Solitary water wave interactions for the Forced Korteweg-de Vries equation
SSolitary water wave interactions for the Forced Korteweg-de Vriesequation
M. V. Flamarion and R. Ribeiro-Jr UFRPE/Rural Federal University of Pernambuco, UACSA/Unidade Acadˆemica do Cabo de Santo Agostinho, BR101 Sul, 5225, 54503-900, Ponte dos Carvalhos, Cabo de Santo Agostinho, Pernambuco, Brazil. UFPR/Federal University of Paran´a, Departamento de Matem´atica, Centro Polit´ecnico, Jardim das Am´ericas,Caixa Postal 19081, Curitiba, PR, 81531-980, Brazil.
Abstract
The aim of this work is to study solitary water wave interactions between two topographicobstacles for the forced Korteweg-de Vries equation (fKdV). Focusing on the details of theinteractions, we identify regimes in which solitary wave interactions maintain two well separatedcrests and regimes where the number of local maxima varies according to the laws 2 → → → → → →
2. It shows that the geometric Lax-categorization of Korteweg-de Vries(KdV) two-soliton interactions still holds for the fKdV equation.
Keywords:
Solitary waves collisions, Korteweg-de Vries equation, Solitons.
The forced Korteweg-de Vries equation (fKdV) has been used as a model to describe atmosphericflows encountering topographic obstacles, flow of water over rocks (Baines [1995]), ship waves andocean waves generated by storms (when a low pressure region moves on the surface of the oceanJohnson [2012]).Solitary waves have a wide range of applications, for instance, in water waves, optical fibers,superconductive electronics, elementary-particle physics, quantum physics and more recent appli-cations in biology and cosmology (Joseph [2016]). It is well known that the Korteweg-de Vriesequation (KdV) is used to describe the propagation and interaction between solitary waves. Study-ing numerical solutions of the KdV equation Zabusky & Kruskal [1965] were the first to observethat solitary waves interact during the collision and return to its initial form. They named this typeof waves as solitons. This study raised interest to investigate further details of soliton interactions.Since then many works have been done on this topic. It is hard to give a comprehensive overview ofcontributions. For the interested reader, we mention a few articles which are seminal in this field.Lax [1968] classified overtaking collisions of two solitons in three categories according to thenumber of crests observed during the interaction. More precisely, he proved that the type ofthe collision can be classified according to the ratio of the initial amplitude of the solitons. Thecategorization given by Lax was verified experimentally by Weidman & Maxworthy [1978] andnumerically by Mirie & Su [1982] for a higher order model. More recently, Craig et al. [2006]presented a work in which is given a broad review on solitary wave interactions. They investigatednumerically and experimentally solitary wave collisions for the Euler equations. Their numericalsimulations show that the collisions of two solitary waves fit into the three geometric categories ofthe KdV two-soliton solutions defined by Lax. However, the algebraic classification based on theratio of the initial amplitudes is within a different range of the one considered by Lax.In this paper we investigate numerically in details the interaction of two solitary wave solutionsof the fKdV. More precisely, we analyse the interaction of these two waves between obstacles. Wefind the three geometric categories described by Lax [1968] for the KdV two-soliton interaction.However, our experiments indicate that an algebraic categorization similar to the one presented byLax is not possible for the fKdV.This article is organized as follows. In section 2 we present the mathematical formulation ofthe non-dimensional fKdV equation. The results are presented in section 3 and the conclusion insection 4.
We consider an inviscid, incompressible, homogeneous fluid on a shallow channel with variabletopography in the presence of a constant current. The flow of the fluid can be classified by the Froudenumber ( F ), which is defined by the ratio of the upstream velocity and the critical speed of shallowwater. When the Froude number is near critical ( F ≈ a r X i v : . [ phy s i c s . f l u - dyn ] F e b mall the weakly nonlinear, weakly dispersive model given by the dimensionless forced Korteweg-deVries equation ζ t + f ζ x − ζζ x − ζ xxx = 12 h x ( x ) , (1)is used to describe the flow over the obstacle (Pratt [1984]; Wu [1987]; Grimshaw & Maleewong[2013]; Milewski [2004]; Flamarion et al. [2019]). Here, ζ ( x, t ) is the free-surface displacementover the undisturbed surface and h ( x ) is the obstacle submerged. The parameter f represents aperturbation of the Froude number, i.e, F = 1 + (cid:15)f , where (cid:15) > M ( t )), with dMdt = 0 , where M ( t ) = (cid:90) + ∞−∞ ζ ( x, t ) dx. When the bottom is flat ( h x = 0) a traveling solitary wave solution for (1) is ζ ( x, t ) = A sech ( k ( x − ct )) , A = 43 k , c = f − A. (2)Notice that when f = A/ x are computed spectrally (Trefethen [2001]). Besides, the time evolution is calculated through theRunge-Kutta fourth-order method (Flamarion et al. [2019]). We investigate collisions of two solitary waves between the osbtacles. For this purpose the initialcondition of (1) is given by a linear sum of two well-separate solitary waves ζ ( x,
0) = S sech ( k ( x − φ )) + S sech ( k ( x + ψ )) , where S = 4 k / S = 4 k /
3, and φ, ψ are positive constants. Our focus is to categorize thecollision of two solitary waves into three types in the same spirit as presented in Lax [1968] for theKdV equation. Studying overtaking ( S > S ) collisions Lax has classified the details of two-solitoninteractions as follows: (A) For any time t the solution of the KdV has two well-defined and separate crests, and it happenswhen S /S < (3 + √ / ≈ . (B) During the collision the number of local maxima varies according to 2 → → → →
2, andfor this case we have (3 + √ / < S /S < (C) In the interaction the number of local maxima changes as 2 → → S /S > β = 20, (cid:15) = 0 .
