Search

In this work, we establish local well-posedness for the Korteweg-de Vries model on a balanced star graph with a structure represented by semi-infinite edges, by considering a boundary condition of δ -type at the unique graph-vertex. Additionally, we extend the linear instability result in Angulo and Cavalcante (Nonlinearity 34:3373–3410, 2021) to one of nonlinear instability. For the proof of local well posedness theory, the principal new ingredient is the utilization of the special solutions by Faminskii in the context of half-lines. As far as we are aware, this approach is being used for the first time in the context of star graphs and can potentially be applied to other boundary classes. In the case of the nonlinear instability result, the principal ingredients are the linearized instability known result and the fact that data-to-solution map determined by the local theory is at least of class C 2 . [ABSTRACT FROM AUTHOR]

Discussed here is a regularized version of the classical Gardner equation u t + u x + u u x + A u 2 u x - u xxt = 0 , that arises in hydrodynamics and plasma physics. This initial-value problem posed on all of R will be considered with bore-like initial data. That is, the initial wave configuration will consist of a moderately smooth function that asymptotes to zero as the spatial variable x → + ∞ , but converges to r > 0 as x → - ∞ . Such initial profiles can arise in internal wave propagation, for example. In their idealized versions set on all of R , they possess an infinite amount of potential energy. This makes the analysis of the initial-value problem a slightly more subtle than the common situation where the initial profile is assumed to be localized, so being modelled by Sobolev-class initial data. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal.53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal.53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 푛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 . [ABSTRACT FROM AUTHOR]

Investigated in this work is some high-order localized waves and their dynamical properties of a typical (3+1)-dimensional shallow water wave model. Firstly, N-solitons and M-order breathers were studied by applying Hirota's bilinear method, and we also investigated the parameter regulation of different structures of high-order breathers. Secondly, we studied high-order lump solutions and their trajectory equations, with a focus on the interaction behaviour between lump waves and n-solitons (where n is any positive integer), including cross-type and Y-type n-solitons, as well as the trajectory equations before and after collision, and provided computational proofs. Thirdly, we also investigated the interaction behaviour and collision trajectory between the lump solution and the breathers. Meanwhile, some structures and properties of the obtained solution are displayed through a large number of two-dimensional and three-dimensional plots. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the solutions of a three-component modified Kadomtsev–Petviashvili (mKP) equation on Grassmannian Gr(N,M). Its solutions are given by v 0 (x , y , t) , v 1 (x , y , t) and v 2 (x , y , t) . The structures of one soliton, Y-type, O-type and P-type solitons are mainly studied. After drawing analysis, it is found that there are only bright solitons in v 0 , and there are bright solitons and dark solitons in v 1 and v 2 . Then we discuss the reason why the kink solution and the table solution appear in the three-component modified Kadomtsev–Petviashvili equation. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate Lax structure and tau function for the large BKP hierarchy, which is also known as Toda hierarchy of B type. Firstly, the large BKP hierarchy can be derived from fermionic BKP hierarchy by using a special bosonization, which is presented in the form of bilinear equation. Then from bilinear equation, the corresponding Lax equation is given, where in particular the relation of flow generator with Lax operator is obtained. Also starting from Lax equation, the corresponding bilinear equation and existence of tau function are discussed. After that, large BKP hierarchy is viewed as sub-hierarchy of modified Toda (mToda) hierarchy, also called two-component first modified KP hierarchy. Finally by using two basic Miura transformations from mToda to Toda, we understand two typical relations between large BKP tau function τ n (t) and Toda tau function τ n Toda (t , - t) , that is, τ n Toda (t , - t) = τ n (t) τ n - 1 (t) and τ n Toda (t , - t) = τ n 2 (t) . Further, we find (τ n (t) τ n - 1 (t) , τ n 2 (t)) satisfies bilinear equation of mToda hierarchy. [ABSTRACT FROM AUTHOR]

We construct a new class of N-dimensional Lie algebras and apply them to integrable systems. In this paper, we obtain a nonisospectral KdV integrable hierarchy by introducing a nonisospectral problem. Then, a coupled nonisospectral KdV hierarchy is deduced by means of the corresponding higher-dimensional loop algebra. It follows that the K symmetries, τ symmetries, and their Lie algebra of the coupled nonisospectral KdV hierarchy are investigated. The bi-Hamiltonian structures of the both resulting hierarchies are derived by using the trace identity. Finally, we derive a multicomponent nonisospectral KdV hierarchy related to the N-dimensional loop algebra, which generalizes the coupled results to an arbitrary number of components. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem to the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/Modified Camassa–Holm equation and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation. We prove the ill-posedness for the Cauchy problem of the gFQXL/gCH-mCH equation in B p , ∞ s with s > max { 2 + 1 / p , 5 / 2 } and 1 ≤ p ≤ ∞ in the sense that the solution map to this equation starting from u 0 is discontinuous at t = 0 in the metric of B p , ∞ s . [ABSTRACT FROM AUTHOR]

For a fractional potential-KdV equation, Lie point symmetries are studied and their infinitesimal generators are investigated to construct conservation laws. Furthermore, the invariants of the obtained generators are used to reduce the studied equation to a nonlinear fractional ordinary differential equation (NFODE) and with Erdélyi–Kober fractional operator the reduced equation is solved explicitly by using the power series approach. [ABSTRACT FROM AUTHOR]

Considered herein is a modified Rotation-Camassa-Holm equation with backward diffusion term, which is regarded as a model describing the velocity dynamics of shallow water waves. We first investigate the equilibrium point types of the unperturbed R-CH system and acquire the homoclinic solution of the unperturbed system via analysis of phase plane and Hamilton function. Furthermore, the existence of solitary wave solutions for the perturbed Rotation-Camassa-Holm equation with a local and nonlocal delay convolution kernel are investigated, respectively. Our refined results are based on the method of dynamical system, including the geometric singular perturbation theory, invariant manifold and Melnikov method, which is the novel aspect. [ABSTRACT FROM AUTHOR]

The focusing modified Korteweg-de Vries (mKdV) equation with multiple high-order poles under the nonzero boundary conditions is first investigated via developing a Riemann-Hilbert (RH) approach. We begin with the asymptotic property, symmetry and analyticity of the Jost solutions, and successfully construct the RH problem of the focusing mKdV equation. We solve the RH problem when 1/S11(k) has a single high-order pole and multiple high-order poles. Furthermore, we derive the soliton solutions of the focusing mKdV equation which corresponding with a single high-order pole and multiple high-order poles, respectively. Finally, the dynamics of one- and two-soliton solutions are graphically discussed. [ABSTRACT FROM AUTHOR]

We study a system described by a type of initial and boundary value problem of the Hirota equation with nonhomogeneous boundary conditions posed on a bounded interval. Firstly, we prove the local well-posedness of the system in the space H s (0 , 1) by using an explicit solution formula and contraction mapping principle for any s ≥ 1 . Secondly, we obtain the global well-posedness in H 1 (0 , 1) and H 2 (0 , 1) by the norm estimation. Especially, the main difficulty is that the characteristic equation corresponding to Hirota equation needs to be solved by construction and that nonlinear terms are taken into consideration. In addition, the norm estimate for global well-posedness of solution in H 1 (0 , 1) and H 2 (0 , 1) are complicated. [ABSTRACT FROM AUTHOR]

This work is devoted to the long time behavior for the damped BBM equation on three dimensional torus below the energy space. First, it is shown that the BBM equation is globally well-posed in H s (T 3) with 1 2 ≤ s ≤ 1 . The result can not be improved by our method since the bilinear estimate used in the proof fails to hold for smaller s. Second, it is proved that there exists a global attractor in H s (T 3) for every 3 4 < s ≤ 1 . The main difficulty lies in proving the existence of a bounded absorbing set in H s (T 3) , since no conservation law is available below H 1 (T 3) . It is overcome by Bourgain's high-low frequency decomposition approach. Finally, some quantitative regularities of the global attractor in terms of the Fourier projection are derived. [ABSTRACT FROM AUTHOR]

