Y. Martel, F. Merle, T. Mizumachi
We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves).
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In this work, we present a stability criteria for the solitary wave solutions to a BBM system that contains coupled nonlinear terms. Using the idea by Bona, Chen and Karakashian and exploiting the accurate point spectrum information of the associated Schrodinger operator, we improve the stability results previously got by Pereira.
This paper sheds new light on the stability properties of solitary wave solutions associated with models of Korteweg-de Vries and Benjamin\&Bona\&Mahoney type, when the dispersion is very lower. Via an approach of compactness, analyticity and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of linear instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and linear instability of the ground states solutions for both models is obtained for som…
We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass $\wp$ functions of both third and fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV--BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstra…
Solitary wave solutions of the delayed KP-BBM equation
针对延迟 KP-BBM 浅水波方程，结合 GSP 理论、不变流形、动力系统方法与 Melnikov 函数，证明不同非线性项扰动下孤立波解的存在性，并以数值模拟验证。
Orbital instability of solitary waves for the generalized BBM equation at the negative critical endpoint
该研究针对广义Benjamin–Bona–Mahony方程，证明在负临界端点速度下，满足特定参数条件的孤立波具有轨道不稳定性，证明结合了余维二coercivity估计、两参数调制与修正局部位势泛函。
This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of the results obtained in arXiv:0910.3204 by the same authors, addressing the same question for the quartic (gKdV) equation. First, we prove that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable (KdV) equation in this regime. Second, we prove that the collision is not perfectly elastic, except in the integrable case (i.e. in the limiting case of the (KdV) equation).
We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to …
Lie Symmetry Analysis and Some New Exact Solutions to the KP-BBM Equation
针对Wazwaz提出的KP-BBM方程，本文结合Painleve检验、Lie对称性分析、Hu-Li-Chen算法及齐次平衡法与Tanh法，构造最优系统、相似约化和新精确解，并分析参数对孤立波的影响。
We consider a system of coupled nonlinear Schr{ö}dinger equations in one space dimension. First, we prove the existence of multi-speed solitary waves, i.e solutions to the system with each component behaving at large times as a solitary wave. Then, we investigate numerically the interaction of two solitary waves supported each on one component. Among the possible outcomes, we find elastic and inelastic interactions, collision with mass extraction and reflexion.
We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularised shallow water equations that include the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We es…
We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-Δ)^s Q + Q - Q^{α+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < α< \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < α< \infty$ for $s \geq 1/2$. Here $(-Δ)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $α=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-Δ)^s…
On the Galerkin / finite-element method for the Serre equations
本文提出了一种高精度数值方案，用于求解描述完全非线性 regime 中 dispersive 浅水波传播的 Serre 方程组。该方案在空间上使用光滑周期样条的 Galerkin/有限元方法，在时间上使用显式四阶 Runge-Kutta 方法。通过与精确的孤立波和 cnoidal 波解的比较，证明了该方案在空间和时间上的最优精度。此外，利用该数值方案详细研究了孤立波、cnoidal 波和 dispersive 破波，并与经典的 Boussinesq 方程进行了比较。结果显示，Serre 方程组中孤立波的相互作用更为不弹性。通过与渐近结果的比较，证明了该数值方案在模拟 dispersive 破波方面的有效性。这些结果可应用于中间或大振幅浅水波的建模，如近岸区域中的情况。
We are concerned with numerical approximations of breather solutions for the cubic Whitham equation which arises as a water-wave model for interfacial waves. The model combines strong nonlinearity with the non-local character of the water-wave problem. The equation is non-integrable as suggested by the inelastic interaction of solitary waves. As a non local model, it generalizes, in the low frequency limit, the well known modified KdV (mKdV) equation which is a completely-integrable model. The mKdV equation has breather solutions, i.e. periodic in time and localized in space biparametric solut…
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