Single soliton solution to the extended KdV equation over uneven depth

Transcription

1 Eur. Phys. J. E 7) : DOI./epje/i Regular Article THE EUROPEAN PHYSICAL JOURNAL E Single soliton solution to the etended KdV equation oer uneen depth George Rowlands, Piotr Rozmej,a, Eryk Infeld 3, and Anna Karczewska Department of Physics, Uniersity of Warwick, Coentry, CV 7A, UK Institute of Physics, Faculty of Physics and Astronomy, Uniersity of Zielona Góra, Szafrana a, 65-6 Zielona Góra, Poland 3 National Centre for Nuclear Research, Hoża 69, -68 Warszawa, Poland Faculty of Mathematics, Computer Science and Econometrics, Uniersity of Zielona Góra, Szafrana a, 65-6 Zielona Góra, Poland Receied August 7 and Receied in final form 8 October 7 Published online: Noember 7 c The Authors) 7. This article is published with open access at Springerlink.com Abstract. In this note we look at the influence of a shallow, uneen rierbed on a soliton. The idea consists in an approimate transformation of the equation goerning wae motion oer an uneen bottom to an equation for a flat one for which the eact solution eists. The calculation is one space dimensional, and so corresponds to long trenches or banks under wide riers or oceans. Introduction Recently, we hae found eact solitonic and periodic wae solutions for water waes moing oer a smooth rierbed. Amazingly they were simple, though goerned by a more eact epansion of the Euler equations with seeral new terms added when compared to the Korteweg de Vries KdV) equation,3 5. Our net step is to consider how these results are modified by a rough rier or ocean bottom. We start with a simple case. The geometry is one space dimensional and the wae a soliton. Een so, approimations rear their head! The consideration of a two-dimensional bump on the bottom, as well as periodic waes propagating oerhead, are planned for a later effort. Here we consider the following equation goerning the eleation of the water surface η/h aboe a flat equlibrium at the surface written in dimensionless ariables): η t + η + α 3 ηη + β 6 η 3 +α η η ) η η ηη 3 +βδ β ) hη) +h η) hη ) ) +β 9 36 η 5 =. ) The first line gies the KdV equation. The first and second line gie the KdV equation both corresponding to een bottom). The last three terms are due to a bottom profile. We emphasize, that ) was deried in,5 under a p.rozmej@if.uz.zgora.pl the assumption that α, β, δ are small positie by definition) and of the same order. As usual, α = A/H, i.e., the ratio of the wae amplitude A to the mean water depth H and β = H/L), where L is the mean waelength. The parameter δ = A h /H is the ratio of the amplitude of the bottom function h) to the mean water depth. Up to this point A, H, L, A h are dimension quantities. Scaling to dimensionless ariables allows us to apply the perturbation approach to the set of Euler equations goerning the model of an ideal fluid. Assuming a flat bottom, the KdV equation is obtained from the first-order perturbation approach. Applying the second-order perturbation approach Marchant and Smyth 3 deried eq. ) limited to the first two lines, the so-called etended KdV equation. Since it is deried from second-order perturbation with respect to small parameters, we call it KdV. Taking into account small bottom fluctuations again using the second-order perturbation approach) led us in, 5 to the KdV equation for an uneen bottom ). In scaled ariables the amplitudes of wae and bottom profiles are equal to one. In, we deried eact soliton and periodic solutions to KdV. These solutions are gien by the same functions as the corresponding KdV solutions but with different coefficients. This paper presents an attempt to describe the dynamics of the eact KdV soliton when it approaches a finite interal of an uneen bottom. We will use the reductie perturbation method introduced by Taniuti and Wei 6. Using two space scales allows us to transform the equation for an uneen bottom ) into the KdV equation with some coefficients altered, that is, the equation for