We introduce a new class of numerical schemes which allow for low-regularity approximations to the expectation E (| u k (t , v η) | 2) , where u k denotes the k-th Fourier coefficient of the solution u of the dispersive equation and v η (x) the associated random initial data. This quantity plays an important role in physics, in particular in the study of wave turbulence where one needs to adopt a statistical approach in order to obtain deep insight into the generic long-time behaviour of solutions to dispersive equations. Our new class of schemes is based on Wick's theorem and Feynman diagrams together with a resonance-based discretisation (Bruned and Schratz in Forum Math Pi 10:E2, 2022) set in a more general context: we introduce a novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair. The character of the scheme draws its inspiration from the treatment of singular stochastic partial differential equations via regularity structures. In contrast to classical approaches, we do not discretise the PDE itself, but rather its expectation. This allows us to heavily exploit the optimal resonance structure and underlying gain in regularity on the finite dimensional (discrete) level. [ABSTRACT FROM AUTHOR]

We prove two conjectures on the Korteweg-de Vries (KdV) and modified KdV (mKdV) hierarchies and Schur Q-functions presented by Yamada. The first one is that the functions defined by Sato and Mori in 1980 coincide with Schur Q-functions indexed by even or odd strict partitions. Mizukawa, Nakajima, and Yamada gave an expression for this function using symmetric functions and Littlewood-Richardson coefficients. We prove that this expression coincides with the Schur Q-function by using the formula of Lascoux, Leclerc, and Thibon. The second one is that Schur Q-functions indexed by strict partitions which have odd parts form a basis for the space of Hirota polynomials of the KdV hierarchy, and that Schur Q-functions indexed by strict partitions which have even parts form a basis for the space of Hirota polynomials of the mKdV hierarchy. This conjecture is verified by rewriting the generating series of the KdV and mKdV hierarchies using the techniques of symmetric functions. [ABSTRACT FROM AUTHOR]

We study the real hyperelliptic solutions of the focusing modified KdV (MKdV) equation of genus three. Since the complex hyperelliptic solutions of the focusing MKdV equation over C are associated with the real gauged MKdV equation, we present a novel construction related to the real hyperelliptic solutions of the gauged MKdV equation. When the gauge field is constant, it can be regarded as the real solution of the focusing MKdV equation, and thus we also discuss the behavior of the gauge field numerically. [ABSTRACT FROM AUTHOR]

This paper investigates a noteworthy phenomenon within the framework of Majda-Biello systems, wherein the inclusion of lower-order terms can enhance the well-posedness of the system. Specifically, we investigate the initial value problem (IVP) of the following system: u t + u xxx = - v v x , v t + α v xxx + β v x = - (u v) x , (u , v) | t = 0 = (u 0 , v 0) ∈ H s (R) × H s (R) , x ∈ R , t ∈ R , where α ∈ R \ { 0 } and β ∈ R . Let s ∗ (α , β) be the smallest value for which the IVP is locally analytically well-posed in H s (R) × H s (R) when s > s ∗ (α , β) . Two interesting facts have already been known in literature: s ∗ (α , 0) = 0 for α ∈ (0 , 4) \ { 1 } and s ∗ (4 , 0) = 3 4 . Our key findings include the following: For s ∗ (4 , β) , a significant reduction is observed, reaching 1 2 for β > 0 and 1 4 for β < 0 . Conversely, when α ≠ 4 , we demonstrate that the value of β exerts no influence on s ∗ (α , β) . These results shed light on the intriguing behavior of Majda-Biello systems when lower-order terms are introduced and provide valuable insights into the role of α and β in the well-posedness of the system. [ABSTRACT FROM AUTHOR]

In this paper we study a class of variable coefficient third order partial differential operators on R n + 1 , containing, as a subclass, some variable coefficient operators of KdV-type in any space dimension. For such a class, as well as for the adjoint class, we obtain a Carleman estimate and the local solvability at any point of R n + 1 . A discussion of possible applications in the context of dispersive equations is provided. [ABSTRACT FROM AUTHOR]

Symmetries of the large BKP hierarchy, also known as Toda hierarchy of B type, are investigated in this paper. We firstly construct symmetries of the large BKP hierarchy by the method of additional symmetries. Then we derive Adler–Shiota–van Morebeke formula to link the actions of additional symmetries on Lax operators and tau functions. [ABSTRACT FROM AUTHOR]

The integrability of (2+1)-dimensional non-autonomous perturbed potential Kadomtsev–Petviashvili (NpPKP) problem has been demonstrated in this manuscript using Lax pairs and the Bäcklund transformation. The bilinear form of the NpPKP problem has been generated with the help of Hirota polynomial technique. Using test function and N-dimensional Riemann theta function, the kink-like breathers, lump-kink, interaction of lump and shock and the kinky quasiperiodic are explored analytically in light of the vector selections in the solution space. Outcomes of the solutions are plotted as illustration using particular values of the relevant parameters. These findings may aid in our understanding of intriguing physical events and mechanisms in potential systems. [ABSTRACT FROM AUTHOR]

In this article, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method, we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for the associated linearized operator and to do so, we use tools of the Floquet theory. Finally, we prove the spectral stability by analyzing the difference between the number of negative eigenvalues of a convenient linear operator restricted to the space constituted by zero-mean periodic functions and the number of negative eigenvalues of the matrix formed by the tangent space associated to the low order conserved quantities of the evolution model. [ABSTRACT FROM AUTHOR]

In this paper, we prove the norm inflation and get the ill-posedness for the modified Camassa-Holm equation in B ∞ , 1 0 . Therefore we completed all well-posedness and ill-posedness problem for the modified Camassa-Holm equation in all critical spaces B p , 1 1 p with p ∈ [ 1 , ∞ ] . [ABSTRACT FROM AUTHOR]

In this article we apply quaternionic linear algebra and quaternionic linear system theory to develop the inverse scattering transform theory for the nonlinear Schrödinger equation with nonvanishing boundary conditions. We also determine its soliton solutions by using triplets of quaternionic matrices. [ABSTRACT FROM AUTHOR]

The current study examines the (2 + 1)-dimensional fractional Chaffee–Infante (FCI) model, which is a nonlinear evolution equation that characterizes the processes of pattern generation, reaction-diffusion, and nonlinear wave propagation. The construction of analytical solutions involves the use of analytical methods, namely the Khater III and improved Kudryashov schemes. The He’s Variational Iteration method is employed as a numerical approach to validate the accuracy of the obtained solutions. The main objective of this study is to get novel analytical and numerical solutions for the FCI model, with the intention of gaining a deeper understanding of the system’s dynamics and its possible implications in the fields of fluid mechanics, plasma physics, and optical fiber communications. The study makes a valuable contribution to the area of nonlinear science via the use of innovative analytical and numerical methodologies in the FCI model. This research enhances our comprehension of pattern creation, reaction–diffusion phenomena, and the propagation of nonlinear waves in diverse physical scenarios. [ABSTRACT FROM AUTHOR]

The Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation is often used in dealing with long-wave propagation interactions, and is widely used in mathematics, physics, and engineering. This paper proposes a new extended (3+1)-dimensional KdV-CBS equation, and it’s never been studied. Additionally, we verify the integrability of the equation based on the Painlevé test. By employing Hirota’s method, a bilinear auto-Bäcklund transformation, the multiple-soliton solutions, and the soliton molecules of the equation are derived. New exact solutions of the equation are constructed utilizing the power series expansion method and (G ′ / G) -expansion method. These exact solutions are also presented graphically. Finally, the conservation laws of the equation are obtained. Our results are helpful for understanding nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and possesses certain properties which are instrumental for the convergence analysis. Assuming the initial data u 0 ∈ H 1 + α (R) , where α ∈ [ 1 , 2) , our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent α . Furthermore, we demonstrate that the Crank–Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates. [ABSTRACT FROM AUTHOR]

We consider compatible pairs of local Hamiltonian structures of Dubrovin–Novikov type and the construction of nonlocal Hamiltonian structures of Ferapontov type. We prove that the third and fourth Hamiltonian structures of Ferapontov type cannot contain more terms in the nonlocal part than the number of field variables. We also give local Lagrangian representations for both these nonlocal Hamiltonian structures. Thus, we demonstrate that any hydrodynamic-type system equipped with a compatible pair of local Hamiltonian structures of Dubrovin–Novikov type simultaneously possesses four local Lagrangian representations. [ABSTRACT FROM AUTHOR]

