Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term

In this work we formally derive the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations. The combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term are then investigated to obtain dark soliton solutions. The solitary wave ansatz is used to carry out the analysis for both models.     

广义KdV方程孤子解的存在参数区

§ 1　IntroductionThe classic Kd V equation has the formut=32gh uux +32 αux +13σuxxx . (1 )Where g,h,α,andσ are physical parameters.After some transformations,various ver-sions of the modified Kd V equation may appear.The Kd V equation is a classic soliton e-quation and is well discussed.Different methods have been created to find the precise orapproximate solitons or solitary waves.The following Kd V equation is the mostcommonone:ut-6 uux +uxxx =0 . (2 )Considering traveling waves an…     

New exact kink solutions, solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation

In this short letter, new exact solutions including kink solutions, soliton-like solutions and periodic form solutions for a combined version of the potential KdV equation and the Schwarzian KdV equation are obtained using the generalized Riccati equation mapping method.     

Solitary waves and their linear stability in weakly coupled KdV equations

We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers. We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.”     

Comments on “New exact kink solutions, solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation”

In this paper, we demonstrate that 14 solutions from 34 of the combined KdV and Schwarzian KdV equation obtained by Li [Z.T. Li, Appl. Math. Comput. 215 (2009) 2886-2890] are wrong and do not satisfy the equation. The other a number of exact solutions are equivalent each other.     

New kinds of solitons and periodic solutions to the generalized KdV equation

In this work, the sine‐cosine method, the tanh method, and specific schemes that involve hyperbolic functions are used to study solitons and periodic solutions governed by the generalized KdV equation. New solutions are determined by using the hyperbolic functions schemes. The study introduces new approaches to handle nonlinear PDEs in the solitary wave theory. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 247–255, 2007     

A novel convenient finite difference method for shallow water waves derived by fifth-order Kortweg and De-Vries-type equation

In this research, two new finite difference schemes are derived and presented for estimating a solution to the fifth-order Kortweg and De-Vries equation. The global conservation law on any time–space regions which yields a three-level linear-implicit algorithm with a diagonal system is exactly preserved by the intended finite difference method. Theoretically verified and numerically proved, the created schemes are unconditionally stable and have the second-order accuracy both in time and space. The obtained results guarantee that the novel idea offers a new aspect to analyze the wave behavior.     

Solitary wave solutions of the improved KdV equation by VIM

The variational iteration method (VIM) is applied to solve numerically the improved Korteweg-de Vries equation (IKdV). A correction function is constructed with a general Lagrange multiplier that can be identified optimally via the variational theory. This technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the IKdV equation. Propagation of single, interaction of two, and three solitary waves, and also birth of solitons have been discussed. Three invariants of motion have been evaluated to determine the conservation properties of the problem. This procedure is promising for solving other nonlinear equations.     

Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Bethuel et al.  and  and Chiron and Rousset [3] gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the NLS equation can approximately be described via solutions of the KdV equation associated with the wave train. Here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of ,  and . Our error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like method, and energy estimates.     

Orbital stability of standing waves for semilinear wave equations with indefinite energy

The orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole Rn with n?2. The stability is determined by the convexity on ω of the lowest energy d(ω) of standing waves with frequency ω. The arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows.     

Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV

By considering the factorizations (flags) and associated (simultaneous) second order Darboux transformations of the square and cube of an arbitrary second order Schrödinger operator, we generate commuting ordinary differential operators of orders four and six with a singular elliptic spectrum. This procedure generates true rank 2 commutative algebras. Under the KdV flow, each such factorization (flag) leads to an integrable equation for which the corresponding Darboux transformation generates a Lax-type operator as one of a commuting pair of orders four and six with singular elliptic spectrum. Hence, these integrable equations are Darboux conjugates of KdV.     

