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This paper deals with the stability for a weakly coupled wave equations with a boundary dissipation of fractional derivative type. We have proved well posedness and polynomial stability using the semigroup theory and a sharp result provided by Borichev and Tomilov. [ABSTRACT FROM AUTHOR]

In this paper, we study the coupled nonlinear Schrödinger–Korteweg–de Vries system with periodic potential. By using the variational method and Nehari manifold, we obtain the existence of non-trivial ground state solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Kawahara equation in a bounded interval and with a delay term in one of the boundary conditions. Using two different approaches, we prove that this system is exponentially stable under a condition on the length of the spatial domain. Specifically, the first result is obtained by introducing a suitable energy functional and using Lyapunov's approach, to ensure that the energy of the Kawahara system goes to 0 exponentially as t → ∞ . The second result is achieved by employing a compactness–uniqueness argument, which reduces our study to prove an observability inequality. Furthermore, the novelty of this work is to characterize the critical lengths phenomenon for this equation by showing that the stability results hold whenever the spatial length is related to the Möbius transformations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Korteweg-de Vries equation posed on the periodic domain T . We show that the Korteweg-de Vries equation is globally approximately controllable by a two dimensional external force. The proof is based on the Agrachev-Sarychev approach in geometric control theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the generalized Camassa–Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761–781, 2005). We prove that the solution map of the generalized Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result includes the present work Li et al. (Differ Equ 269:8686–8700, 2020) on Camassa–Holm equation with Q = 1 and extends the previous non-uniform continuity in Sobolev spaces Mi and Mu (Monatsh Math 176:423–457, 2015) to Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the local-in-time existence and uniqueness of solution to the Degasperis-Procesi equation in B ∞ , 1 1 (R) . Moreover, we prove that the data-to-solution of this equation is continuous but not uniformly continuous in B ∞ , 1 1 (R) . [ABSTRACT FROM AUTHOR]

Well-posedness issues for the initial value problem (IVP) associated to the three-wave mixing Schrödinger system with quadratic nonlinearities posed on d-dimensional Zoll manifold M with initial data in H s (M) are investigated. New bilinear Strichartz’s type estimates for the interacting evolution groups are obtained considering that the system parameter η be a rational number. As a consequence, bilinear estimates for the interacting nonlinearities are obtained so as to prove the local well-posedness results with initial data in H s (M) whenever s > 1 4 if d = 2 and s > d - 2 2 if d ≥ 3 . Using some conserved quantities combined with the sharp Gagliardo-Nirenberg inequality the global well-posedness results are also obtained in dimensions d = 2 , 3 for initial data with Sobolev regularity s ≥ 1 . The present work improves the earlier results for the 2 × 2 Schrödinger system in Nogueira and Panthee (J Math Anal Appl 494:124574, 2021) relaxing the restrictions on the system parameter and consequently the local and the global well-posedness results there. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the long-time behavior stability of solitary waves for the weakly damped Korteweg–de Vries equation. More concretely, solutions of the dissipative system with the initial values near a c 0 -speed solitary wave, are approximated by a long curve on the family of solitary waves with the time-varying speed | c (t) − c 0 | being small, in the long-time period (i.e., 0 ⩽ t ⩽ O (1 ϵ τ) ). Meanwhile, the approximation difference in a suitably weighted space H a 1 (R) is of the order of the damping coefficient and of some kind of exponential weight form. As a comparison, we also study the long-time behavior stability, i.e., for 0 ⩽ t < + ∞ , the solutions are approximated by a long curve on the family of solitary waves with the exponential decay speed c (t) = c 0 e − β t (0 < β ⩽ 1 ), when the initial values are near a c 0 -speed solitary wave. However, here, the approximation difference merely defined in H 1 (R) depends on the damping coefficient ϵ and the exponential decay coefficient β. [ABSTRACT FROM AUTHOR]

The reductive perturbation technique is used to obtain the Modified KP equation at critical densities distinguished by the MKPE in plasma ion pair with fast electron positron. The new structures reveal that super-solitary and period waveforms are derived via the mathematical analysis of Jacobi-elliptic function expansions (MJEFE) for MKPE. The electrostatic new structures such as super-soliton, cnoidals, shock-like, super-dispersive and superperiodic plasma waves which existed at critical densities have been introduced. The positrons (electron) nonthermality supports on nonlinear dispersive new structures have been discussed. Also, many of the obtained electrostatic solutions are important and may be applicable in ionosphere plasma observations. [ABSTRACT FROM AUTHOR]

