A note on solitary waves solutions of classical wave equations

Solitary wave solutions to a class of Whitham-Boussinesq systems

In this note we study solitary wave solutions of a class of Whitham-Boussinesq systems which includes the bi-directional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnström, Groves and Wahlén. In that paper the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni, Groves and Wahlén, together with the concentration-compactness principle.

A Note On 3Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation"

The goal of this note is to construct a class of traveling solitary wave solutions for the compound Burgers-Korteweg-de Vries equation by means of a hyperbolic ansatz.

No pure capillary solitary waves exist in 2D finite depth

We prove that the 2D finite depth capillary water wave equations admit no solitary wave solutions. This closes the existence/non-existence problem for solitary water waves in 2D, under the classical assumptions of incompressibility and irrotationality, and with the physical parameters being gravity, surface tension and the fluid depth.

Notes on numerical analysis and solitary wave solutions of Boussinesq/Boussinesq systems for internal waves

In this paper a three-parameter family of Boussinesq systems is studied. The systems have been proposed as models of the propagation of long internal waves along the interface of a two-layer system of fluids with rigid-lid condition for the upper layer and under a Boussinesq regime for the flow in both layers. We first present some theoretical properties of well-posedness, conservation laws and Hamiltonian structure of the systems. Then the corresponding periodic initial-value problem is discretized in space by the spectral Fourier Galerkin method and for each system, error estimates for the s…

Notes on the Boussinesq-Full dispersion systems for internal waves: Numerical solution and solitary waves

In this paper we study some theoretical and numerical issues of the Boussinesq/Full dispersion system. This is a a three-parameter system of pde's that models the propagation of internal waves along the interface of two-fluid layers with rigid lid condition for the upper layer, and under a Boussinesq regime for the upper layer and a full dispersion regime for the lower layer. We first discretize in space the periodic initial-value problem with a Fourier-Galerkin spectral method and prove error estimates for several ranges of values of the parameters. Solitary waves of the model systems are the…

Solitary wave solutions of a generalization of the mKdV equation

We consider a generalization of the mKdV equation, which contains dissipation terms similar to those contained in both the Benjamin-Bona-Mahoney equation and the famous Camassa-Holm and Degasperis-Procesi equations. Our objective is the construction of classical (solitons) and non-classical (peakons and cuspons) solitary wave solutions of this equation.

Solitary waves in a Whitham equation with small surface tension

Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator i…

Stability of the Solitary Manifold of the Perturbed Sine-Gordon Equation

We study the perturbed sine-Gordon equation $θ_{tt}-θ_{xx}+\sin θ= F(\varepsilon,x)$, where $F$ is of differentiability class $C^n$ in $\varepsilon$ and the first $k$ derivatives vanish at $0$, i.e., $\partial_\varepsilon^l F(0,\cdot)=0$ for $0\le l\le k $. We construct implicitly a virtual solitary manifold by deformation of the classical solitary manifold in $n$ iteration steps. Our main result establishes that the initial value problem with an appropriate initial state $\varepsilon^n$-close to the virtual solitary manifold has a unique solution which follows up to time $1/(\tilde C\varepsil…

关于混合色散广义四阶非线性薛定谔方程孤立波的轨道稳定性

On the orbital stability of solitary waves for the fourth order nonlinear Schrödinger equation

本文研究混合色散广义四阶非线性薛定谔方程孤立驻波的轨道稳定性，通过数值方法构造光滑曲线，确定非线性项阈值功率α₀≈4.8，明确α在不同区间时显式孤立波的稳定与不稳定特性。

Solitary wave solutions of a Whitham-Boussinesq system

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…

Existence and stability of interfacial capillary-gravity solitary waves with constant vorticity

In this paper, we consider capillary-gravity waves propagating on the interface separating two fluids of finite depth and constant density. The flow in each layer is assumed to be incompressible and of constant vorticity. We prove the existence of small-amplitude solitary wave solutions to this system in the strong surface tension regime via a spatial dynamics approach. We then use a variant of the classical Grillakis--Shatah--Strauss (GSS) method to study the orbital stability/instability of these waves. We find an explicit function of the parameters (Froude number, Bond number, and the depth…

Traveling waves and transverse instability for the fractional Kadomtsev-Petviashvili equation

