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Asymptotic evaluation of three-dimensional integrals with singularities in application to wave phenomena

A. V. Shanin, A. Yu. Laptev

TL;DR

The paper develops a 3D extension of stationary-phase analysis for Fourier-type integrals with integrand singularities by deforming the integration domain into complex space, ensuring $\mathrm{Im}\,G>0$ away from a finite set of special points. It introduces a locality principle: the leading asymptotics decompose into contributions from neighborhoods of these special points, with precise topological and gradient conditions determining when each point contributes. Leading terms are derived for several point types, including stationary-phase points in the bulk, stationary-phase on singularities, crossings of singularities, and conical points, with nested or Gamma-type integral reductions where applicable. The method is demonstrated on Kelvin waves behind a moving body, where two families of waves arise from stationary points on the intersection of singularities, yielding the classical Kelvin angle and capturing transient behavior. The approach provides a general, physically interpretable framework for 3D wave phenomena described by oscillatory integrals with singular integrands, applicable to deep-water wave problems and beyond.

Abstract

We consider a three-dimensional Fourier integral in which the exponent in the exponential factor is the product of some phase function and a large parameter. The asymptotics of this integral is sought when the large parameter tends to infinity. In the one-dimensional case, the asymptotics of such an integral is constructed by the points of stationary phase and singularities of the integrand. The three-dimensional case is more complicated: special points such as points of stationary phase in the domain, on singularity, on the crossing of singularities, points of triple crossing of singularities, and also conical points of the singularities, can contribute to the asymptotics. For all these types of singularities, topological conditions for the existence of nonzero asymptotics are constructed, and the asymptotics themselves are derived. The proposed technique is tested on the example of the classical problem of Kelvin waves on the surface of a deep fluid behind a towed body.

Asymptotic evaluation of three-dimensional integrals with singularities in application to wave phenomena

TL;DR

The paper develops a 3D extension of stationary-phase analysis for Fourier-type integrals with integrand singularities by deforming the integration domain into complex space, ensuring away from a finite set of special points. It introduces a locality principle: the leading asymptotics decompose into contributions from neighborhoods of these special points, with precise topological and gradient conditions determining when each point contributes. Leading terms are derived for several point types, including stationary-phase points in the bulk, stationary-phase on singularities, crossings of singularities, and conical points, with nested or Gamma-type integral reductions where applicable. The method is demonstrated on Kelvin waves behind a moving body, where two families of waves arise from stationary points on the intersection of singularities, yielding the classical Kelvin angle and capturing transient behavior. The approach provides a general, physically interpretable framework for 3D wave phenomena described by oscillatory integrals with singular integrands, applicable to deep-water wave problems and beyond.

Abstract

We consider a three-dimensional Fourier integral in which the exponent in the exponential factor is the product of some phase function and a large parameter. The asymptotics of this integral is sought when the large parameter tends to infinity. In the one-dimensional case, the asymptotics of such an integral is constructed by the points of stationary phase and singularities of the integrand. The three-dimensional case is more complicated: special points such as points of stationary phase in the domain, on singularity, on the crossing of singularities, points of triple crossing of singularities, and also conical points of the singularities, can contribute to the asymptotics. For all these types of singularities, topological conditions for the existence of nonzero asymptotics are constructed, and the asymptotics themselves are derived. The proposed technique is tested on the example of the classical problem of Kelvin waves on the surface of a deep fluid behind a towed body.

Paper Structure

This paper contains 10 sections, 2 theorems, 138 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem statement and basic assumptions
  3. Scheme of the method. Description of the deformation of the integration domain
  4. Bypass of the singularities of $\sigma_j$ by the initial domain $\Gamma$
  5. The principle of locality
  6. Points that do not contribute to the integral evaluation
  7. Points contributing to the integral asymptotics (special points)
  8. Special points on the boundary of the integration domain
  9. An example: Kelvin waves
  10. Conclusion

Key Result

Proposition 1

Let $D$ be a small ball in $\mathbb{R}^3$ centered at $\xi^* \in \sigma_j$, and let $\Gamma$ be described by a small enough smooth vector field $\eta$. The following condition is sufficient for $\Gamma \cap D$ not to intersect $\sigma_j$:

Figures (7)

  • Figure 1: Contours $\hat{\gamma}_+$ and $\hat{\gamma}_-$
  • Figure 2: Contours $\gamma_+$ and $\gamma_-$ for the stationary phase point
  • Figure 3: Contours $\tilde{\gamma}_+$ and $\tilde{\gamma}_-$ for the stationary phase point
  • Figure 4: Cones $K_\pm$
  • Figure 5: Curves $L_\pm$
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2