In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme is three-level and linear implicit and gives second- and fourth-order accuracy in time and space, respectively. It is rigorously proved by using the discrete energy method that the proposed difference scheme is uniquely solvable, stable and convergent. Numerical examples are given to confirm the stability and convergence of the numerical solution with fourth-order accuracy and the effectiveness of the present scheme for handling the single and multi-solitary waves for a long time. [ABSTRACT FROM AUTHOR]

We study additional non-isospectral symmetries of multicomponent constrained N = 2 supersymmetric Kadomtsev—Petviashvili (KP) hierarchies. These symmetries are shown to form an infinite-dimensional non-Abelian superloop superalgebra. [ABSTRACT FROM AUTHOR]

In this work, the ( 2 + 1 )-dimensional breaking soliton equation is investigated, which can be used to describe certain properties of exact solutions. We construct the lump soliton and more general lump soliton with more arbitrary parameters via using Hirota bilinear method. Furthermore, the lump off solution is presented by considering a stripe soliton solution generated completely with the lump solution. The lump part is cut by soliton part before and after a special time. Finally, according to a pair of stripe solitons, we construct the special rogue waves by cutting the lump soliton. Our results show that the occurrence of rogue wave time and location can be captured by tracking the motion path of the lump solution and confirming when and where it collides with visible solitons. [ABSTRACT FROM AUTHOR]

This paper is devoted to the spatial analyticity of the solution of the BBM equation on the real line with an analytic initial data. It is shown that the analytic radius has a lower bound like t - 2 3 as the time t goes to infinity, which is an improvement of previous results. The main new ingredient is a higher order almost conservation law in analytic spaces. This is proved by introducing an equivalent analytic norm with smooth symbol and establishing some algebra identities of higher order polynomials. [ABSTRACT FROM AUTHOR]

The article deals with a class of Boussinesq system coupling two higher-order Benjamin–Bona–Mahony (BBM)-type equations. Introducing appropriate damping mechanisms, we study the asymptotic behavior in time of the corresponding damped models. This is done both in the case of internal and boundary damping. We first prove the global well-posedness of the systems and the convergence towards a solution which is null on a band. Then, applying a unique continuation property it follows that the origin is asymptotically stable for the damped models. Our proofs rely on the approach developed in Rosier (J Math Study 49:195–204, 2016) to study similar problems for the scalar BBM equation. [ABSTRACT FROM AUTHOR]

We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical 'Pitman's transformation' of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural 'carrier process' for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation. [ABSTRACT FROM AUTHOR]

This research explores a (2 + 1)-D generalized Camassa–Holm–Kadomtsev–Petviashvili model. We use a probable transformation to build bilinear formulation to the model by Hirota bilinear technique. We derive a single lump waves, multi-soliton solutions to the model from this bilinear form. We present various dynamical properties of the model such as one, two, three and four solitons. The double periodic breather waves, periodic line rogue wave, interaction between bell soliton and double periodic rogue waves, rogue and bell soliton, rogue and two bell solitons, two rogues, rogue and periodic wave, double periodic waves, two pair of rogue waves as well as interaction of double periodic rogue waves in a line are established. Among the results, most of the properties are unexplored in the prior research. Furthermore, with the assistance of Maple software, we put out the trajectory of the obtained solutions for visualizing the achieved dynamical properties. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of ground-state solution for the coupled nonlinear Schrödinger–Korteweg–de Vries system - Δ u + λ 1 u = u 3 + β u v in R N , - Δ v + λ 2 v = 1 2 v 2 + 1 2 β u 2 in R N , under the constrains ∫ R N | u | 2 d x = a , ∫ R N | v | 2 d x = b , where N = 1 , 2 , β > 0 and a , b > 0 are prescribed. In the system, the parameters λ 1 and λ 2 are unknown and will correspond to the Lagrange multipliers. Based on the variational method and rearrangement techniques, we prove that there exists a positive ground state solution for the system by the precompactness of the minimizing sequences, up to translation. [ABSTRACT FROM AUTHOR]

