Dispersive Riemann problems for the Benjamin–Bona–Mahony equation
Dispersive Riemann problems for the Benjamin–Bona–Mahony equation
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DOI：
10.1111/sapm.12426
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发表时间：
2020-12
影响因子：
2.7
通讯作者：
M. Shearer;G. El;M. Hoefer;T. Congy
中科院分区：
文献类型：
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作者：
M. Shearer;G. El;M. Hoefer;T. Congy
利用渐近方法和数值模拟研究了Benjamin - Bona - Mahony （BBM）方程ut+uux=uxxt的光滑阶跃初值问题或色散Riemann问题的长时间动力学问题。BBM方程的色散黎曼问题的解目录比相关的可积Korteweg‐de Vries方程ut+uux+uxxx=0的解目录要丰富得多。发现初始平滑阶跃的过渡宽度对动力学有显著影响。窄宽度会产生稀疏和色散激波（DSW），并伴随产生两相线性波串、孤立波脱落和膨胀激波。窄的初始宽度和宽的初始宽度都会产生两相非线性波串或DSW内爆，并对对称数据产生一种新的色散Lax激波。用BBM方程的近似自相似解描述了色散松弛激波，该方程在t→∞时的极限是平稳的不连续弱解。通过在色散松散激波的数据中引入轻微的不对称性，观察到非相干孤立波列的产生。进一步的不对称性导致DSW内爆状态有效地描述了一对耦合的非线性Schrödinger方程。BBM方程中的非定域性、非线性和色散之间的复杂相互作用是色散黎曼问题的丰富的非经典色散流体动力解的基础。
Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equation ut+uux=uxxt are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation ut+uux+uxxx=0 . The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit as t→∞ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem.