关于Boussinesq方程的多维伪局部解
On the Pseudolocalized Solutions in Multi-dimension of Boussinesq Equation
摘要：
考虑一类新的三维方程的Boussinesq范式的解决方案。相应的配置文件不是本地化的功能在这个意义上的可积性的平方在一个无限的区域。对于新类型的解，梯度和Hessian/Laplacian是平方可积的。在线性极限情况下，得到了伪局部解的解析表达式;在非线性情况下，用球对称微分算子的特殊近似进行数值处理，这种近似允许自动识别坐标系原点的行为条件。渐近边界条件源于伪局部化轮廓在无穷远处的$1/r$行为。设计了一种特殊的近似，使我们能够获得更小的计算框的适当行为。对于二次非线性和三次非线性都得到了伪局部化解。
英文摘要：
A new class of solutions of three-dimensional equations from the Boussinesq paradigm are considered. The corresponding profiles are not localized functions in the sense of the integrability of the square over an infinite domain. For the new type of solutions, the gradient and the Hessian/Laplacian are square integrable. In the linear limiting case, analytical expressions for the profiles of the pseudolocalized solutions are found. The nonlinear case is treated numerically with a special approximation of the differential operators with spherical symmetry that allows for automatic acknowledgement of the behavioral conditions at the origin of the coordinate system. The asymptotic boundary conditions stem from the $1/r$ behavior at infinity of the pseudolocalized profile. A special approximation is devised that allows us to obtain the proper behavior for much smaller computational box. The pseudolocalized solutions are obtained for both quadratic and cubic nonlinearity.
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