-bution, Freeman and Johnson [22] derived, by the use of asymptotic expansion, a KdVequation with the coefficients modified to include the effect of shear. The wave prop-agation controlled by the Camassa–Holm (CH) equation in the presence of vorticity(with detailed results for the constant vorticity case) was studied in [30, 32, 36]. Thetwo-component (integrable and non-integrable) shallow water model equations withconstant vorticity that we analyse in our paper, that is, the two-component Camassa–Holm equations, the Zakharov–It¯o system and the Kaup–Boussinesq equations, werederived by asymptotic expansion in Ivanov [33]. We focus on the travelling wave solu-tions on a constant shear current, whose study was presumably initiated in [3, 5, 19, 42].Burns [5] was the first to consider the propagation of small-amplitude shallow waterwaves on the surface of an arbitrary shear flow. The travelling wave solutions to theweakly nonlinear Benjamin model [3] were computed recently in [40]. However, most ofthe works focus on the weakly nonlinear regime. Two notable exceptions are [42], where5 / 27 . Dutykh and D. Ionescu-Kruse the full Euler equations on shear flow were solved using the Boundary Integral Equa-tion Method (BIEM), and [9], where fully nonlinear weakly dispersive Green–Naghdiequations were derived. In our previous study [20] we provided a complete phaseplane analysis of all possible travelling wave solutions which may arise in several two-component systems which model the propagation of shallow water waves, namely, theGreen–Naghdi equations, the integrable two-component Camassa–Holm equations anda new two-component system of Green–Naghdi type. In the capillary-gravity regime,a phase plane analysis of the solitary waves propagating in shallow water modelled bythe Green–Naghdi equations with surface tension, was done in [10, 11].The significance of the results in the present study is the inclusion of vorticity. Wewill consider in turn the two-component Camassa–Holm equations, the Zakharov–It¯osystem and the Kaup–Boussinesq equations. For each model we derive the most generalordinary differential equation describing the whole family of travelling wave solutions.We provide a complete phase plane analysis of all possible solitary and periodic wavesolutions which may arise in these models. By appropriately choosing the constantsof integration, for each model we obtain the equations describing the solitary wavesolutions. For the first two systems, the solitary wave solutions are restricted to theinterval r , c c ` s , with c the constant speed of propagation of the nonliner shallowwater wave described by the models and c ` the constant speed of the linear shallowwater wave on the constant shear current obtained by Burns. We get the pulse-typewave solutions and the front wave solutions. For the Zakharov–It¯o system we underlinethe occurrence of the pulse and anti-pulse solutions, in the case c c ` ą
2. Mathematical models
The wave motion to be discussed will be assumed to occur in two dimensions ona shallow water over a flat bottom. We consider a Cartesian coordinate system
O x y with the axis
O y directed vertically upwards (in the direction opposite to the gravityvector ~g “ p , ´ g q ) and the horizontal axis O x along the flat impermeable bottom y “ x -direction, and thatthe physical variables depend only on x and y . The total water depth is given by thefunction y “ H p x, t q def : “ d ` η p x, t q , where d is the constant water depth and η p x, t q is the free surface elevation above the still water level. The sketch of the fluiddomain with free surface is given in Figure 1. By using a suitable set of scaled variables,we can set d “ g “ y ; we denote this component by u p x, t q . The particularity of the present study is that we consider the shallow water6 / 27 Mathematical models
Figure 1.
