Lump solutions and interaction phenomena of the (3 + 1)-dimensional nonlinear evolution equations

Mao, Jin-Jin; Tian, Shou‐Fu; Yan, Xuehua; Zhang, Tiantian

doi:10.1108/hff-02-2019-0160

Cited by 14 publications

(6 citation statements)

References 52 publications

(59 reference statements)

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“…Relatively recently it has been demonstrated that the 3D KP equation also admits a Hirota representation [29], [30]. However, so far all built 3D KP solutions based on this approach turn out to be only 2D, see also [31], [32], [33], [34]. Therefore, one of the main tasks is the problem of constructing exact three-dimensional solutions that have a physical meaning.…”

Section: Discussionmentioning

confidence: 99%

“…Second, at the same time P Ω , as a positive value, decreases. It follows that the relation in (32) increases, and, consequently, |u| max increases. This process, similar to the evaporation of water droplets, leads to the formation of weak singularities, when the energy E of the acoustic waves captured in the singularity (coinciding with P up to a constant factor) formally vanishes at the moment of collapse.…”

Section: Collapsementioning

confidence: 92%

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Instability of solitons and collapse of acoustic waves in media with positive dispersion

Kuznetsov

2022

Preprint

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This article is a brief review of the results of studying the collapse of sound waves in media with positive dispersion, which is described in terms of the three-dimensional Kadomtsev-Petviashvili (KP) equation [1] . The KP instability of one-dimensional solitons in the long-wavelength limit is considered using the expansion for the corresponding spectral problem. It is shown that the KP instability also takes place for two-dimensional solitons in the framework of the three-dimensional KP equation with positive dispersion. According to B.B. Kadomtsev [2] this instability belongs to the self-focusing type. The nonlinear stage of this instability is a collapse. One of the collapse criteria is the Hamiltonian unboundedness from below for a fixed momentum projection coinciding with the L2-norm. This fact follows from scaling transformations, leaving this norm constant. For this reason, collapse can be represented as the process of falling a particle to the center in a self-consistent unbounded potential. It is shown that the radiation of waves from a region with a negative Hamiltonian, due to its unboundedness from below, promotes the collapse of the waves. This scenario was confirmed by numerical experiments [3,4]. Two analytical approaches to the study of collapse are presented: using the variational method and the quasiclassical approximation. In contrast to the nonlinear Schrödinger equation (NLSE) with a focusing nonlinearity, a feature of the quasiclassical approach to describing acoustic collapse is that this method is proposed for the three-dimensional KP equation as a system with hydrodynamic nonlinearity. Within the framework of the quasiclassical description, a family of self-similar collapses is found. The upper bound of this family corresponds to a strong collapse, in which the energy captures into the singularity is finite. The existence of such a regime is also confirmed based on the variational approach. The other boundary of the collapsing hierarchy coincides with the self-similar solution of the three-dimensional KP equation, which describes the fastest weak collapse.

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Section: Discussionmentioning

confidence: 99%

Section: Collapsementioning

confidence: 92%

Instability of solitons and collapse of acoustic waves in media with positive dispersion

Kuznetsov

2022

Preprint

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“…In this article, we will consider the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation as follows [44][45][46][47] b a b g g…”

Section: Introductionmentioning

confidence: 99%

“…Wang et al have presented the rogue wave solution and solitary wave solution of equation (1) by using the Bell's polynomials, and the interactions between them have been studied [44]. Mao et al have obtained the lump solutions of equation (1) by Hirota's bilinear method, and they have investigated the interaction between a lump and a stripe soliton [45].…”

Section: Introductionmentioning

confidence: 99%

Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids

Zhang¹,

Wang²,

Liu³

et al. 2021

Commun. Theor. Phys.

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In this article, we consider the (3+1)-dimensional generalized Kadomtsev–Petviashvili (GKP) equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the (oscillating-) W- and M-shaped waves, rational W-shaped waves, multi-peak solitary waves, (quasi-) Bell-shaped and W-shaped waves and (quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.

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“…In soliton theory, the exact solutions occupy a vital position in understanding the dynamic behavior and wave propagation characteristics that the nonlinear evolution equations (NLEEs) can describe some nonlinear phenomena in optics, fluids, plasma physics and other fields, so that seeking for the exact solutions is significantly importance (Hirota, 1971; Akhmediev et al , 2013; Qu et al , 2000; Hu et al , 2004; Depassier, 2006; Zheng, 2016). So far, a variety of powerful methods have been established to find exact solutions of NLEEs, such as the inverse scattering transformation (Ablowitz and Clarkson, 1991), Hirota bilinear method (Hirota, 2004; Ma and Fan, 2011), Darboux transformation (Matveev and Salle, 1991; Shi and Zhao, 2018), Lie symmetry method (Kang and Xian, 2016; Kaur and Wazwaz, 2018; Huang and Chen, 2017), the direct method (Ma, 2015; Kudryashov, 2020; Zayed et al , 2019; Mao et al , 2019; Darvishi et al , 2017; Mao, 2018; Wazwaz, 2017; Wazwaz, 2019; Xian et al , 2020) and so on.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic breather waves, rouge waves and multi-soliton waves for a (2+1)-dimensional Mel’nikov equation

Liu

2020

HFF

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Purpose The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets. Design/methodology/approach The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method. Findings The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures. Research limitations/implications This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations. Originality/value The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

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Lump solutions and interaction phenomena of the (3 + 1)-dimensional nonlinear evolution equations

Cited by 14 publications

References 52 publications

Instability of solitons and collapse of acoustic waves in media with positive dispersion

Instability of solitons and collapse of acoustic waves in media with positive dispersion

Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids

Homoclinic breather waves, rouge waves and multi-soliton waves for a (2+1)-dimensional Mel’nikov equation

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