Recent zbMATH articles in MSC 37K40https://zbmath.org/atom/cc/37K402021-05-28T16:06:00+00:00WerkzeugOptical solitons and other solutions to Kaup-Newell equation with Jacobi elliptic function expansion method.https://zbmath.org/1459.370602021-05-28T16:06:00+00:00"Ahmed, Hamdy M."https://zbmath.org/authors/?q=ai:ahmed.hamdy-m"Rabie, Wafaa B."https://zbmath.org/authors/?q=ai:rabie.wafaa-b"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: In this paper, we studied Kaup-Newell (KN) equation in coupled vector form without four-wave mixing terms in birefringent fibers. We employed Jacobi elliptic function expansion method in order to demonstrate sub-pico-second optical soliton solutions. Beside bright and dark solitons, Jacobi elliptic function solutions and hyperbolic solutions are obtained. Moreover, the graphs for some solution are presented.Nonlinear localized waves resonance and interaction solutions of the \((3 + 1)\)-dimensional Boiti-Leon-Manna-Pempinelli equation.https://zbmath.org/1459.350842021-05-28T16:06:00+00:00"Wu, Juanjuan"https://zbmath.org/authors/?q=ai:wu.juanjuan"Liu, Yaqing"https://zbmath.org/authors/?q=ai:liu.yaqing"Piao, Linhua"https://zbmath.org/authors/?q=ai:piao.linhua"Zhuang, Jianhong"https://zbmath.org/authors/?q=ai:zhuang.jianhong"Wang, Deng-Shan"https://zbmath.org/authors/?q=ai:wang.dengshanSummary: This paper deals with localized waves in the \((3+1)\)-dimensional Boiti-Leon-Manna-Pempinelli equation in the incompressible fluid. Based on Hirota's bilinear method, \(N\)-soliton solutions related to Boiti-Leon-Manna-Pempinelli equation are constructed. Novel nonlinear wave phenomena are obtained by selecting appropriate parameters to \(N\)-soliton solutions, and time evolutions of different kinds of solitary waves are investigated in detail. Rich elastic interactions are illustrated analytically and graphically. More specifically, the inelastic interactions, i.e., fusion and fission of solitary waves, are constructed by choosing special parameters on kink solitons and breathers. The analysis of the influence of parameters on propagation is revealed in three tables. The results have potential applications in fluid mechanics.Existence and stability of Klein-Gordon breathers in the small-amplitude limit.https://zbmath.org/1459.390462021-05-28T16:06:00+00:00"Pelinovsky, Dmitry E."https://zbmath.org/authors/?q=ai:pelinovsky.dmitry-e"Penati, Tiziano"https://zbmath.org/authors/?q=ai:penati.tiziano"Paleari, Simone"https://zbmath.org/authors/?q=ai:paleari.simoneSummary: We consider a discrete Klein-Gordon (dKG) equation on \(\mathbb{Z}^d\) in the limit of the discrete nonlinear Schrödinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength \(\epsilon\). By using the classical Lyapunov-Schmidt method, we show existence and linear stability of the KG breather from existence and linear stability of the corresponding dNLS soliton. Nonlinear stability, for an exponentially long time scale of the order \(\mathcal{O}(\exp(\epsilon^{-1}))\), is obtained via the normal form technique, together with higher order approximations of the KG breather through perturbations of the corresponding dNLS soliton.
For the entire collection see [Zbl 1457.35005].Soliton elastic interactions and dynamical analysis of a reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon.https://zbmath.org/1459.370612021-05-28T16:06:00+00:00"Wang, Hao-Tian"https://zbmath.org/authors/?q=ai:wang.haotian"Wen, Xiao-Yong"https://zbmath.org/authors/?q=ai:wen.xiaoyongSummary: Under investigation in this paper is a discrete reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon, which may have some prospective applications in modern nanoribbon. First, we construct the infinitely many conservation laws and discrete \(N\)-fold Darboux transformation for this system based on its known Lax pair. Then bright-bright multi-soliton and breather solutions in terms of determinants are obtained by means of the resulting Darboux transformation. Moreover, we investigate soliton interactions through asymptotic analysis and analyze some important physical quantities such as amplitudes, wave numbers, wave widths, velocities, energies and initial phases. Finally, the dynamical evolution behaviors are discussed via numerical simulations. It is found that soliton interactions in this system are elastic, and their evolutions are stable