2 Page of 5 Eur. Phys. J. E 7) : the flat bottom. This transformation is approimate but the analytical solution to the resulting equation is known. This approimate analytic description will be compared with eact numerical calculations. KdV soliton een bottom) In this section we briefy reiew the eact soliton solution to the KdV equation gien in. Assume the form of a soliton moing to the right, η, t) =η t). So, η t = η and the KdV equation, that is ) without the last row, becomes an ordinary differential equation ODE): )η + α 3 ηη + β 6 η α η η + αβ η η ) ηη 3 Integration gies )η + α 3 η + β 6 η 3 8 α η 3 + αβ 8 η ) ηη + β 9 36 η 5 =. ) + β 9 36 η =. 3) Then the solution is assumed in the same form as the KdV solution ηy) =A sech By), ) where A =, since in dimensionless ariables the amplitude is already rescaled. Howeer, for further consideration it is conenient to keep the general notation. The insertion of the postulated form of the solution ) and the use of properties of hyperbolic functions gie 3) in the polynomial form C sech By)+C sech By)+C6 sech 6 By) =, 5) which requires the simultaneous anishing of all coefficients C, C, C6. These three conditions are as follows: )+ 3 B β B β =, 6) 3Aα B β + Aα B β 9 3 B β =, 7) ) Aα) 3 8 Aα B β B β =. 8) Denoting z = βb αa one obtains 8) as a quadratic equation with respect to z with solutions and z = z = < 9).599 >. ) Since B = α β za, only z proides a real B alue. In principle sech of imaginary argument can be epressed by a quotient of epressions gien by hyperbolic functions of real arguments. Howeer, these epressions are singular for some alues of arguments and therefore physically irreleant.) Equations 7) and 8) are consistent only when α = α s = 35 35) Then 6) determines elocity =+ 3 α sz α sz ).56. ) 3 Variable depth Equation ) can be written in the form where fη, h) is gien by fη, h) =η + 3α η α t + fη, h) =, ) η3 ) η η + β η β η +βδ β hη + h η η h. 3) We treat h as slowly arying and introduce two space scales and = ɛ) which are treated as independent until the end of the calculation 6 We also introduce h = hɛ) =h ), ɛ. ) y = aɛ)d t, 5) where a is as yet undefined. To first order in ɛ one has η = η y, )+ɛη y, )+..., 6) t = ɛ +..., 7) = a ) + ɛ + ɛa ) +..., 8) ) η = a η a +ɛ +a η +a η +... Now 9) n η n = an n η + Oɛ). ) n

3 Eur. Phys. J. E 7) : Page 3 of 5 We hae fη, h) =η + 3α η α 3 8 a 8 η3 ) η a η + β 6 a η β a η +βδ h )η a h ) η β + Oɛ) = f η,h)+oɛ). ) From ), 7) and ) to lowest order we hae + a f η,h) = ) and, since a = a ), we obtain η af )=. 3) We restrict the consideration to a single soliton, so η asy ± and so does f. The integration of 3) yields η a )f = ) to lowest order. Let us introduce ζ = y/a ) which remains constant in our approimation. Now, and from ), ), 5) we obtain a ))η 3α η a + α 8 η3 a ) 3 αβ 8 ζ η η ζ η +βδ h ) β + η ζ a β 6 = a ζ, 5) Diiding by a) yields δh ) η + 3α a η α ) 3 8 ζ η η ζ a 9 36 β η ζ a η a =. 6) ζ 8 η β η ζ + β 6 6 δh) η =. 7) ζ This should be compared to 3) or to eq. ) of. Remember that at this stage δh )istobetreatedasa constant with respect to integration oer ζ. The only difference is that the coefficient δh + a ) appears instead of ) in the first term and the coefficient 6 δh) appears instead of in the last term. Following we obtain η = A sech Bζ), ζ = a ) a )d t. 8) In eqs. 6), 7) ), 5) and ) of ) we replace βb but not β B or αβab since we modify only first-order terms) by β 6 δh) B. 9) Now z = z = is as in 9). Hence η = Ā B ) sech a )d t a ) with and 3) a + δh = βδh, q = b B, b = 3z 76 3 z 3) Ā = A + qδh), B = B + qδ h/), where A, B, are gien by eqs. 3) 3) in. Thus ) a = + β δh. 3) At this stage we take = ɛ and δh = δh). So a)d = d. 33) + β)δh) Assume δh) is nonzero only for L,L. For <L, η = A sech B t)), δh, a =. For >L, δh, a = and ) η = A sech B L L a)d + t). 3) There is a change of phase as the pulse passes through the region where δh. The alteration in the phase is gien by L L d /+β)δh β +/ L L δh)d. 35) If this integral is zero the phase is unaltered. This can happen if a deeper region is followed by a shallower region of appropriate shape or ice ersa. 3. Eamples In the following figures we present the time eolution of the approimate analytic solution 3) to the KdV equation with uneen bottom ) for seeral parameter alues of the system. These eolutions are compared with eact numerical solutions of ). In both cases the initial conditions were the eact solutions of the KdV equation.

