阶跃初值条件解的完全分类:流体力学中广义Gardner方程的分析与数值验证

The Complete Classification of Solutions to the Step Initial Condition:Analysis and Numerical Verification for the Generalized Gardner Equation in Fluid Mechanics

该文中,通过Whitham调制理论研究了广义Gardner方程的初始不连续性的演化,该方程可以描述地形上分层流体的跨临界流动.首先,通过雅可比椭圆函数表示的周期波推导出不同极限情况下的线性谐波,孤子和非线性三角波.随后通过有限间隙积分方法得到了基于黎曼不变量的Whitham特征速度与调制系统.由于广义Gardner方程的调制系统既不是严格的椭圆型也不是严格的双曲型,这使得与KdV方程相比,不同区域当中的动力学演化行为更加多样化.此外,对正负三次非线性项情况下的所有波结构进行了完整的分类,包括色散冲击波,稀疏波,三角冲击波,扭结及其组合波结构,并通过数值模拟验证了结果的正确性.最后分析了一定条件下线性项和非线性项的系数对阶跃初值问题的影响.

In this paper,we investigate the evolution of the initial discontinuity for the generalized Gardner equation through the Whitham modulation theory,which the generalized Gardner equation can describe the transcritical fow of stratified fuids over topography.Firstly,we derive the linear harmonic wave,soliton and nonlinear trigonometric wave in different limiting cases via the periodic waves represented by the Jacobi elliptic functions.Then we obtain the Whitham characteristic velocities and modulation system based on the Riemann invariants by the finite-gap integration method.Since the modulation system of the generalized Gardner equation is neither strictly elliptic nor hyperbolic type,which makes the dynamical evolution behavior more varied in different regions compared to the KdV equation.Furthermore,we perform a complete classification for all wave structures in the cases of positive and negative cubic nonlinear terms,including the dispersive shock wave,rarefaction wave,trigonometric dispersive shock wave,solibore and their combinations.In addition,the correctness of the results is verified by numerical simulations,and the numerical solutions are in good agreement with the analytical solutions.Finally,the influences of the coefficients of the linear and nonlinear terms on the step initial value problem under certain conditions are analyzed.