against a small noise in a short period of time. Results obtained in this paper may have some prospective applications for understanding some physical phenomena.Lattice Boltzmann model for the interaction of \((2+1)\)-dimensional solitons in generalized Gross-Pitaevskii equation.https://zbmath.org/1459.370622021-05-28T16:06:00+00:00"Wang, Huimin"https://zbmath.org/authors/?q=ai:wang.huimin"Yan, Guangwu"https://zbmath.org/authors/?q=ai:yan.guangwuSummary: In this paper, a new lattice Boltzmann model for the interaction of two solitons in \((2+1)\)-dimensional generalized Gross-Pitaevskii equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales are obtained. By selecting the appropriate higher-order moments of equilibrium distribution functions, the macroscopic equation can be recovered. A numerical example is given to test the scheme. Numerical experiments demonstrate the lattice Boltzmann method is an appropriate tool to simulate the interaction of \((2+1)\)-dimensional solitons.Lax pair, infinitely many conservation laws and solitons for a \((2 + 1)\)-dimensional Heisenberg ferromagnetic spin chain equation with time-dependent coefficients.https://zbmath.org/1459.823382021-05-28T16:06:00+00:00"Lan, Zhongzhou"https://zbmath.org/authors/?q=ai:lan.zhongzhou"Gao, Bo"https://zbmath.org/authors/?q=ai:gao.boSummary: Under investigation in this paper is a \((2 + 1)\)-dimensional Heisenberg ferromagnetic spin chain equation with time-dependent coefficients. Based on the symbolic computation, Lax pair and infinitely many conservation laws are constructed. Multi-soliton solutions are derived by virtue of the Darboux transformation. Propagation and interaction properties of the solitons are discussed. Amplitude of the soliton is determined by the spectral parameter and wave number, while the velocity is related to both these parameters and the time-dependent coefficients. Elastic interactions between the two solitons are displayed, and their amplitudes keep unchanged after the interaction except for the phase shifts.Application of the Riemann-Hilbert method to the vector modified Korteweg-de Vries equation.https://zbmath.org/1459.370562021-05-28T16:06:00+00:00"Wang, Xiu-Bin"https://zbmath.org/authors/?q=ai:wang.xiubin"Han, Bo"https://zbmath.org/authors/?q=ai:han.boSummary: Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable systems. For the direct scattering problem, the spectral analysis is performed for the equation, from which a Riemann-Hilbert problem is well constructed. For the inverse scattering problem, the Riemann-Hilbert problem corresponding to the reflection-less case is solved. Furthermore, as applications, three types of multi-soliton solutions are found. Finally, some figures are presented to discuss the soliton behaviors of the vmKdV equation.Dynamics of solitons in the fourth-order nonlocal nonlinear Schrödinger equation.https://zbmath.org/1459.353432021-05-28T16:06:00+00:00"Gadzhimuradov, T. A."https://zbmath.org/authors/?q=ai:gadzhimuradov.t-a"Agalarov, A. M."https://zbmath.org/authors/?q=ai:agalarov.a-m"Radha, R."https://zbmath.org/authors/?q=ai:radha.ram|radha.ramakrishnan|radha.ramaswamy|radha.repaka"Tamil Arasan, B."https://zbmath.org/authors/?q=ai:tamil-arasan.bSummary: We consider the fourth-order nonlocal nonlinear Schrödinger equation and generate the Lax pair. We then employ Darboux transformation to generate dark and antidark soliton solutions. The highlight of the results is that one ends up generating a two-soliton solution characterized by one spectral parameter alone, a property which has never been witnessed so far.Bilinear approach to soliton and periodic wave solutions of two nonlinear evolution equations of mathematical physics.https://zbmath.org/1459.353262021-05-28T16:06:00+00:00"Cao, Rui"https://zbmath.org/authors/?q=ai:cao.rui"Zhao, Qiulan"https://zbmath.org/authors/?q=ai:zhao.qiulan"Gao, Lin"https://zbmath.org/authors/?q=ai:gao.linSummary: In the present paper, the potential Kadomtsev-Petviashvili equation and (\(3+1\))-dimensional potential-YTSF equation are investigated, which can be used to describe many mathematical and physical backgrounds, e.g., fluid dynamics and communications. Based on Hirota bilinear method, the bilinear equation for the (\(3+1\))-dimensional potential-YTSF equation is obtained by applying an appropriate dependent variable transformation. Then N-soliton solutions of nonlinear evolution equation are derived by the perturbation technique, and the periodic wave solutions for potential Kadomtsev-Petviashvili equation and (\(3+1\))-dimensional potential-YTSF equation are constructed by employing the Riemann theta function. Furthermore, the asymptotic properties of periodic wave solutions show that soliton solutions can be derived from periodic wave solutions.Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation.https://zbmath.org/1459.353392021-05-28T16:06:00+00:00"Abhinav, Kumar"https://zbmath.org/authors/?q=ai:abhinav.kumar"Guha, Partha"https://zbmath.org/authors/?q=ai:guha.partha"Mukherjee, Indranil"https://zbmath.org/authors/?q=ai:mukherjee.indranilSummary: The non-holonomic deformation of the nonlinear Schrödinger equation, uniquely obtained from both the Lax pair and \textit{B. A. Kupershmidt}'s bi-Hamiltonian [Phys. Lett., A 372, No. 15, 2634--2639 (2008; Zbl 1220.35153)] approaches, is compared with the quasi-integrable deformation of the same system [\textit{L. A. Ferreira} et al., J. High Energy Phys. 2012, No. 9, Paper No. 103, 35 p. (2012; Zbl 1397.81095)]. It is found that these two deformations can locally coincide only when the phase of the corresponding solution is discontinuous in space, following a definite phase-modulus coupling of the non-holonomic inhomogeneity function. These two deformations are further found to be not gauge equivalent in general, following the Lax formalism of the nonlinear Schrödinger equation. However, the localized solutions corresponding to both these cases converge asymptotically as expected. Similar conditional correspondence of non-holonomic deformation with a non-integrable deformation, namely due to locally scaled amplitude of the solution to the nonlinear Schrödinger equation, is further obtained.Cross soliton and breather soliton for the \((3+1)\)-dimensional Yu-Toda-Sasa-Fukuyama equation.https://zbmath.org/1459.350822021-05-28T16:06:00+00:00"Pu, Zhiqiang"https://zbmath.org/authors/?q=ai:pu.zhiqiang"Pan, Zhigang"https://zbmath.org/authors/?q=ai:pan.zhigangSummary: Cross-soliton solution, breather soliton, periodic solitary solution, and doubly periodic solution are obtained by using an extended homoclinic test approach with perturbation parameter \(u_{0}\) and complexity of parameters, respectively. Dynamical feature of cross soliton flow including degeneracy of soliton with different directions, retroflexion of breather soliton for YTSF equation is investigated using the parameter perturbation method. Result shows that the value range of constant equilibrium solution can determine the dynamics of cross soliton for a higher dimensional nonlinear system.Solitons and periodic waves for the \((2+1)\)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics.https://zbmath.org/1459.350762021-05-28T16:06:00+00:00"Deng, Gao-Fu"https://zbmath.org/authors/?q=ai:deng.gao-fu"Gao, Yi-Tian"https://zbmath.org/authors/?q=ai:gao.yitian"Su, Jing-Jing"https://zbmath.org/authors/?q=ai:su.jingjing"Ding, Cui-Cui"https://zbmath.org/authors/?q=ai:ding.cui-cui"Jia, Ting-Ting"https://zbmath.org/authors/?q=ai:jia.tingtingSummary: Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. Under investigation in this paper is the \((2+1)\)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. Via the Pfaffian technique and certain constraint on the real constant \(\alpha\), the \(N\) th-order Pfaffian solutions are derived. One- and two-soliton solutions are obtained via the \(N\) th-order Pfaffian solutions. Based on the Hirota-Riemann method, one- and two-periodic wave solutions are constructed. With the help of the analytic and graphic analysis, we notice that: (1) of the one soliton, amplitude is irrelevant to \(\gamma \), a real constant coefficient in the equation, velocity along the \(x\) direction is independent of \(\gamma\), while velocity along the \(y\) direction is proportional to \(\gamma\); (2) one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton; (3) one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart; (4) periodic behaviors for the two-periodic wave exist along the \(x\) and \(y\) directions, respectively; (5) under certain limiting conditions, one-periodic wave solutions approach to the one-soliton solutions and two-periodic wave solutions approach to the two-soliton solutions.