4 Page of 5 Eur. Phys. J. E 7) :. β=δ=.5.6. β=δ=.5 analytic numeric.8. η,t) Fig.. Profiles of the soliton as gien by 3) with β = δ =.5. The shape of the trapezoidal bottom is shown not in scale). Consecutie times are t n = n, n =,,, 3,...,3. η,t) β=δ= Fig.. Profiles of the numerical solution of eq. ) obtained with the same initial condition. Time instants are the same as in fig.. Therefore in all the presented eamples α = α s and the amplitude of the initial soliton is equal to. In fig. we present the approimate solution 3) for the case when the soliton moes oer a trapezoidal eleation with L =5andL = 5. We took β = δ =.5. For smaller δ the effects of an uneen bottom are ery small, for larger δ second-order effects not present in the analytic approimation) cause stronger oerlaps of different profiles. We compare this approimate solution of ) to a numerical simulation obtained with the same initial condition. The eolution is shown in fig.. We see that the approimate solution has the main properties of the soliton motion as goerned by eq. ). Howeer, since the numerical solution contains higher-order terms depending on the shape of h the eact motion as obtained from numerics shows additional small amplitude structures known from earlier papers, for eample, 5. This is clearly seen in fig. 3 where profiles obtained in analytic and numeric calculations are compared at time instants t =, 5,, 5,, 5, 3 on a wider interal of. Allnumerical results were obtained with calculations performed on a wider interal 3, 7 with periodic boundary conditions. Details of numerics are described in,, 5. In figs. 6 we present results analogous to those presented in figs. 3 but with a different shape of the bot- η,t) Fig. 3. Comparison of the wae profiles shown in figs. and for time instants t =, 5,, 5,, 5, 3. Consecutie profiles are ertically shifted by.. η,t) β=δ= Fig.. Profiles of the soliton as gien by 3) with β = δ =.. The shape of the parabolic bottom is shown not in scale). Consecutie times are t n = n, n =,,, 3,...,3. η,t) β=δ= Fig. 5. Profiles of the numerical solution of eq. ) obtained with the same initial condition. Time instants are the same as in fig.. tom bump and larger alues of β = δ =.. In this case the bump is chosen as an arc of parabola h) = 5) / between the same L =5andL = as in the trapezoidal case. In the approimate analytic solution, the KdV soliton changes its amplitude and elocity only oer bottom fluctuations. When the bottom bump is passed it comes back to its initial shape only the phase may be changed).

5 Eur. Phys. J. E 7) : Page 5 of 5 η,t) β=δ=. analytic numeric We hae to emphasise that this behaiour is generic, and looks similar for different shapes of bottom bumps and different alues of β, δ parameters. This was obsered in our earlier papers, 7, 8 in which initial conditions were in the form of the KdV soliton. Conclusions Fig. 6. Comparison of the wae profiles shown in figs. and 5 for time instants t =, 5,, 5,, 5, 3. η,t) β=δ=.5 3 t= t=6 t=3 t=7 t= t=5 t=9 Fig. 7. Long-time numerical eolution with a trapezoidal bottom bump for β = δ =.5. We hae deried a simple formula which gies an approimate description of a soliton encountering an uneen rierbed. The model reproduces the known increase in amplitude when passing oer a shallower region, as well as the change in phase. Howeer, the full dynamics of the soliton motion is much richer, with the uneen bottom causing low amplitude soliton radiation both ahead and after the main wae. This behaiour was obsered in our earlier papers,7,8 in which initial conditions were in the form of the KdV soliton, whereas in the present cases the KdV soliton, that is, the eact solution of the KdV equation has been used. Author contribution statement All the authors were inoled in the preparation of the manuscript. All the authors hae read and approed the final manuscript. Open Access This is an open access article distributed under the terms of the Creatie Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, proided the original work is properly cited. This is not the case for the eact numerical eolution of the same initial KdV soliton when it eoles according to the second-order equation ). This is clearly isible in figs. 3 and 6. What is this motion for much longer periods? In order to answer this question one has to perform numerical calculations on a much wider interal of. Such results are presented in fig. 7. The interaction of a soliton with the bottom bump creates two wae packets of small amplitudes. The first moes with a higher frequency faster than the soliton and is created when the soliton enters the bump, while second moes slower with a lower frequency and appears when the soliton leaes it. After some time both are separated from the main wae. Since periodic boundary conditions were used in the numerical algorithm, the head of the wae packet radiated forward traelled for t>7 larger distance than the interal chosen for the calculation and is seen on the left side of the wae profile. References. A. Karczewska, P. Rozmej, E. Infeld, Phys. Re. E 9, 97 ).. E. Infeld, A. Karczewska, G. Rowlands, P. Rozmej, Eact solitonic and periodic solutions of the etended KdV equation, arxi: T.R. Marchant, N.F. Smyth, J. Fluid Mech., 63 99).. G.I. Burde, A. Sergyeye, J. Phys. A: Math. Theor. 6, 755 3). 5. A. Karczewska, P. Rozmej, L. Rutkowski, Phys. Scr. 89, 56 ). 6. T. Taniuti C.C. Wei, J. Phys. Soc. Jpn., 9 968). 7. A. Karczewska, P. Rozmej, E. Infeld, Phys. Re. E 9, 53 5). 8. A. Karczewska, P. Rozmej, E. Infeld, G. Rowlands, Phys. Lett. A 38, 7 7).