We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$. This model has solitary waves with speeds $-1<c<1$. When $|c|$ approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.
In a recent paper, we developed an inverse scattering approach to the Boussinesq equation in the case when no solitons are present. In this paper, we extend this approach to include solutions with solitons.
In this paper the permanent profile waves governed by a Boussinesq-type wave equation are analysed. The model involves displacement-type nonlinearities and dispersion terms. Physically such a model equation describes longitudinal waves (density change) in biomembranes which have an internal structure composed by lipid molecules. The possible solutions are constructed and analysed. The phase plane analysis and numerical simulation reveal a novel phenomenon: the possible existence of a soliton doublet.
We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by $\frac{x}{t}\in (\frac{1}{\sqrt{3}},1)$, in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction…
This work oversees with the coupled Schrödinger-Boussinesq equations with conformable derivative, which have lots of applications in laser and plasma. The said equations are reduced to a coupled stationary form using complex travelling wave transformation. Next Painlevé test applied to derived the integrable cases of the reduced equation, after that using RCAM derived the solution of reduced equations integrable and nonintegrable cases. Few theorems have been presented and proved to ensure their boundedness. All presented boundedness cases have been checked and explained by plotting the soluti…
In this work we present a solution of the Boussinesq equation. The derived formulas include solitons, Schwartz class solutions and solutions, possessing singularities on a closed set Z of the (x,t) domain, obtained from the zeros of the tau function. The idea for solving the Boussinesq equation is identical to the (unified) idea of solving the KdV and the evolutionary NLS equations: we use a theory of evolutionary vessels. But a more powerful theory of non-symmetric evolutionary vessels is presented, inserting flexibility into the construction and allowing to deal with complex-valued solutions…
We are concerned with conservation laws and integral relations associated with rational solutions of the Boussinesq equation, a soliton equation solvable by inverse scattering which was first introduced by Boussinesq in 1871. The rational solutions are logarithmic derivatives of a polynomial, are algebraically decaying and have a similar appearance to rogue-wave solutions of the focusing nonlinear Schrödinger equation. For these rational solutions the constants of motion associated with the conserved quantities are zero and they have some interesting integral relations which depend on the tota…
We study rational solutions of the Boussinesq equation, which is a soliton equation solvable by the inverse scattering method. These rational solutions, which are algebraically decaying and depend on two arbitrary parameters, are expressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the focusing nonlinear Schrödinger (NLS) equation and have an interesting structure. Further rational solutions of the Kadomtsev-Petviashvili I (KPI) equation are derived in two ways, from rational solutions of the NLS equation a…
We study the local well-posedness of the initial-value problem for the nonlinear "good" Boussinesq equation with data in Sobolev spaces \textit{$H^s$} for negative indices of $s$.
We study the well-posedness of the initial-value problem for the periodic nonlinear "good" Boussinesq equation. We prove that this equation is local well-posed for initial data in Sobolev spaces \textit{$H^s(\T)$} for $s>-1/4$, the same range of the real case obtained in Farah \cite{LG4}.
In this paper we study the regularity properties of the "good" Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. Moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data, obtaining half derivative smoothing of the nonlinear term in some cases. Our paper improves the result in [Himonas-Mantzavinos 2015], being the first result that constructs solutions for the initial and boundary value problem of the "good" Boussinesq equation below the $L^2$ space. …
We develop an inverse scattering transform formalism for the "good" Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a $3 \times 3$ matrix Riemann-Hilbert problem. The Riemann-Hilbert problem is formulated in terms of two reflection coefficients whose definitions involve only the initial data, and it has a form which makes it suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.
This paper concerns the local well-posedness for the "good" Boussinesq equation subject to quasi-periodic initial conditions. By constructing a delicately and subtly iterative process together with an explicit combinatorial analysis, we show that there exists a unique solution for such a model in a small region of time. The size of this region depends on both the given data and the frequency vector involved. Moreover the local solution has an expansion with exponentially decaying Fourier coefficients.
