Rational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systems

Chen, Junchao; Chen, Yong; Feng, Bao Feng; Maruno, Ken Ichi

doi:10.1016/j.physleta.2015.02.040

Cited by 130 publications

(79 citation statements)

References 70 publications

(99 reference statements)

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“…Thus, the solution is a breather. Besides, the point (Γ, 1I ) = (0, ) is a critical point of the breather solution u defined in (23), where Γ = 1R + . Below, we can determine the type of breather u by the following Hessian matrix:…”

Section: The Breather Solution Of the Boussinesq Equationmentioning

confidence: 99%

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Novel high‐order breathers and rogue waves in the Boussinesq equation via determinants

Liu

Wazwaz

2019

Math Methods in App Sciences

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By virtue of the bilinear method and the Kadomtsev–Petviashvili (KP) hierarchy reduction technique, wider classes of high‐order breather and semirational and rogue wave solutions to the Boussinesq equation are derived. These solutions are presented explicitly in terms of Gram determinants, whose matrix elements have simply algebraic expressions. The breather and rogue wave solutions are derived from two different types of tau functions of a bilinear equation in the single‐component KP hierarchy. By taking a long wave limit of high‐order breather solutions, a range of hybrid solutions consisting of solitons, breathers, and one fundamental rogue wave are generated. For the rational rogue waves, some typical patterns such as Peregrine‐type, triple, and sextuple rogue waves are put forward by modifying the input parameters. Besides, a new rogue wave pattern of third‐order rogue waves is found, which features a mixture of a triangular pattern of three fundamental rogue waves and a fundamental pattern of second‐order rogue wave. These results may help understand the protean rogue wave manifestations in areas ranging from water waves to fluid dynamics.

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Section: The Breather Solution Of the Boussinesq Equationmentioning

confidence: 99%

“…In the previous studies, high‐order rogue waves in the NLS equation have been discussed, which are also localized in both space and time and could exhibit higher main peaks in fundamental pattern. In addition to the NLS equation, families of nonlinear soliton equations have been verified possessing rogue wave solutions …”

Section: Introductionmentioning

confidence: 99%

Novel high‐order breathers and rogue waves in the Boussinesq equation via determinants

Liu

Wazwaz

2019

Math Methods in App Sciences

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“…The bilinear method is also applicable in the search for higher order RWs , and the case of second‐order RW is utilized as an illustrative example. We start with an expansion scheme typically used for dark soliton:

\begin{matrix} true \\ right f & = & left \sum_{μ = 0, 1} exp (\sum_{j = 1}^{4} μ_{j} φ_{j} + \sum_{i < j} μ_{i} μ_{j} m_{i j}), \\ right g & = & left \sum_{μ = 0, 1} exp [\sum_{j = 1}^{4} μ_{j} ((), φ_{j} + θ_{j}) + \sum_{i < j} μ_{i} μ_{j} m_{i j}], \\ right h & = & left \sum_{μ = 0, 1} exp [\sum_{j = 1}^{4} μ_{j} ((), φ_{j} + ψ_{j}) + \sum_{i < j} μ_{i} μ_{j} m_{i j}], \end{matrix}

where the coefficients are expressed in terms of the wave numbers ( p j ), frequencies

(w_{j})

, and phase factors (

ζ^{(j)}

) as follows:

φ_{j} = p_{j} x - w_{j} t + ζ^{((), j)},

\begin{matrix} true \\ right prefixexp ((), m_{i j}) & = & left \begin{matrix} left (w_{j} p_{i} - w_{i<...>}) \end{matrix} \end{matrix}

…”

Section: Second‐order Rogue Wavesmentioning

confidence: 99%

“…The bilinear method is also applicable in the search for higher order RWs [35,36], and the case of second-order RW is utilized as an illustrative example. We start with an expansion scheme typically used for dark soliton:…”

Section: Derivation Of Second-order Rogue Wavesmentioning

confidence: 99%

Rogue Waves for an Alternative System of Coupled Hirota Equations: Structural Robustness and Modulation Instabilities

Chan

Chow

2017

Stud Appl Math

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The Hirota system consists of two coupled, third‐order partial differential equations with cubic nonlinearities. Rogue wave (RW) modes for such coupled Hirota equations different from those usually considered in the literature are derived by the bilinear method. RWs exist in the defocusing regime (dispersion and nonlinearities of different signs), with an existence criterion closely related to the condition for baseband modulation instability (MI). This connection is examined further through numerical simulations. RW‐like structures can emerge from a plane wave background perturbed by random noise only if baseband MI is present. Furthermore, the development of an individual RW will not be “masked” by the growth of the noise if the MI itself is not too strong. Even more illuminating information is revealed with disturbance of a specified wavelength (or a narrow range of frequency input). A disturbance with wave numbers in the passband will not generate any coherent structures, but those in the baseband may generate an Akhmediev breather or a RW‐like structure.

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“…Most recently, this method is used to obtain the N-dark soliton [15] and bright-dark mixed N-soliton [16] solutions of the multi-component Yajima-Oikawa (YO) system. In some other recent works, the KP hierarchy reduction technique has also been applied to derive rogue wave solutions of integrable systems [34][35][36], see also the literatures [37][38][39].…”

Section: Introductionmentioning

confidence: 99%

Bright-Dark MixedN-Soliton Solution of Two-Dimensional Multicomponent Maccari System

Han

Chen

2017

Zeitschrift Für Naturforschung A

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By virtue of the KP hierarchy reduction technique, we construct the general bright-dark mixed N-soliton solution to the multi-component Mel'nikov system comprised of multiple (say M) short-wave components and one longwave component with all possible combinations of nonlinearities including all-positive, all-negative and mixed types. Firstly, the two-bright-one-dark (2-b-1-d) and one-bright-two-dark (1-b-2-d) mixed N-soliton solutions in short-wave components of the three-component Mel'nikov system are derived in detail. Then we extend our analysis to the M-component Mel'nikov system to obtain its general mixed N-soliton solution. The formula obtained unifies the allbright, all-dark and bright-dark mixed N-soliton solutions. For the collision of two solitons, the asymptotic analysis shows that for a M-component Mel'nikov system with M ≥ 3, inelastic collision takes place, resulting in energy exchange among the short-wave components supporting bright solitons only if the bright solitons appear at least in two short-wave components. Whereas, the dark solitons in the short-wave components and the bright solitons in the long-wave component always undergo elastic collision which just accompanied by a position shift.

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Rational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systems

Cited by 130 publications

References 70 publications

Novel high‐order breathers and rogue waves in the Boussinesq equation via determinants

Novel high‐order breathers and rogue waves in the Boussinesq equation via determinants

Rogue Waves for an Alternative System of Coupled Hirota Equations: Structural Robustness and Modulation Instabilities

Bright-Dark MixedN-Soliton Solution of Two-Dimensional Multicomponent Maccari System

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