Sketch of the fluid domain: free surface flow on a shear current. waves travelling over the constant shear current U p y q “ Ω y , where Ω “ const isthe vorticity. For Ω ą x ´ coordinate, for Ω ă c ą c in the x ´ direction. Thus,we consider that: H p x, t q “ H p ξ q , u p x, t q “ u p ξ q , ξ def : “ x ´ c t . (2.1)If we look for periodic waves, the functions H p ξ q and u p ξ q have to be periodic. Forsolitary waves the profile p H, u q has to tend to a constant state p , q at infinity, whileall the derivatives tend to p , q , that is, H Ñ , H p n q Ñ , n ě , ξ Ñ 8 , (2.2) u p n q Ñ , n ě , ξ Ñ 8 . (2.3) In the shallow water regime, a system which models the wave-shear current interac-tions was derived in [33]: u t ` u u x ´ u txx ´ u x u xx ´ u u xxx ` H H x ´ Ω u x “ , (2.4) H t ` “ H u ‰ x “ , (2.5)This is an integrable bi-Hamiltonian system [33]. Wave-breaking criteria and a sufficientcondition guaranteeing the existence of a global solution are presented in [24]. Thewell-posedness of this system was studied in [21].For Ω “ . Dutykh and D. Ionescu-Kruse after integration, we get u “ c H ´ K H , (2.6)and ´ c u ` u ` c u ´ p u q ´ u u ` H ´ Ω u “ K , (2.7)where the prime denotes the usual derivative operation with respect to ξ and K , K P R are some integration constants. We multiply (2.7) by 2 u (2.6) “ K H H , weintegrate this equation once again: ´ c u ` u ` c p u q ´ u p u q ` K H ´ Ω u “ K u ` K , (2.8) K an integration constant. We use in (2.7) the expression (2.6) for u along with itsfirst derivative and finally we get for the unknown H the following first order implicitordinary differential equation: p H q “ H ¨ R p H q def “ : P p H q (2.9)where R p H q is the following polynomial function in H : R p H q def : “ „ ´ K H ` ´ c Ω ` c K ` K K ¯ H ` ´ c ´ c ´ K K ¯ H ´ ´ c ´ Ω K ¯ H ` . (2.10)Thus, the key point is to understand the solutions to the equation (2.9) since the restof the information can be recovered from them. Taking into account the conditions (2.2), (2.3), the values of the integration constantsin (2.6) and (2.9) are: K “ c , K “ , K “ c . Thus, the ODE (2.9) which describes the solitary wave (SW) solutions becomes p H q “ P p H q “ H c ¨ Q p H q , (2.11)where Q p H q is a polynomial function in H defined as Q p H q def : “ ´ H ` p c Ω ` q H ` p c ´ c ´ q H ` c p Ω ´ c q H ` c “ p H ´ q ¨ p c ` c Ω H ´ H q“ p H ´ q ¨ p c c ` ´ H q ¨ p H ´ c c ´ q , (2.12)with c ˘ given by: c ˘ def : “ ` Ω ˘ ? ` Ω ˘ , (2.13)8 / 27 Mathematical models -5 0 50123456
Figure 2.
The graph of the inequality c c ` ą against the vorticity Ω . c ´ being always negative and c ` always positive. We recognize that c ˘ are the solutionsto the equation c ´ Ω c ´ “ , for the speed of the linear shallow water waves on the constant shear flow. This equationis the Burns condition [5] ż d y ` U p y q ´ c ˘ “ , (2.14)with U p y q “ Ω y . Thompson [41] and Biésel [4] obtained early this dispersion relationfor a constant shear flow. Burns [5] obtained the general formula (2.14) for the speedof linear shallow water waves for general shear profiles U p y q . For more details aboutthe Burns condition see also [3, 22, 35].We return to the equation (2.11). From the decomposition (2.12), it follows thatreal-valued solutions exist only if c c ´ ď H ď c c ` . The wave height H being positive, we have a slightly stricter condition:0 ď H ď c c ` . (2.15)According to the asymptotic behaviour of H , it follows that1 ď c c ` . (2.16)The graph of this inequality is presented in Figure 2.We give a description of the solitary wave profiles for the CH2 model with constantvorticity, by performing a phase plane analysis of the equation (2.11) for c c ` ą c c ` “ . Dutykh and D. Ionescu-Kruse presented in Figure 3. The homoclinic orbits in the phase portrait lead to the pulse-type wave solutions and the heteroclinic orbits correspond to the front (or kink-type)wave solutions. We highlight the fact that two fronts tend only algebraically ∗ to theequilibrium state H “ H p ξ q « ` ξ ´ a as ξ Ñ 8 , a ą The analysis of periodic wave solutions in the CH2 model with constant vorticity isdone along the lines of our previous study [20]. The sixth order polynomial in (2.9) has0 as double root. The leading coefficient of the forth order