In this paper, we explore the effect of sine-type higher-order nonlinearity on the dispersive dynamics by considering the Cauchy problem for a sine-type Camassa-Holm (alias sine-CH) equation, which is a higher-order generalization of the remarkable CH equation, and also admits the peakon solution. Some main results are presented containing the local well-posedness for strong solutions in subcritical or critical Besov spaces, Hölder continuity of the data-to-solution map, the blow-up criterion and the precise blow-up quantity in Sobolev space, and a sufficient condition with regard to the initial data ensuring the occurance of the wave-breaking phenomenon. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to find the analytical solutions of the Korteweg–de Vries (KdV) equation with dual power law nonlinearity using the exp-function method. The KdV equation is used to study the shallow water wave behavior of long wavelengths and small amplitude. With the help of the exp-function method, a variety of exact wave solutions of the KdV equation are obtained including kink, anti-kink, and bell-shaped traveling wave solutions. The various kinds of solutions to the considered nonlinear KdV equation may help to explore the relevant nonlinear phenomena in fluid dynamics. The presented results are novel and the KdV equation with dual power law nonlinearity has been evaluated using the presented technique for the first time in this work. In addition, some of the exact solutions have been shown by contour planes, 3D plots, and 2D graphics for visualizing the underlying dynamics of the results. [ABSTRACT FROM AUTHOR]

This paper aims to establish the existence of travelling waves for a generalized KdV–Burgers–Kuramoto equation via utilising geometric singular perturbation theory. Firstly, we explore the existence results of orbits for the equation without delay and perturbation by employing Argument Principle. Secondly, the existence of travelling waves for the equation with two types of special delay convolution kernels are proved with the aid of combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality theorem. Finally, asymptotic behaviors of traveling waves are given with the method of the asymptotic theory. [ABSTRACT FROM AUTHOR]

A class of non-local integrable equations is proposed as a new generalization of the constrained 1st modified KP hierarchy (i.e. Kupershmidt–Kiso version), which is called the generalized k-modified intermediate long wave ( gILW k for short) hierarchy. Then the bilinear formulations for the gMILW k hierarchy are constucted. Based on the bilinear formulations, rational and soliton solutions are obtained by using the boson-fermion correspondence. [ABSTRACT FROM AUTHOR]

On the basis of the spectral analysis of the 4×4 matrix Lax pair, the initial value problem of the coherently-coupled nonlinear Schrödinger system is transformed into a 4×4 matrix Riemann-Hilbert problem. By using the nonlinear steepest decent method, the long-time asymptotics of the solution of the initial value problem for the coherently-coupled nonlinear Schrödinger system is obtained through deforming the Riemann-Hilbert problem into a solvable model one. [ABSTRACT FROM AUTHOR]

A Fourier/stability analysis of the third-order Korteweg–de Vries equation is presented subject to a class of local discontinuous Galerkin discretization using high-degree Lagrange polynomials. The selection of stability parameters involved in the method is made on the basis of the study of the higher frequency eigenmodes and the Fourier analysis. Explicit analytical dispersion relation and group velocity are obtained and the stability study of the discrete frequency is performed. The emergence of gaps in the imaginary part of the computed frequency is observed and studied for the first time to our knowledge. Further, a superconvergent result is demonstrated for the discrete frequency by obtaining an explicit analytical asymptotic formula for the latter. [ABSTRACT FROM AUTHOR]

The (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation is a significant physical model. By using the long wave limit method and confining the conjugation conditions on the interrelated solitons, the general M-lump, high-order breather, and localized interaction hybrid solutions are investigated, respectively. Then we implement the numerical simulations to research their dynamical behaviors, which indicate that different parameters have very different dynamic properties and propagation modes of the waves. The method involved can be validly employed to get high-order waves and study their propagation phenomena of many nonlinear equations. [ABSTRACT FROM AUTHOR]

We revisit the symmetry structure of integrable PDEs, looking at the specific example of the KdV equation. We identify four nonlocal symmetries of KdV depending on a parameter, which we call generating symmetries. We explain that since these are nonlocal symmetries, their commutator algebra is not uniquely determined, and we present three possibilities for the algebra. In the first version, three of the four symmetries commute; this shows that it is possible to add further (nonlocal) commuting flows to the standard KdV hierarchy. The second version of the commutator algebra is consistent with Laurent expansions of the symmetries, giving rise to an infinite-dimensional algebra of hidden symmetries of KdV. The third version is consistent with asymptotic expansions for large values of the parameter, giving rise to the standard commuting symmetries of KdV, the infinite hierarchy of “additional symmetries,” and their traditionally accepted commutator algebra (though this also suffers from some ambiguity as the additional symmetries are nonlocal). We explain how the three symmetries that commute in the first version of the algebra can all be regarded as infinitesimal double Bäcklund transformations. The four generating symmetries incorporate all known symmetries of the KdV equation, but also exhibit some remarkable novel structure, arising from their nonlocality. We believe this structure to be shared by other integrable PDEs. [ABSTRACT FROM AUTHOR]

