Description of tsunami propagation based on the Maslov canonical operator

Functions of Noncommuting Operators in an Asymptotic Problem for a 2D Wave Equation with Variable Velocity and Localized Right-hand Side

“…satisfying (2.4). Such functions model elliptic-shaped sources of various eccentricity and various direction of axes with a wavy relief depending on the differential operator P (see [11,15]). Figure 1 shows the graph of V (y) rotated by an angle of π/10 and of its Fourier transform.…”

“…We consider the Cauchy problem with zero initial data for a 2D wave equation with variable velocity and with right-hand side localized near the origin in space and decaying in time. One physical interpretation of this problem is that it describes, in the linear approximation, the propagation of tsunami waves generated by local vertical displacements of the ocean bottom (see [4][5][6][7][8][9][17][18][19] and also [11][12][13][14][15][16] and the bibliography therein). Normally, the diameter of the region where these displacements occur (some tens to a hundred of kilometers) is much smaller than the distance traveled by the waves (thousands of kilometers), and their ratio, µ, can serve as a small parameter.…”

“…This corresponds to a right-hand side of the form δ ′ (t)v(x), where δ(t) is the Dirac delta function, and the problem is immediately equivalent, via Duhamel's principle, to the Cauchy problem for the homogeneous wave equation with initial data v(x) for the unknown function itself and zero initial data for its t-derivative. Fairly explicit asymptotic solution formulas suitable for easy implementation in Wolfram Mathematica [27] were constructed and analyzed for the latter problem in [11][12][13][14][15][16] on the basis of a generalization of Maslov's canonical operator [1,20]. Now assume we wish to take into account the fact that the ocean bottom displacement evolves in time rather than happens instantaneously.…”

“…The transient part is apparently not so important in tsunami wave analysis, but nevertheless it might be useful from the viewpoint of satellite registration of tsunami waves [17][18][19]. Since, as was mentioned above, the asymptotic formulas for the solution of the Cauchy problem with localized initial data for the homogeneous wave equation are already known from [11][12][13][14][15][16], we see that the only remaining thing is to compute f (L)v for all these functions f (ξ). It is here that noncommutative analysis comes fully into play.…”

Asymptotic solutions of the linear shallow-water equations with localized initial data

“…We present these formulae in Sect. 3; the majority of them were presented in different papers and in various forms [5][6][7][8][9]. Their derivation is not simple.…”