polynomial R p H q in (2.10)is smaller than zero and its constant term is greater than zero, thus, by Viète formulas,this polynomial has at least one positive root and one negative root. Although theparameter space is now p c , Ω , K , K , K q , the discussion on the possible numberand location of the roots turn out to be the same as in the case without vorticity.Consequently, the phase portraits are topologically exactly the same. So, we refer to[20, Figures 9, 11, 13] for the description of all possible periodic wave solutions in theCH2 model with constant vorticity. The Zakharov–It¯o (ZI) system with constant vorticity represents a two-componentgeneralization of the classical KdV equation. This system, deduced in the shallow waterregime in [33], is formally integrable and it matches the ZI system [29, 45]. Its form isthe following: u t ´ Ω u x ` u x x x ` u u x ` H H x “ , (2.17) H t ` “ H u ‰ x “ . (2.18)From the governing equations (2.17), (2.18) one can derive the ‘total energy’ conserva-tion equation using the methods explained in [8]: ` p H ` u q ˘ t ` “ u ` H u ` u u x x ´ u x ´ Ω u ‰ x “ . The last relation can be used in theoretical investigations, but also in numerical studiesto check the discretization scheme accuracy by tracking the conservation of the energyin time.By substituting the travelling wave solution (2.1) into the ZI system with vorticity(2.17), (2.18) and by integrating, we obtain again the equation (2.6) and the following ∗ In order to see better the asymptotic behaviour of the travelling solution while approaching H “ H “ c c ` “ H “ p ´ H q ∼ p ´ H q . After integrating this relation we obtain the desired conclusion 1 ´ H ∼ ξ as ξ Ñ 8 . A similar reasoning shows that the decay to H “ H “
10 / 27
Mathematical models
P(H) H HOH ' H + + - - (a) P(H) H HOH ' H + - + - (b) Figure 3.
The phase-portrait and the solitary wave profiles in the CH2 modelwith constant vorticity, for: (a) c c ` ą ; (b) c c ` “ . We highlight thefact that two fronts tend only algebraically to the equilibrium state H “ . ODE: p H q “ H ¨ „ ´ K H ` ´ c Ω ` c K ` K K ¯ H ` ´ c ´ c ´ K K ¯ H ` p Ω ´ c q H ` K “ K H ¨ „ ´ K H ` ´ c Ω ` c K ` K K ¯ H ` ´ c ´ c ´ K K ¯ H ´ ´ c ´ Ω K ¯ H ` “ K H ¨ R p H q def “ : P p H q , (2.19)11 / 27 . Dutykh and D. Ionescu-Kruse with K , , P R some integration constants and R p H q the polynomial (2.10) obtainedin the CH2 case. From (2.2), (2.3), we obtain for the integration constants K , , the values: K “ c , K “ , K “ c . Substituting these values into the equation (2.19) yields the following ODE whichdescribes the solitary wave solutions: p H q “ Hc ¨ Q p H q (2.12) “ Hc p H ´ q ¨ p c c ` ´ H q ¨ p H ´ c c ´ q , with c ˘ defined in (2.13). A necessary condition for the existence of the solitary wavesis (2.15). We get the condition (2.16) too. We distinguish in our study the followingsituations: c c ` ą c c ` “ c c ` ą For the periodic wave solutions to the ZI system (2.17), (2.18) with constant vor-ticity, we return to the general ODE (2.19). The values taken by the parameters p c, Ω , K , K , K q , will have a direct influence on the nature and location of the poly-nomial R p H q in (2.19). The fifth order polynomial P p H q in (2.19) has 0 as single rootand it is written as a factorization into K H and the forth order polynomial R p H q .The leading coefficient of R p H q being smaller than zero and its constant term greaterthan zero, by Viète formulas, this polynomial has at least one positive root and onenegative root. For distinct roots, the following situations are possible: ‚ The polynomial R p H q has two real roots, H ă H ą K ă P p H q , the corresponding phase-plane portrait and the wave profile look likein Figure 5. For K ą ‚ The polynomial R p H q has four real roots, H ă ă H ă H ă H .Then, the phase plane portraits for K ą K ă ‚ The polynomial R p H q has four real roots, H ă H ă H ă H ą K ą K ă K ă Mathematical models
P(H) H HOH ' H + - + - (a) P(H) H H O H ' H -- + + (b) Figure 4.
The phase-portrait and the solitary wave profiles in the ZI modelwith constant vorticity, for: (a) c c ` ą ; (b) c c ` “ . We highlight thefact that two fronts tend only algebraically to the equilibrium state H “ .
13 / 27 . Dutykh and D. Ionescu-Kruse
P(H) HHO H ' H OO
Figure 5.