This study undertakes a comprehensive analytical and numerical investigation of the nonlinear Kairat model, a significant evolution equation that governs a wide range of physical phenomena, including shallow water waves, plasma physics, and optical fibers. The Kairat model effectively describes the propagation of nonlinear waves in shallow water, capturing the intricate interplay between nonlinearity and dispersion. It exhibits similarities with well-known nonlinear evolution equations such as the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, thereby offering insights into their common underlying dynamics. To achieve the objectives of this research, we employ the modified Khater (MKhat) and unified (UF) methodologies to derive exact solutions for the Kairat model. Furthermore, the trigonometric-quantic-B-spline (TQBS) scheme is utilized as a numerical technique to verify the accuracy of these derived solutions and validate their applicability within the domain of shallow water wave propagation. This investigation yields a collection of innovative and precise analytical solutions, elucidating the complex nonlinear behavior of the Kairat model and its effectiveness in capturing the dynamics of shallow water waves. Moreover, these analytical solutions are corroborated through numerical simulations conducted using the TQBS scheme, ensuring their reliability and practical significance in understanding and predicting shallow water wave phenomena. The significance of this endeavor lies in its contribution to a deeper understanding of the dynamics of the Kairat model and its potential applications in fields such as coastal engineering, oceanography, and related disciplines. The integration of analytical and numerical techniques offers new perspectives and methodologies for exploring nonlinear evolution equations, potentially benefiting researchers in applied mathematics, physics, and engineering. In summary, this comprehensive analytical and numerical investigation provides novel insights, precise solutions, and a robust foundation for further exploration of the physical implications and applications of the Kairat model in the context of shallow water wave propagation. [ABSTRACT FROM AUTHOR]

This paper is systematically concerned with the solitary waves on the constrained manifold preserved the L 2 -momentum conservation for the generalized singular perturbed KdV equation with L 2 -subcritical, critical and supercritical nonlinearities, which is a long-wave approximation to the capillary-gravity waves in an infinitely long channel with a flat bottom. First, using the profile decomposition in H 2 and the optimal Gagliardo–Nirenberg inequality, we prove the existence of subcritical ground state solitary waves and describe their asymptotic behavior. Second, we obtain some sufficient conditions for the existence and non-existence of critical ground states, and then prove the existence of critical and supercritical ground state solitary waves on the Derrick–Pohozaev manifold by utilizing the new minimax argument and the numerical simulation of the best Gagliardo–Nirenberg embedding constant. Meanwhile, we use the moving plane method to obtain the existence of positive and radially symmetric solutions. Furthermore, we study the concentration behavior of the critical ground state solutions. Finally, the spectral stability of the ground state solitary wave solutions is discussed by using the instability index theorem. [ABSTRACT FROM AUTHOR]

A nonlinear shallow water wave equation containing the famous Degasperis-Procesi and Fornberg-Whitham equations is investigated. The well-posedness of short time solutions is established to illustrate that the solution map of the equation is continuous in the critical Besov space B ∞ , 1 1 (R) . Using the methods to construct high and low frequency functions, we prove that the solution map of the equation is non-uniform continuous dependence on the initial data in B ∞ , 1 1 (R) . [ABSTRACT FROM AUTHOR]

In this study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

In this note, we prove the local well-posedness in the energy space of the k-generalized Zakharov–Kuznetsov equation posed on R × T for any power non-linearity k ≥ 2 . Moreover, we obtain global solutions under a precise smallness assumption on the initial data by proving a sharp Gagliardo Nirenberg type inequality. [ABSTRACT FROM AUTHOR]

This article describes the solution procedure of the fractional Pade-II equation and generalized Zakharov equation(GSEs) using the sine-cosine method. Pade-II is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high, and low-frequency waves. Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed. Graphical simulations of the numerical solutions are flaunted by MATLAB. [ABSTRACT FROM AUTHOR]

This study employs advanced computational methodologies to investigate the nonlinear complex fractional generalized Zakharov equations, herein referred to as NCFGZ . The primary objective is to elucidate precise and groundbreaking solitary wave solutions. These equations find wide–ranging applications in diverse domains, including plasma physics, nonlinear optics, and wave dynamics. The pursuit of highly accurate solitary wave solutions is of paramount significance, as it contributes to both theoretical comprehension and practical real-world applications. By harnessing cutting-edge computational techniques along with conformable fractional derivative (CFD) and leveraging the capabilities of Mathematica 13.1 software, this research significantly advances theoretical knowledge and provides valuable practical insights. The computational approaches employed in this research outperform conventional methods in terms of their efficacy, demonstrating a high degree of precision and efficiency in solving complex fractional differential equations. These findings underscore the pivotal role that precise solitary wave solutions play in forecasting and comprehending the behavior of nonlinear systems governed by the generalized Zakharov, or G Z , equations. Furthermore, this study yields profound implications for the symmetry theorem of solitons, enhancing our understanding of symmetry properties within the context of G Z equations and their broader influence on wave phenomena. [ABSTRACT FROM AUTHOR]

