Özet
Dispersive shock waves (DSWs) in the three dimensional Benjamin–Ono (3DBO) equation are studied with step-like initial condition along a paraboloid front. By using a similarity reduction, the problem of studying DSWs in three space one time (3+1) dimensions reduces to finding DSW solution of a (1+1) dimensional equation. By using a special ansatz, the 3DBO equation exactly reduces to the spherical Benjamin–Ono (sBO) equation. Whitham modulation equations are derived which describes DSW evolution in the sBO equation by using a perturbation method. These equations are written in terms of appropriate Riemann type variables to obtain the sBO-Whitham system. DSW solution which is obtained from the numerical solutions of the Whitham system and the direct numerical solution of the sBO equation are compared. In this comparison, a good agreement is found between these solutions. Also, some physical qualitative results about DSWs in sBO equation are presented. It is concluded that DSW solutions in the reduced sBO equation provide some information about DSW behavior along the paraboloid fronts in the 3DBO equation.
| Orijinal dil | İngilizce |
|---|---|
| Makale numarası | 102502 |
| Dergi | Wave Motion |
| Hacim | 94 |
| DOI'lar | |
| Yayın durumu | Yayınlandı - Mar 2020 |
Bibliyografik not
Publisher Copyright:© 2020 Elsevier B.V.
Finansman
I would like to thank you the referees for their constructive comments which led to the improvement of this paper. This research was supported by the Istanbul Technical University Office of Scientific Research Projects (ITU BAPSIS), Turkey, under grant TGA-2018-41318 . We thank D.E. Baldwin for MATLAB codes of version of the ETDRK4 method that we use in the study. Appendix A We use a numerical method for the direct numerical simulations which can be used for problems with fixed boundary conditions. But, left boundary condition for sBO is function of t . For this reason, we first transform (2.11) by (A.1) f = R ( t ) ϕ to the following variable coefficient BO (vBO) equation (A.2) ϕ t + R ( t ) ϕ ϕ η + ϵ H ϕ η η = 0 . Here R ( t ) = t 0 t + t 0 . Eq. (A.2) has the left boundary condition fixed at ϕ − = 1 , while the right boundary condition ϕ + = 0 stays the same as in the original sBO equation. In order to solve Eq. (A.2) numerically (see also [36–38] ) we differentiate with respect to η and define ϕ η = z to get (A.3) z t + R ( t ) z ϕ η + ϵ H z η η = 0 . Transforming to Fourier space gives (A.4) z t ̂ = L z ̂ + R ( t ) N z ̂ , t where z ̂ = F ( z ) is the Fourier transform of z , L z ̂ ≡ i ϵ s g n ( k ) k 2 z ̂ and (A.5) N z ̂ , t = − i k F ϕ − + ∫ − L η F − 1 z ̂ d η ′ F − 1 z ̂ . where L is a large positive constant. The only difference from the classical BO case is that for sBO the nonlinear term N has a time dependent coefficient. To solve the above ODE system in Fourier space we use a modified version of the exponential-time-differencing fourth-order Runge–Kutta (ETDRK4) method [37,38] . For the required spectral accuracy of the ETDRK4 method, the initial condition for z must be smooth and periodic. However, the step initial condition (2.3) for u or equivalently f leads to z ( η , 0 ) = − δ ( η ) , where δ represents the Dirac delta function. Therefore we regularize this initial condition with the analytic function [36] (A.6) z η , 0 = − K ̃ 2 sech 2 K ̃ η , where K ̃ > 0 is large. Thus Eq. (A.3) can be solved numerically via Eqs. ( (A.4) – (A.5) ) on a finite spatial domain [ − L , L ] , where F represents the discrete Fourier transform. Appendix B I would like to thank you the referees for their constructive comments which led to the improvement of this paper. This research was supported by the Istanbul Technical University Office of Scientific Research Projects (ITU BAPSIS), Turkey, under grant TGA-2018-41318. We thank D.E. Baldwin for MATLAB codes of version of the ETDRK4 method that we use in the study.
| Finansörler | Finansör numarası |
|---|---|
| Istanbul Technical University Office of Scientific Research Projects | |
| Istanbul Teknik Üniversitesi | TGA-2018-41318 |