The phase-portrait and the wave profiles for the ZI model withconstant vorticity, in the case the polynomial R p H q has only two real roots, H ă and H ą and the constant K ă . A derivation of the Kaup–Bousinesq (KB) system: u t ` ” u ` H ı x “ , (2.20) H t ´ u x x x ` C p Ω q “ p H ´ q u ‰ x “ , (2.21)with C p Ω q def : “ ` ` Ω ` ? ` Ω ˘ loooooooooooomoooooooooooon (2.13) “ p c ` q “ ` p c ` q , (2.22)as a model of shallow water waves in the presence of a linear shear current is presentedin [33]. This system is integrable iff Ω “ ` ’ sign. However, this yields a linearly ill-posed model, see [2]. The inverse scatteringfor the KB equations was developed further in [25]. For other studies on the KB systemsee, for example, the papers [7, 23, 34, 37] and the references therein.By using direct integration, we obtain below all travelling wave solutions to the KBsystem. We substitute (2.1) into the KB system (2.20), (2.21) and by integrating once,we get H “ K ` c u ´ u , (2.23)14 / 27 Mathematical models
P(H) HHOH ' HOO (a)
P(H) HHOH ' HOO (b)
Figure 6.
The phase-portrait and the wave profiles for the ZI model withconstant vorticity in the case the polynomial R p H q has four real roots: H ă and ă H ă H ă H , and the constant: (a) K ą ; (b) K ă . and ´ c H ´ u ` C p Ω q p H ´ q u “ K , (2.24)where the prime denotes the usual derivative operation with respect to ξ and K , K P R are some integration constants. We replace (2.23) into (2.24), and we get adifferential equation in u only. We multiply this equation by 2 u , we integrate again15 / 27 . Dutykh and D. Ionescu-Kruse P(H) HHOH ' HOO (a)
P(H) HHOH ' HOO (b)
Figure 7.
The phase-portrait and the wave profiles for the ZI model withconstant vorticity in the case the polynomial R p H q has four real roots: H ă H ă H ă and H ą , and the constant: (a) K ą ; (b) K ă .
16 / 27
Mathematical models and we get the following ODE: p u q “ ´ C p Ω q u ` c “ ` C p Ω q ‰ u ` “ p K ´ q C p Ω q ´ c ‰ u ´ p c K ` K q u ` K “ : P p u q , (2.25) K P R being an integration constant. With the conditions (2.2) and (2.3) in view, the integration constants K , , in (2.23)– (2.25) take the values: K “ , K “ ´ c , K “ , (2.26)and the ODE (2.25) becomes: p u q “ u ” ´ C p Ω q u ` c “ ` C p Ω q ‰ u ´ c ı def “ : u Q p u q . (2.27)In the integrable case, that is, for Ω “ C p Ω q “ q , thisequation becomes p u q “ u ´ ´ u ` c u ´ c ¯ “ ´ u p u ´ c q , and has the solutions u “ u “ c , which yields H ” Q p u q ě C p Ω q ą Q p u q has two real roots u ˘ such that: u ´ ď u ď u ` . The polynomial Q p u q in (2.27) has two real roots iff16 c ` ` C p Ω q ˘ ´ c C p Ω q ě , which yields C p Ω q ď
12 or C p Ω q ě . Hence, with the notation (2.22) in view, we get the following restriction on the constantvorticity: Ω ě . (2.28)The solution of the separable differential equation (2.27) is obtained by integration.We denote by U def : “ u . (2.29)17 / 27 . Dutykh and D. Ionescu-Kruse Then, we get the integral I : “ ż d u d u „ ´ C p Ω q u ` c “ ` C p Ω q ‰ u ´ c “ ´ ż d U c ´ c U ` c “ ` C p Ω q ‰ U ´ C p Ω q . (2.30)Since ´ c ă ą
32 , the integral (2.30) can be calculated ( c.f. [1, Chapter 3])if: | ´ c U ` c “ ` C p Ω q ‰ | ă ? c a ´ C p Ω q ` C p Ω q , that is, U ă U ă U ,U : “ “ ` C p Ω q ‰ ´ ? a ´ C p Ω q ` C p Ω q c ą U : “ “ ` C p Ω q ‰ ` ? a ´ C p Ω q ` C p Ω q c ą , (2.32)and has the expression: I “ ´ c arcsin ? r´ c U ` ` C p Ω qs a ´ C p Ω q ` C p Ω q . (2.33)Therefore, for constant vorticity Ω ą
32 , in the interval1 U ă u ă U , (2.34)with U , U ą ´ c arcsin ? r´ c ` r ` C p Ω qs u p ξ qs u p ξ q a ´ C p Ω q ` C p Ω q “ ξ , which yields: u p ξ q “ ? c ? r ` C p Ω qs ` a ´ C p Ω q ` C p Ω q sin r c ξ s , (2.35)and, by (2.23), the function H has the expression: H p ξ q “ ` ? c ? r ` C p Ω qs ` a ´ C p Ω q ` C p Ω q sin r c ξ s (2.36) ´ c “ ? r ` C p Ω qs ` a ´ C p Ω q ` C p Ω q sin r c ξ s ‰ .