We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space $H^1\times H^2$, existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in w…
We prove that the "good" Boussinesq model with the periodic boundary condition is locally well-posed in the space $H^{s}\times H^{s-2}$ for $s > -3/8$. In the proof, we employ the normal form approach, which allows us to explicitly extract the rougher part of the solution. This also leads to the conclusion that the remainder is in a smoother space $C([0,T], H^{s+a}), where $0 <= a < \min (2s+1, 1/2)$. If we have a mean-zero initial data, this implies a smoothing effect of this order for the non-linearity. This is new even in the previously considered cases $s > -1/4$.
In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in $H^s(\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtained from reduction of the problem into a quadratic nonlinear Schrödinger equation and the contraction argument in suitably modified $X^{s,b}$ spaces. The proof of the crucial bilinear estimates in these spaces, especially in the lowest regularity, rely on some bilinear estimates for one dimensional perio…
We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial zeroes. This new difficulty is solved by the use of "good unknowns " and of normal form techniques.
The aim of this article is to prove new ill-posedness results concerning the nonlinear "good" Boussinesq equation, for both the periodic and non-periodic initial value problems. Specifically, we prove that the associated flow map is not continuous in Sobolev spaces $H^s$, for all $s<-1/2$.
In this article we present ill-posedness results for generalized Boussinesq equations, which incorporate also the ones obtained by the authors for the classical "good" Boussinesq equation (arXiv:1202.6671). More precisely, we show that the associated flow map is not smooth for a range of Sobolev indices, thus providing a threshold for the regularity needed to perform a Picard iteration for these problems.
The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in [Dinvay, Dutykh, Kalisch 2018], where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers [Dinvay 2018], [Dinvay, Selberg, Tesfahun 2019] the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approac…
The Boussinesq $abcd$ system is a 4-parameter set of equations posed in $\mathbb{R}_t \times \mathbb{R}_x$, originally derived by Bona, Chen and Saut as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation. Among many particular regimes, depending each of them in terms of the value of the parameters $(a,b,c,d)$ present in the equations, the "generic" regime is characterized by the setting $b,d>0$ and $a,c<0$. The system is hamiltonian if also $b=d$. Th…
In this paper we address the temperature patch problem of the 2D viscous Boussinesq system without heat diffusion term. The temperature satisfies the transport equation and the initial data of temperature is given in the form of non-constant patch, usually called the temperature front initial data. Introducing a good unknown and applying the method of striated estimates, we prove that our partially viscous Boussinesq system admits a unique global regular solution and the initial $C^{k,γ}$ and $W^{2,\infty}$ regularity of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ …
We study the Gross-Pitaevskii equation (nonlinear Schroedinger equation) with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L^2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.
In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*} {{array}{ll} (\partial_{t}+u\cdot\nabla)u-κΔ_{h} u+\nabla Π=ρe_{3},\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}, (\partial_{t}+u\cdot\nabla)ρ=0, \text{div}u=0. {array}. {equation*} Under the assumption that the support of the axisymmetric initial data $ρ_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\fracρr$ for large time by taking advantage …
Freely floating cylinder on a 3D fluid governed by the Boussinesq equations in the axisymmetric without swirl case
Univ Rennes, IRMAR UMR 6625（雷恩大学，IRMAR UMR 6625）；Centre Inria de l’Université de Rennes (MINGuS)（雷恩大学Inria中心（MINGuS））；ENS Rennes（雷恩高等师范学院）；ENS de Physique de Lyon（里昂物理高等学院）
Primarily motivated by the stability analysis of nonlinear waves in second-order in time Hamiltonian systems, in this paper we develop an instability index theory for quadratic operator pencils acting on a Hilbert space. In an extension of the known theory for linear pencils, explicit connections are made between the number of eigenvalues of a given quadratic operator pencil with positive real parts to spectral information about the individual operators comprising the coefficients of the spectral parameter in the pencil. As an application, we apply the general theory developed here to yield sp…
