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Long time behavior of solutions to the generalized Boussinesq equation

Amin Esfahani, Gulcin M. Muslu

TL;DR

The paper analyzes the Cauchy problem for the generalized Boussinesq equation $u_{tt} = {(u - \alpha u_{xx} + u_{tt} - \kappa u_{xxtt} + f(u))}_{xx}$ arising in surface-tension–driven water waves, addressing long-time behavior via a multi-space well-posedness theory, decay estimates, and solitary-wave analysis. It develops local well-posedness in Sobolev, Bessel potential, and modulation spaces, derives linear decay and dispersive bounds, and establishes asymptotic decay in various norms. The study proves nonexistence of solitary waves for several parameter regimes using Pohozaev identities, and simultaneously constructs solitary waves numerically with the Petviashvili iteration, complemented by a Fourier pseudo-spectral time-evolution scheme. Numerical experiments illuminate gap-interval dynamics between global existence and blow-up, revealing threshold amplitudes that separate global persistence from finite-time blow-up and demonstrating how higher-order dispersion and nonlinearity shape long-time behavior.

Abstract

In this paper, we study the generalized Boussinesq equation as a model for the water wave problem with surface tension. Initially, we investigate the initial value problem within Sobolev spaces, deriving conditions under which solutions are either global or experience blow-up in time. Subsequently, we extend our analysis to Bessel potential and modulation spaces, determining the asymptotic behavior of solutions. We establish the non-existence of solitary waves for certain parameters using Pohozaev-type identities. Additionally, we numerically generate solitary wave solutions of the generalized Boussinesq equation through the Petviashvili iteration method. To further examine the time evolution of solutions, we propose employing the Fourier pseudo-spectral numerical method. Our investigation extends to the gap interval, where neither global existence nor blow-up results have been theoretically established. We find that our numerical results effectively fill these gaps, supplementing the theoretical findings.

Long time behavior of solutions to the generalized Boussinesq equation

TL;DR

The paper analyzes the Cauchy problem for the generalized Boussinesq equation arising in surface-tension–driven water waves, addressing long-time behavior via a multi-space well-posedness theory, decay estimates, and solitary-wave analysis. It develops local well-posedness in Sobolev, Bessel potential, and modulation spaces, derives linear decay and dispersive bounds, and establishes asymptotic decay in various norms. The study proves nonexistence of solitary waves for several parameter regimes using Pohozaev identities, and simultaneously constructs solitary waves numerically with the Petviashvili iteration, complemented by a Fourier pseudo-spectral time-evolution scheme. Numerical experiments illuminate gap-interval dynamics between global existence and blow-up, revealing threshold amplitudes that separate global persistence from finite-time blow-up and demonstrating how higher-order dispersion and nonlinearity shape long-time behavior.

Abstract

In this paper, we study the generalized Boussinesq equation as a model for the water wave problem with surface tension. Initially, we investigate the initial value problem within Sobolev spaces, deriving conditions under which solutions are either global or experience blow-up in time. Subsequently, we extend our analysis to Bessel potential and modulation spaces, determining the asymptotic behavior of solutions. We establish the non-existence of solitary waves for certain parameters using Pohozaev-type identities. Additionally, we numerically generate solitary wave solutions of the generalized Boussinesq equation through the Petviashvili iteration method. To further examine the time evolution of solutions, we propose employing the Fourier pseudo-spectral numerical method. Our investigation extends to the gap interval, where neither global existence nor blow-up results have been theoretically established. We find that our numerical results effectively fill these gaps, supplementing the theoretical findings.
Paper Structure (12 sections, 33 theorems, 194 equations, 17 figures)

This paper contains 12 sections, 33 theorems, 194 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Well-posedness in the Bessel potential spaces
  4. Asymptotic behavior
  5. Modulation spaces
  6. Solitary wave solutions
  7. The Petviashvili iteration method
  8. Accuracy test
  9. Numerical generation of solitary waves for the gBq equation
  10. Fourier pseudo-spectral numerical method for the gBq equation
  11. Time evolution of the single solitary wave
  12. Global and Blow-up Solutions

Key Result

Lemma 2.1

The following inequalities hold for all $\varphi\in H^s(\mathbb{R})$ and $\psi\in H^{s-2}(\mathbb{R})$:

Figures (17)

  • Figure 1: The solitary wave solution constructed by Petviashvili method for \ref{['gBE']} and the exact solitary wave solution of the HBq equation for the speed $c = 1.1272$ and the variation of the $Error(n)$, $|1-M_n|$ and $RES$ with the number of iterations in semi-log scale.
  • Figure 2: The solitary wave solution generated by the Petviashvili's method for \ref{['gBE']} with the parameters $\alpha=2$, $c=\sqrt{3}, \kappa=\beta=1$ (top left), the close-up look at the profile (top right) and the variation of the $Error(n)$, $|1-M_n|$ and $RES$ with the number of iterations in semi-log scale (bottom).
  • Figure 3: The solitary wave solution generated by the Petviashvili's method for \ref{['gBE']} with the parameters $\alpha=2$, $c=1.3, \kappa=\beta=1$ (left panel) and the variation of the $Error(n)$, $|1-M_n|$ and $RES$ with the number of iterations in semi-log scale (right panel).
  • Figure 4: The solitary wave solution generated by the Petviashvili's method for \ref{['gBE']} with the parameters $\alpha=0.5, c=\sqrt{2.1}, \kappa=0.1, \beta=1$ (left panel), the close-up look at the profile (right panel).
  • Figure 5: The solitary wave solution obtained by the Fourier pseudo-spectral scheme for \ref{['gBE']} with $\alpha=0$ and the exact solitary wave solution for the HBq equation
  • ...and 12 more figures

Theorems & Definitions (54)

  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.2
  • Theorem 2.4
  • ...and 44 more