58 research outputs found

    Exploring the Dynamics of Nonlocal Nonlinear Waves: Analytical Insights into the Extended Kadomtsev-Petviashvili Model

    Full text link
    The study of nonlocal nonlinear systems and their dynamics is a rapidly increasing field of research. In this study, we take a closer look at the extended nonlocal Kadomtsev-Petviashvili (enKP) model through a systematic analysis of explicit solutions. Using a superposed bilinearization approach, we obtained a bilinear form of the enKP equation and constructed soliton solutions. Our findings show that the nature of the resulting nonlinear waves, including the amplitude, width, localization, and velocity, can be controlled by arbitrary solution parameters. The solutions exhibited both symmetric and asymmetric characteristics, including localized bell-type bright solitons, superposed kink-bell-type and antikink-bell-type soliton profiles. The solitons arising in this nonlocal model only undergo elastic interactions while maintaining their initial identities and shifting phases. Additionally, we demonstrated the possibility of generating bound-soliton molecules and breathers with appropriately chosen soliton parameters. The results of this study offer valuable insights into the dynamics of localized nonlinear waves in higher-dimensional nonlocal nonlinear models.Comment: 22 pages, 10 figures; submitted to journa

    Rational solutions of the Boussinesq equation and applications to rogue waves

    Get PDF
    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 Schr\"{o}dinger (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 and from rational solutions of the Boussinesq equation. It is shown that the two families of rational solutions of the KPI equation are fundamentally different

    Physical Mechanisms of the Rogue Wave Phenomenon

    No full text
    A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin-Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey - Stewartson system, the Korteweg - de Vries equation, the Kadomtsev - Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon

    Multi-breather solutions to the Sasa-Satsuma equation

    Get PDF
    General breather solution to the Sasa-Satsuma (SS) equation is systematically investigated in this paper. We firstly transform the SS equation into a set of three Hirota bilinear equations under proper plane wave background. Starting from a specially arranged tau-function of the Kadomtsev-Petviashvili hierarchy and a set of eleven bilinear equations satisfied, we implement a series steps of reduction procedure, i.e., C-type reduction, dimension reduction and complex conjugate reduction, and reduce these eleven equations to three bilinear equations for the SS equation. Meanwhile, general breather solution to the SS equation is found in determinant of even order. The one- and two-breather solutions are calculated and analyzed in details

    Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation

    Get PDF
    The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates N N -soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation
    • …
    corecore