We take a closer look at the Riemann–Hilbert problem associated to one-gap solutions of the Korteweg–de Vries equation. To gain more insight, we reformulate it as a scalar Riemann–Hilbert problem on the torus. This enables us to derive deductively the model vector-valued and singular matrix-valued solutions in terms of Jacobi theta functions. We compare our results with those obtained in recent literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present an extension of the results in [8]. We establish a more general proof for the moving kernel formula to prove the spectral stability of periodic traveling wave solutions for the regularized Benjamin–Bona–Mahony type equations. As applications of our analysis, we show the spectral instability for the quintic Benjamin–Bona–Mahony equation and the spectral (orbital) stability for the regularized Benjamin–Ono equation. [ABSTRACT FROM AUTHOR]

Starting from the 4 × 4 Lax pair associated with a new type coupled nonlinear Schrödinger system and its spectral analysis, a Riemann–Hilbert problem is constructed, and the solution of Cauchy problem of the new type coupled nonlinear Schrödinger system is transformed into the solution of the Riemann–Hilbert problem. Based on the basic Riemann–Hilbert problem, the Deift–Zhou deformations to the contour of the Riemann–Hilbert problem are considered. With the aid of the Deift–Zhou nonlinear steepest descent method, the basic Riemann–Hilbert problem is transformed into a model Riemann–Hilbert problem and we derive the leading-order asymptotics of its solution according to the asymptotic expansions of the parabolic cylinder function. Finally, we obtain the long-time asymptotic behavior of solutions to the Cauchy problem of the new type coupled nonlinear Schrödinger system. [ABSTRACT FROM AUTHOR]

In this work, we deal with the exponential stability of the nonlinear Korteweg–de Vries equation on a finite star-shaped network in the presence of delayed internal feedback. We start by proving the well-posedness of the system and some regularity results. Then, we state an exponential stabilization result using a Lyapunov function by imposing small initial data and a restriction over the lengths. In this part also, we are able to obtain an explicit expression for the decay rate. Then, we prove a semi-global exponential stability result, which is based on an observability inequality working directly on the nonlinear system. Next, we study the case where it may happen that a control domain with delay is outside the control domain without delay. In that case, we obtain also a local exponential stabilization result. Finally, we present some numerical simulations to illustrate the stabilization. [ABSTRACT FROM AUTHOR]

A generalized Liouville's formula is established for linear matrix differential equations involving left and right multiplications. Its special cases are used to determine the localness of characteristics of symmetries and solutions to Riemann-Hilbert problems in soltion theory. [ABSTRACT FROM AUTHOR]

In the present paper we prove the validity of the Camassa-Holm equation as a long wave limit to the double dispersion equation which describes the propagation of bidirectional weakly nonlinear and dispersive waves in an infinite elastic medium. First we show formally that the right-going wave solutions of the double dispersion equation can be approximated by the solutions of the Camassa-Holm equation in the long wave limit. Then we rigorously prove that the solutions of the double dispersion and the Camassa-Holm equations remain close over a long time interval, determined by two small parameters measuring the effects of nonlinearity and dispersion. [ABSTRACT FROM AUTHOR]

This paper is concerned with the third-order nonlinear Schrödinger equation, a version of the center manifold theorem is established which is suitable for the third-order nonlinear Schrödinger equation with critical lengths. [ABSTRACT FROM AUTHOR]

In this paper we construct Darboux transformations and Bäcklund transformations for the supersymmetric two-boson hierarchy. Three Darboux transformations and associated Bäcklund transformations are obtained. For one of them, we also work out the corresponding nonlinear superposition formula. We build, for both the supersymmetric two-boson equation and the N = 2 , a = 4 supersymmetric KdV equation, their one-soliton solutions and present their semi-discrete versions. [ABSTRACT FROM AUTHOR]

We relate the scattering theory of the focusing AKNS system with equally sized nonvanishing boundary conditions to that of the matrix Schrödinger equation. This (shifted) Miura transformation converts the focusing matrix nonlinear Schrödinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method of solving the Marchenko integral equations by separation of variables to derive the multisoliton solutions of this nonlocal equation, thus proposing a method to solve the reflectionless matrix NLS equation. [ABSTRACT FROM AUTHOR]