01 and f = 0 .
34. Besides, inorder to avoid radiation from the topography we sum a term r ( x ) to the initial condition, where r is the steady solution of the uniform flow.We start considering the collision of two well-separeted solitary waves that initially have thesame amplitude. Details of the wave profile are given in figure 1 (top). Initiallly two solitary wavespropagate downstream. When the right wave reaches the obstacle its amplitude increases and thewave reflects back upstream. Then the waves collide mimicking a counterpropagating collision. Asthe right wave approaches the wave with smaller crest the larger wave begins to shrink and thesmaller one begins to grow until the two waves interchange their roles (see figure 1 (bottom-right).Throughout the interaction there are two well-defined and separate crests as shown in figure 1(bottom-left). This behaviour is similar to case (A) of Lax classification. Figure 2 displays thecontinuation of figure 1(top). After a series of collisions both waves escape out. We point out thatthe numerical method conserves mass and the relative error is:max ≤ t ≤ | M ( t ) − M (0) || M (0) | = O (10 − ) . Figure 3 (top) displays the collision of two well-separate solitary waves that initially have differ-ent amplitudes. Differently from the previous case there is a period of time in the interaction that2
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Figure 1: Top: Collision of two well-separate solitary waves – category (A) . Bottom (left): Cresttrajectories. Bottom (right): The local maxima of the solution as a function of time. Parameters S = S = 0 . φ = 16 and ψ = 12.Figure 2: Continuation of figure 1 (top). Parameters S = S = 0 . φ = 16 and ψ = 12.only one crest exists. The interaction is characterized by an absorption of the smaller wave and itsreemission later, along with a phase lag in the trajectories of the crest, see figure 3 (bottom).Lastly, we show a collision that presents features similar to the cases (A) and (B) simultaneously,see figure 4. The smaller wave is first swallowed, then expelled by the larger one. This dynamic isvery similar to the description given previously in case (C) . However, during the collision there isa central region consisting of two crests. This behaviour is described in great details in a serie ofsnapshots depicted in figure 5.For the KdV equation the transition between two categories is determined by the ratio of theamplitudes of the two separated solitary waves given initially. However, for the fKdV is not pos-sible to estimate a similar condition regarding the ratio of the amplitudes as shown in table 1.Nevertheless the fKdV equations still holds the geometric features of the Lax categorization.3
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Figure 3: Top: Collision of two well-separate solitary waves – category (C) . Bottom (left): Cresttrajectory. Bottom (right): The local maxima of the solution as a function of time. Parameters S = 0 . S = 0 . φ = 16 and ψ = 45. Figure 4: Top: Collision of two well-separate solitary waves – category (B) . Bottom (left): Cresttrajectory. Bottom (right): The local maxima of the solution as a function of time. Parameters S = 0 . S = 0 . φ = 16 and ψ = 45. In this paper we have investigated solitary wave collisions for the fKdV equation. Through apseudospectral numerical method, we showed that the geometric Lax characterisation for the KdVtwo-soliton interaction still holds for the fKdV, i.e. solitary wave interactions maintain two wellseparated crests in regime (A) , the larger solitary wave absorbs the smaller one and the numberof local maxima varies according to the law 2 → → → → (B) or the numberof local maxima changes as 2 → →
2, case (C) . Although there are a number of theoretical and4
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Figure 5: Snapshots of the interaction of the two well-separate solitary waves of figure 4 during thecollision – category (B) . S S max { S , S } / min { S , S } category0.60 0.30 2.00 A A B B B B C C B B A Table 1: Classification of the collision for different values of S and S .numerical works on collisions for the KdV equation, as far as we know there are no articles focusedon collision details for the fKdV equation. The authors are grateful to IMPA-National Institute of Pure and Applied Mathematics for theresearch support provided during the Summer Program of 2020 to Prof. Paul Mileswki (Universityof Bath) for his constructive comments and suggestions which improved the manuscript. M.F. isgrateful to Federal University of Paran´a for the visit to the Department of Mathematics. R.R.-Jr isgrateful to University of Bath for the extended visit to the Department of Mathematical Sciences.
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