Of concern are traveling wave solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension breaking bifurcation. Moreover, the line solitary wave solutions and their transverse (in)stability are discussed. Analogous to the classical Kadmomtsev--Petviashvili (KP) equation, the fKP equation comes in two versions: fKP-I and fKP-II. We show that the line solitary waves of fKP-I equation are transversely linearly instable. We also perform numerical experiments to observe the (in)stability dynamics of line…

Construction of solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle for the $L^2$-supercritical case

We consider the focusing $L^2$-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solution of NLS blowing up at several blow-up point together with a topological argument…

Global bifurcation of solitary waves to the Boussinesq $abcd$ system

The Boussinesq $abcd$ system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters $(a,b,c,d)$ satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to $b \ne d$). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the glo…

Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities

A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational…

大振幅稳态重力水波与一般涡度

Large-Amplitude Steady Solitary Water Waves with General Vorticity

研究有限深度流体表面稳态孤立重力波，处理一般涡度分布和悬垂波形，通过共形映射将问题转化为椭圆系统和边界值问题，利用中心 manifold 方法证明小振幅波存在，基于解析全局分支定理获得大振幅波。

Structural stability of finite dispersion-relation preserving schemes

The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numeric…

Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

We study the linear stability of localized modes in self-interacting spinor fields, analyzing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov--Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov--Kolokolov condition $dQ/dω=0$ and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross--Neveu model and the generalized massive Thirring model in the charge-subcritical, critical, and s…

Long time dynamics of highly concentrated solitary waves for the nonlinear Schroedinger equation

In this paper we study the behavior of solutions of a nonlinear Schroedinger equation in presence of an external potential, which is allowed to be singular at one point. We show that the solution behaves like a solitary wave for long time even if we start from a unstable solitary wave, and its dynamics coincide with that of a classical particle evolving according to a natural effective Hamiltonian.

Comment on "Orbital stability of solitary wave solutions for an interaction equation of short and long dispersive waves"

J. Angulo and J. F. Montenegro (J. Differential Equations 174 (2001), no. 1, 181-199) published a paper about nonlinear stability of solitary waves for an interaction system between a long internal wave and a short surface wave in a two layer fluid considering that the fluid depth of the lower layer is sufficiently large in comparison with the wavelength of the internal wave. In this note, we show that in a critical step during the proof of Lemma 2.4 in the above mentioned paper, there is a claim used by the authors which fails to be true. Lemma 2.4 is crucial for the proof of Lemma 2.7, and f…

Transverse instability of periodic and generalized solitary waves for a fifth-order KP model

We consider a fifth-order Kadomtsev-Petviashvili equation which arises as a two-dimensional model in the classical water-wave problem. This equation possesses a family of generalized line solitary waves which decay exponentially to periodic waves at infinity. We prove that these solitary waves are transversely spectrally unstable and that this instability is induced by the transverse instability of the periodic tails. We rely upon a detailed spectral analysis of some suitably chosen linear operators.

退化情形下带吸引δ势的一维NLS孤立波的不稳定性

Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case

本文针对退化情形下带吸引δ势的一维NLS，通过构造不稳定二阶近似和精细Virial恒等式，克服经典方法失效问题，证明了孤立波的轨道不稳定性。

Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations

Some effects of surface tension on fully-nonlinear, long, surface water waves are studied by numerical means. The differences between various solitary waves and their interactions in subcritical and supercritical surface tension regimes are presented. Analytical expressions for new peaked travelling wave solutions are presented in the case of critical surface tension. The numerical experiments were performed using a high-accurate finite element method based on smooth cubic splines and the four-stage, classical, explicit Runge-Kutta method of order four.

Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity

We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimiser of the wave energy $\mathcal H$ subject to the constraint $\mathcal I=2μ$, where $\mathcal I$ is the wave momentum and $0< μ\ll 1$. Since $\mathcal H$ and $\mathcal I$ are both conserved quantities a standard argument asserts the stability of the set $D_μ$ of minimisers: solutions starting near $D_μ$ remain close to $D_μ$ in a suit…

Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves

In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy subject to the constraint that the momentum is fixed. We prove the existence of a minimiser of the energy subject to the constraint that the momentum is fixed and small. The existence of a small-amplitude solitary wave is thus as…

Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials

We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order $α>1$. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when $α$ is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV e…

A variational approach to solitary gravity-capillary interfacial waves with infinite depth

We present an existence and stability theory for gravity-capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy $\mathcal{E}$ subject to the constraint $\mathcal{I}=2μ$, where $\mathcal{I}$ is the wave momentum and $0< μ< μ_0$, where $μ_0$ is chosen small enough for the validity of our calculations. Since $\mathcal{E}$ and $\mathcal{I}$ are both conserved quantities a standard argument asserts the …

Solitary waves for the Hartree equation with a slowly varying potential

We study the Hartree equation with a slowly varying smooth potential, $V(x) = W(hx)$, and with an initial condition which is $ε\le \sqrt h$ away in $H^1$ from a soliton. We show that up to time $|\log h|/h$ and errors of size $ε+ h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer-Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the resul…

A variational reduction and the existence of a fully-localised solitary wave for the three-dimensional water-wave problem with weak surface tension

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $β$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<β<\frac{1}{3}$. A classical vari…

A dimension-breaking phenomenon for water waves with weak surface tension

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength t…

A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations

In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schrödinger equations.

Title Scattering of Solitons for Schrödinger Equation Coupled to a Particle

We establish soliton-like asymptotics for finite energy solutions to the Schrödinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold, converges to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto solitary manifold in the Hilbert phase space.

Klein-Gordon-Boussinesq系统的数学性质

Mathematical properties of Klein-Gordon-Boussinesq systems

本文从初值问题适定性、行波解（含孤立波存在性与剖面性质）及KdV近似有效性三方面，分析Klein-Gordon-Boussinesq（KGB）系统的数学性质，为长波极限近似研究提供模型支撑。

关于次线性Fujita问题具有精确代数速率和过渡指数的时间渐近界的一个注记

A note on time-asymptotic bounds with a sharp algebraic rate and a transitional exponent for the sublinear Fujita problem

本文为次线性指数的Fujita经典演化问题建立了精确的时间渐近代数速率界，并识别出一个与经典Fujita临界爆破指数存在简单互反关系的过渡稳定性指数。

关于一维波动方程的导数半线性模型经典解存在性的注记

Note on the existence of classical solutions of derivative semilinear models for one dimensional wave equation

本文补充了Takamura关于一维非线性波动方程的综述，研究具有乘积型导数非线性项的半线性波动方程经典解的长时间存在性，通过逐点估计的经典迭代论证扩展了单分量情形，为验证一般理论的最优性提供了更多模型。

改进的无旋孤立水波弗鲁德数上限

An improved upper bound for the Froude number of irrotational solitary water waves

本文通过新策略严格改进了Starr的弗鲁德数上限，证明了孤立波底部速度不超过47%的传播速度。

Solutions of the focusing nonradial critical wave equation with the compactness property

Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. In a previous paper, we proved that any solution which is bounded in the energy space converges, along a sequence of times and in some weak sense, to a solution with the compactness property, that is a solution whose trajectory stays in a compact subset of the energy space up to space translation and scaling. It is conjectured that the only solutions with the compactness property are stationary solutions and solitary waves that are Lorentz transforms of the former. In this note we prove this conjecture with an ad…

A note on the sublinear Sobolev inequality

In this note we show that the classical Sobolev inequality cannot in unrestricted form hold for exponents $p \in (0,1)$.

Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis

Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of nonlinear and dispersive water waves of significant interest such as solitary and tsunami waves. The initial-boundary value problem on a finite interval for this family of systems is studied both theoretically and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via the unified transform of Fokas. The corresponding analysis reveals two types of admissible boundary conditions, thereby suggesting appropriate boundary conditions for the nonlinear N…

An index theorem for the stability of periodic traveling waves of KdV type

We consider periodic solutions to equations of Korteweg-Devries type. While the stability theory for periodic waves has received much some attention the theory is much less developed than the analogous theory for solitary wave stability, and appears to be mathematically richer. We prove an index theorem giving an exact count of the number of unstable eigenvalues of the linearized operator in terms of the number of zeros of the derivative of the traveling wave profile together with geometric information about a certain map between the constants of integration of the ordinary differential equati…

关于Serre方程的Galerkin/有限元方法

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。