单边算子有界性和KdV型色散方程的适定性研究

数学和物理中许多重要问题均可归结为算子在某些函数空间中的有界性质.奇异积分算子有界性质的研究是调和分析理论的核心课题之一,由此发展起来的各种方法和技巧已广泛应用于偏微分方程的研究.借助奇异积分算子在Lebesgue空间或Morrey型空间中建立的时空估计和半群理论,可以得到非线性色散方程在低阶Sobolev空间中Cauchy问题的适定性.本文首次定义一类单边振荡奇异积分算子并研究该类算子的经典加权有界性质.受经典交换子刻画理论的启发,本文首次引入Morrey空间的交换子刻画理论.利用不同于常规极大函数的方法得到两类象征函数在Morrey空间中的交换子刻画.以上结果为偏微分方程的研究提供了新的工具.最后,结合能量方法和数论知识,本文解决几类KdV型色散方程的适定性问题.     

Stability of small periodic waves for the nonlinear Schrödinger equation

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x−ct for some c∈R. In this paper we investigate the stability of the small amplitude traveling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude traveling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.     

On the stability of the solitary waves to the (generalized) Kawahara equation

In this paper we investigate the orbital stability of solitary waves to the (generalized) Kawahara equation (gKW) which is a fifth order dispersive equation. For some values of the power of the nonlinearity, we prove the orbital stability in the energy space $H^2(\mathbb{R})$ of two branches of even solitary waves of gKW by combining the well-known spectral method introduced by Benjamin [4] with continuity arguments. We construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gKW found by Dey et al. [9]. The second family consists of even traveling waves with low speeds. They are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gKdV with speed 1.     

Solitary waves of the generalized KdV equation with distributed delays

In this paper, we study an integro-differential equation based on the generalized KdV equation with a convolution term which introduces a time delay in the nonlinearity. Special attention is paid to the existence of solitary wave solutions. Motivated by [M.J. Ablowitz, H. Seger, Soliton and Inverse Scattering Transform, SIAM, Philadelphia, 1981; C.K.R.T. Jones, Geometrical singular perturbation theory, in: R. Johnson (Ed.), Dynamical Systems, in: Lecture Notes in Math., vol. 1609, Springer, New York, 1995; T. Ogawa, Travelling wave solutions to perturbed Korteweg-de Vries equations, Hiroshima Math. J. 24 (1994) 401-422], we prove, using the linear chain trick and geometric singular perturbation analysis, that the solitary wave solutions persist when the average delay is suitably small, for a special convolution kernel.     

KdV方程和Kupersmidt方程的递推求解公式

根据Ｐａｉｎｌｅｖｅ分析的已知结果,给出了ＫｄＶ方程和Ｋｕｐｅｒｓｃｈｍｉｄｔ方程的新的递推求解公式。     

应用(1/G)-展开法求解五阶KdV方程

本篇论文首次提出(1/G) -展开法,用于求解非线性演化方程的行波解.将该法应用于五阶KdV方程的求解,当参数满足一定条件时,该方程可化为Sawada-Kotera (SK)方程、Caudrey-Dodd-Gibbon(CDG)方程、Kaup-Kupershmidt (KK)方程、Lax方程和Ito方程.其解可被表示为...     

Orbital stability of solitary waves on a ferrofluid jet

We prove the orbital stability of small-amplitude axisymmetric solitary waves on the surface of an incompressible, inviscid ferrofluid jet. The ferrofluid surrounds a current-carrying rod and is subject to the azimuthal magnetic field generated by the rod. We show that under appropriate assumptions on the magnitude of the magnetic intensity in the ferrofluid, both the trivial flow and the solitary waves with strong surface tension are conditionally orbitally stable, while the conditional orbital stability of solitary waves with near-critical surface tension can be deduced from properties of the corresponding dispersive PDE model equation. The arguments are based on the recent orbital stability results for internal waves by Chen and Walsh (2022) and an improved version of the Grillakis–Shatah–Strauss method introduced by Varholm et al. (2020).