Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Abstract

a b s t r a c tThe classical group analysis approach used to study the symmetries of integro-differential equations in a semiclassical approximation is considered for a class of nearly linear integrodifferential equations. In a semiclassical approximation, an original integro-differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis.The approach is illustrated by the calculation of the Lie symmetries in explicit form for a special case of the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov population equation.

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

“…However, for nearly linear equations [18] a wide class of symmetry operators can be constructed by solving linear determining equations for operators of this type much as symmetry operators are found for linear PDEs. We have illustrated this situation with the example of the generalized multidimensional Gross-Pitaevskii equation (2.1).…”

“…The intertwining operator D(t, C , C) presented, according to (3.12), as 18) where, according to (3.11),…”

Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

“…Note that this equation yieldsṠ(t, D 1 ) = 0 for S(t, D 1 ) in (18) and u (0,0) (x, t) ∈ P D 1 t . We can call (39) the semiclassically reduced nonlocal Fisher-KPP Equation (10) for the leading term u (0,0) (x, t) of semiclassical asymptotics (37).…”

“…Extending the scope of the Lie-group methods to integro-differential equations by invoking various means of bringing IDEs to PDEs (see, e.g., [13][14][15][16][17]) made it possible to apply the group and symmetry analysis to non-local RD models with long-range interactions, including non-local generalizations of the Fisher-KPP equation [18].…”

“…Approximate methods and approaches are also promising for the development of analytical methods to study complex and especially non-local RD models. We will focus on the Wentzel-Kramers-Brillouin (WKB)-Maslov method of semiclassical asymptotics [19][20][21], which has been effectively applied to the non-local generalization of the Fisher-KPP equation (see [6,7,18,22,23] and references therein):…”