This study presents a new approach for accurately obtaining solitary wave solutions in the Boiti–Leon–Pempinelli (BLP) system, which was developed by Italian mathematicians Luca Boiti, Franco Leon, and Guido Pempinelli in the late 1980 s. The BLP system is very significant in physics and is employed in several fields. It is used to simulate the behavior of nonlinear media in electromagnetic and acoustic wave propagation, as well as to understand the features of Bose–Einstein condensates. This research thoroughly examines the acquired solutions, with a particular focus on crucial factors such as the maximum amplitude, duration, and velocity of the waves. The accurate solitary wave solutions play a crucial role in enhancing the comprehension of these physical phenomena, establishing the foundation for possible practical uses. The suggested modified exponential function approach outperforms previous numerical strategies in terms of efficacy and dependability, demonstrating the significant potential for future study in this field. The results of our investigation are graphically shown using meticulously designed graphs, providing valuable insights into the behavior of various solutions inside the (x, t) domain. This visual depiction facilitates the identification of key characteristics such as velocity, magnitude, and form, hence improving the clarity of our study findings. [ABSTRACT FROM AUTHOR]

We present a new exact solution to the defocusing modified Korteweg–de Vries equation to describe the interaction of a dark soliton and a traveling periodic wave. The solution (which we refer to as the dark breather) is obtained by using the Darboux transformation with the eigenfunctions of the Lax system expressed in terms of the Jacobi theta functions. Properties of elliptic functions including the quarter-period translations in the complex plane are applied to transform the solution to the simplest form. We explore the characteristic properties of these dark breathers and show that they propagate faster than the periodic wave (in the same direction) and attain maximal localization at a specific parameter value which is explicitly computed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonexistence of the periodic peakon and peakon for a highly nonlinear shallow-water model, which has been recently derived from the full governing equations for two dimensional flow with the Coriolis effect or with constant vorticity, under a larger scaling than the Camassa-Holm (CH) one. Note that the so obtained model not only has CH-type terms, but also exhibits cubic order nonlinearities. Thus it is interesting to study how the higher power nonlinear terms affect the existence of the (periodic) peakon. [ABSTRACT FROM AUTHOR]

We show that the uniform radius of spatial analyticity σ (t) of solutions at time t to the fifth order, KdV-BBM type equation cannot decay faster than 1 / t for large t, given initial data that is analytic with fixed radius σ 0 . This improves a recent result by Belayneh, Tegegn and the third author [2], where they obtained a 1/t decay of σ (t) for large time t. [ABSTRACT FROM AUTHOR]

The long-term behavior of low regularity solutions to the damped BBM equation with a distribution force on the torus is studied. Since the energy equation fails to hold for the low regularity solutions, the existence of a bounded absorbing set is not a trivial. This difficulty is overcome by splitting the solution into five parts, where some parts decay exponentially in gradually higher regularity spaces, the final remainder belongs the energy space and thus enjoys the dissipative effect. In this way, the existence of a global attractor is proved in the sharp low regularity space. Moreover, the attractor is shown to have a finite fractal dimension based on the quasi-stable estimate method. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for a system of nonlinear Schrödinger equations with quadratic interactions and initial data belonging to a class of analytic Gevrey functions. Here, we present a local well-posedness result in the analytic Gevrey class G σ , s × G σ , s by proving some bilinear estimates in Bourgain's space with exponential weight. Furthermore, we prove that the obtained solution can be extended to any time T > 0 , as long as the radius of the spatial analyticity σ is bounded below by c T - 2 , if 0 < a < 1 / 2 , or c T - 4 , if a > 1 / 2 . [ABSTRACT FROM AUTHOR]