18 / 27
Mathematical models -6 -4 -2 0 2 4 600.511.522.5 (a) -6 -4 -2 0 2 4 600.511.522.533.54 (b)
Figure 8.
Analytical expressions (2.36) and (2.35) , respectively, for differentvalues of the constant vorticity Ω and of the speed of propagation c . HO OuuuH
Figure 9.
Multi-pulse travelling wave solutions with two troughs.
Thus, surprisingly, for the values (2.26) of the integration constants, which are obtainedunder the conditions (2.2) and (2.3), we got some periodic solutions. We point outhowever that the velocity u has the expression (2.35) only in the interval (2.34) whichis situated above ξ “
0. We plot in Figure 8 the analytical expressions (2.35) and(2.36) for Ω “ . . c “ . . . c . Chen [7] found numerically multi-pulse travelling wave solutions to the (KB) system,solutions which consist of an arbitrary number of troughs; a multi-pulse travelling wavesolution with two troughs is plotted in Figure 9. Let us take the following values for the integration constants K , , in (2.23) – (2.25): K “ , K “ ´ c , K “ , (2.37)19 / 27 . Dutykh and D. Ionescu-Kruse which will mean that the wave profile p H, u q tends to the constant state p , q atinfinity. Then, the ODE (2.25) becomes: p u q “ u ” ´ C p Ω q u ` c “ ` C p Ω q ‰ u ` C p Ω q ´ c ı def “ : u Q p u q . (2.38)A necessary condition for the existence of these waves is Q p u q ě C p Ω q ą Q p u q in (2.38) has two real roots u ˘ suchthat: u ´ ď u ď u ` . By using the notation (2.29), the solution of the equation (2.38) is obtained by inte-gration: I : “ ż d u d u „ ´ C p Ω q u ` c “ ` C p Ω q ‰ u ` C p Ω q ´ c “ ´ ż d U ? α U ` β U ` γ , (2.39)where the constants α , β and γ are: α def : “ “ C p Ω q ´ c ‰ , β def : “ c ` ` C p Ω q ˘ , γ def : “ ´ C p Ω q . If α ą c ă C p Ω q , ∗ (2.40)since γ ă β ´ α γ ą
0. In this case, we do not have any restrictionon the value of the constant vorticity Ω and the result of the integral in (2.39) is ( c.f. [1, Chapter 3]): I “ ´ ? α ln r ? α ¨ a α U ` β U ` γ ` α U ` β s . (2.41)From the notation (2.29) we conclude that, I “ ´ ? α ln „ ? α ¨ ? γ u ` β u ` α ` α ` β uu . Therefore, the solution of the differential equation (2.38) has the implicit form2 ? α ¨ a γ u p ξ q ` β u p ξ q ` α ` α ` β u p ξ q u p ξ q “ e ´? α ξ , with ξ “ x ´ c t . After algebraic manipulations we arrive to the explicit expressionof the function u : u p ξ q “ e ´? α ξ ` e ´? α ξ ´ β ˘ α ´ γ . (2.42) ∗ For Ω “ c ă
20 / 27
Mathematical models -10 -5 0 5 10-2-1.5-1-0.500.511.522.5 (a) ξ -10 -5 0 5 10 u ( ξ ) Ω = 1, c = 0 . Ω = − c = 0 . Ω = 0, c = 0 . (b) Figure 10.
