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On the Long Time Existence of a Fractional KdV-BBM Type Equation

Goksu Oruc

TL;DR

We address the Cauchy problem for a fractional KdV-BBM equation with $0<α<1$, aiming to extend the lifespan of small-data solutions from $O(1/ε)$ to $O(1/ε^2)$ using a normal-form transformation and a carefully constructed modified energy that remains almost equivalent to the $H^{N+α/2}$ norm. The analysis combines Fourier techniques to cancel quadratic nonlinearities and energy estimates to control the cubic remainder, yielding a rigorous $T \gtrsim 1/ε^2$-existence result. Numerically, the paper explores long-time behavior with a pseudospectral method and finds evidence of global behavior for $α>1/3$ and potential blow-up for small $α$ with large data, while small data remain globally regular and display radiation and soliton formation. The results illuminate how dual fractional dispersive terms influence lifespan and dynamics, providing both analytical and numerical insights into smooth solution existence and long-time behavior. The work has implications for fractional dispersive models where simultaneous KdV- and BBM-type dispersions interact.

Abstract

We consider a fractional Korteweg de Vries-Benjamin Bona Mahony (KdV-BBM) type equation including both fractional dispersive terms of fractional KdV and fractional BBM equations. We aim to enhance the existence time of solutions with small initial data $|| u_0||_{H^{N+α/2}}= ε$ from $\frac{1}ε$ to $\frac{1}{ε^2}$. The proof relies on the combination of a modified energy method with Fourier techniques. In addition, the long time existence issues are investigated numerically. Numerical observations of the lifespan give an evidence of existence of solutions beyond the hyperbolic time scale. This study provides a detailed analysis from both analytical and numerical aspects for the existence of smooth solutions.

On the Long Time Existence of a Fractional KdV-BBM Type Equation

TL;DR

We address the Cauchy problem for a fractional KdV-BBM equation with , aiming to extend the lifespan of small-data solutions from to using a normal-form transformation and a carefully constructed modified energy that remains almost equivalent to the norm. The analysis combines Fourier techniques to cancel quadratic nonlinearities and energy estimates to control the cubic remainder, yielding a rigorous -existence result. Numerically, the paper explores long-time behavior with a pseudospectral method and finds evidence of global behavior for and potential blow-up for small with large data, while small data remain globally regular and display radiation and soliton formation. The results illuminate how dual fractional dispersive terms influence lifespan and dynamics, providing both analytical and numerical insights into smooth solution existence and long-time behavior. The work has implications for fractional dispersive models where simultaneous KdV- and BBM-type dispersions interact.

Abstract

We consider a fractional Korteweg de Vries-Benjamin Bona Mahony (KdV-BBM) type equation including both fractional dispersive terms of fractional KdV and fractional BBM equations. We aim to enhance the existence time of solutions with small initial data from to . The proof relies on the combination of a modified energy method with Fourier techniques. In addition, the long time existence issues are investigated numerically. Numerical observations of the lifespan give an evidence of existence of solutions beyond the hyperbolic time scale. This study provides a detailed analysis from both analytical and numerical aspects for the existence of smooth solutions.

Paper Structure

This paper contains 11 sections, 4 theorems, 69 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Governing Equation
  3. Preliminaries
  4. Conservation Laws
  5. Solitary Waves
  6. Existence in Large Time Scale
  7. The Normal Form Transform
  8. The Modified Energy
  9. The Energy Estimates
  10. Numerical Results
  11. Results and Discussion

Key Result

Theorem 1.1

For $0<\alpha <1$ and $N\geq 2,$ there exists $\epsilon_0>0$, such that for any initial data $u(0,x)=u_0(x)$ satisfying there exists a unique solution $u \in {C}([0,T]; H^{N+\alpha/2}(\mathbb{R}))$ of fKdVBBM with $T \gtrsim \frac{1}{\epsilon^2}$ and

Figures (7)

  • Figure 1: The propagation of solitary wave to the fractional KdV-BBM equation \ref{['fKdVBBM']} for $\alpha=0.5$.
  • Figure 2: Variation in the $L_{\infty}$ norm of the solution to the fractional KdV-BBM equation \ref{['fKdVBBM']} for $\alpha=0.5$ and $\delta=20$ (upper left panel), modulus of the Fourier coefficients for the corresponding solution (upper right panel) and variation of the energy identiy $I_1$ in time (bottom panel)
  • Figure 3: The profile of the solution to the fractional KdV-BBM equation \ref{['fKdVBBM']} for $\alpha=0.9$ and $\delta=0.1$ at t=50 (upper left panel), variation in the $L_{\infty}$ norm of the corresponfding solution (upper right panel), modulus of the Fourier coefficients for the corresponding solution (bottom panel)
  • Figure 4: The profile of the solution to the fractional KdV-BBM equation \ref{['fKdVBBM']} for $\alpha=0.2$ and $\delta=1.1$ at several times
  • Figure 5: The variation in the $L_{\infty}$ norm of the wave profile of the fractional KdV-BBM equation \ref{['fKdVBBM']} for $\alpha=0.2$ and $\delta=1.1$ with time (left panel), modulus of the Fourier coefficients for the corresponding solution (right panel)
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • proof : Proof of Theorem \ref{['theorem']}