One-trough travelling wave solutions to the KB system based onanalytical formulas (2.42) and (2.43) . Taking into account (2.23), the function H has the expression: H p ξ q “ ` c e ´? α ξ ` e ´? α ξ ´ β ˘ α ´ γ ´ e ´ ? α ξ « ` e ´? α ξ ´ β ˘ α ´ γ ff . (2.43)We plot in Figure 10 the solutions (2.42) and (2.43) for different values of the constantvorticity Ω and of the speed of propagation c . We get one-trough travelling wavesolutions.If α ă c ą C p Ω q , ∗ (2.44)the integral (2.39) can be calculated ( c.f. [1, Chapter 3]) only if: β ´ αγ “ c ` ` C p Ω q ˘ ` C p Ω q “ C p Ω q ´ c ‰ ą , and | α U ` β | ă a β ´ α γ . (2.45)In this case, its expression is: I “ ´ ?´ α arcsin 2 α U ` β ? β ´ αγ . (2.46)Therefore, the solution of the differential equation (2.38) has the implicit form ´ ?´ α arcsin 2 α ` β u p ξ q u p ξ q ? β ´ αγ “ ξ , which yields: u p ξ q “ ´ αβ ` ? β ´ αγ sin r?´ α ξ s . (2.47) ∗ For Ω “
0, the condition (2.44) becomes c ą .
21 / 27 . Dutykh and D. Ionescu-Kruse -6 -4 -2 0 2 4 6-4-3-2-101234567 (a) ξ -6 -4 -2 0 2 4 6 u ( ξ ) Ω = 1, c = 1 . Ω = − c = 1 . Ω = 0, c = 1 . (b) Figure 11.
Analytical expressions (2.49) and (2.47) , respectively, fordifferent values of the constant vorticity Ω and of the speed of propagation c . With (2.45) in view, the solution (2.47) is restricted to the interval | α u ` β | ă a β ´ α γ . (2.48)By (2.23), the function H has the expression: H p ξ q “ ` c ´ αβ ` ? β ´ αγ sin r?´ α ξ s´ α “ β ` ? β ´ αγ sin r?´ α ξ s ‰ . (2.49)In this case we obtained the same type of solutions as in Section 2.3.1. We plot inFigure 11 the graphs of the analytical expressions (2.47) and (2.49) for different valuesof Ω and c . In the general ODE (2.25), by Viète formulas, we obtain for the right-going travellingwaves that u ` u ` u ` u “ c p ` C p Ω qq C p Ω q ą , where u , , , are roots of the polynomial P p u q . Thus, we can conclude that at leastone of the roots has to be positive. Further, by the phase plane analysis methods, westudy the qualitatively possible types of periodic solutions to the KB system (2.20),(2.21). Depending on the roots of the fourth order polynomial P p u q defined in (2.25),the following situations are possible: ‚ one positive real root, one negative real root and two complex conjugate roots.We obtain one family of periodic waves with a velocity which changes the sign(see Figure 12). 22 / 27 Mathematical models
O O
Figure 12.
The periodic velocity profile for the KB system in the case thepolynomial P p H q has one positive real root, one negative real root and twocomplex conjugate roots. ‚ two positive real roots and two complex conjugate roots. In this case, we obtainone family of periodic waves with positive velocity. ‚ one positive real root and three negative real roots. Then, we obtain two familiesof periodic waves: one with negative velocity and the another one with a velocitywhich changes the sign. ‚ two positive real roots and two negative real roots. We obtain two families ofperiodic waves: one with negative velocity and one with positive velocity (seeFigure 13). ‚ three real positive roots and one negative real root. We have two families ofperiodic solutions: one with positive velocity and one with a velocity whichchanges the sign (see Figure 14). ‚ four positive real roots. We get two families of periodic solutions with positivevelocities. Acknowledgments
The authors acknowledge the support of this work by CNRS and Le Groupement deRecherche International (GDRI) ECO–Math.
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Figure 13.
The periodic velocity profiles for the KB system in the case thepolynomial P p H q has two positive real roots and two negative real roots. O O
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References
D. Dutykh:
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000Chambéry, France and LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, CampusScientifique, 73376 Le Bourget-du-Lac Cedex, France
E-mail address : [email protected]
URL : D. Ionescu-Kruse:
Simion Stoilow Institute of Mathematics of the Romanian Acad-emy, Research Unit No. 6, P.O. Box 1-764, 014700 Bucharest, Romania