Lump and rogue waves for the variable-coefficient Kadomtsev–Petviashvili equation in a fluid

2018 ◽  
Vol 32 (10) ◽  
pp. 1850086 ◽  
Author(s):  
Xiao-Yue Jia ◽  
Bo Tian ◽  
Zhong Du ◽  
Yan Sun ◽  
Lei Liu
Keyword(s):  
Small Amplitude ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Single Layer ◽  
Rogue Waves ◽  
Variable Coefficients ◽  
The Other ◽  
Transverse Coordinate ◽  
Petviashvili Equation

Under investigation in this paper is the variable-coefficient Kadomtsev–Petviashvili equation, which describes the long waves with small amplitude and slow dependence on the transverse coordinate in a single-layer shallow fluid. Employing the bilinear form and symbolic computation, we obtain the lump, mixed lump-stripe soliton and mixed rogue wave-stripe soliton solutions. Discussions indicate that the variable coefficients are related to both the lump soliton’s velocity and amplitude. Mixed lump-stripe soliton solutions display two different properties, fusion and fission. Mixed rogue wave-stripe soliton solutions show that a rogue wave arises from one of the stripe solitons and disappears into the other. When the time approaches 0, rogue wave’s energy reaches the maximum. Interactions between a lump soliton and one-stripe soliton, and between a rogue wave and a pair of stripe solitons, are shown graphically.

Modulation instability and two types of non-autonomous rogue waves for the variable-coefficient AB system in fluid mechanics and nonlinear optics

2016 ◽  
Vol 30 (01) ◽  
pp. 1550264 ◽  
Author(s):  
Lei Wang ◽  
Feng-Hua Qi ◽  
Bing Tang ◽  
Yu-Ying Shi
Keyword(s):  
Nonlinear Optics ◽  
Variable Coefficient ◽  
Modulation Instability ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Variable Coefficients ◽  
Short Pulses ◽  
Plane Wave Solution ◽  
Double Peak Structure ◽  
Peak Structure

Under investigation in this paper is a variable-coefficient AB (vcAB) system, which describes marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. The modulation instability analysis of solutions with variable coefficients in the presence of a small perturbation is studied. The modified Darboux transformation (mDT) of the vcAB system is constructed via a gauge transformation. The first-order non-autonomous rogue wave solutions of the vcAB system are presented based on the mDT. It is found that the wave amplitude of [Formula: see text] exhibits two types of structures, i.e. the double-peak structure appears if the plane-wave solution parameter [Formula: see text] is equal to zero, while selecting [Formula: see text] yields a single-peak one. Effects of the variable coefficients on the rogue waves are graphically discussed in detail. The periodic rogue wave and composite rogue wave are obtained with different inhomogeneous parameters. Additionally, the nonlinear tunneling of the rogue waves through a conventional hyperbolic nonlinear well and barrier are investigated.

scholarly journals Gramian solutions and soliton interactions for a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in a plasma or fluid

2019 ◽  
Vol 475 (2228) ◽  
pp. 20190122 ◽  
Author(s):  
Su-Su Chen ◽  
Bo Tian
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Nonlinear Phenomena ◽  
Visible Matter ◽  
Petviashvili Equation ◽  
Spatial Coordinates ◽  
Dimensional Variable ◽  
Soliton Fission ◽  
Wave Phenomena

Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation for the nonlinear phenomena in a plasma or fluid. Based on the existing bilinear form, N -soliton solutions in the Gramian are derived, where N  = 1, 2, 3…. With N  = 3, three-soliton solutions are constructed. Fission and fusion for the three solitons are presented. Effects of the variable coefficients, i.e. h ( t ), l ( t ), q ( t ), n ( t ) and m ( t ), on the soliton fission and fusion are revealed: soliton velocity is related to h ( t ), l ( t ), q ( t ), n ( t ) and m ( t ), while the soliton amplitude cannot be affected by them, where t is the scaled temporal coordinate, h ( t ), l ( t ) and q ( t ) give the perturbed effects, and m ( t ) and n ( t ), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We show the three parallel solitons with the same direction.

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

2012 ◽  
Vol 67 (10-11) ◽  
pp. 525-533
Author(s):  
Zhi-Qiang Lin ◽  
Bo Tian ◽  
Ming Wang ◽  
Xing Lu
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Einstein Condensate ◽  
Matter Wave ◽  
Bilinear Method ◽  
Matter Waves ◽  
Bose Einstein Condensate ◽  
Bose Einstein ◽  
Hirota Bilinear

Under investigation in this paper is a variable-coefficient coupled Gross-Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlev´e analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose-Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes.We also present the collisions among three solitons of atomic matter waves.

Localized waves and interaction solutions to an extended (3+1)-dimensional Kadomtsev–Petviashvili equation

2020 ◽  
Vol 34 (06) ◽  
pp. 2050076 ◽  
Author(s):  
Han-Dong Guo ◽  
Tie-Cheng Xia ◽  
Wen-Xiu Ma
Keyword(s):  
Three Dimensional ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Dark Soliton ◽  
Complex Conjugate ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Parameter Constraints ◽  
Localized Waves

In this paper, an extended (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied via the Hirota bilinear derivative method. Soliton, breather, lump and rogue waves, which are four types of localized waves, are obtained. N-soliton solution is derived by employing bilinear method. Then, line or general breathers, two-order line or general breathers, interaction solutions between soliton and line or general breathers are constructed by complex conjugate approach. These breathers own different dynamic behaviors in different planes. Taking the long wave limit method on the multi-soliton solutions under special parameter constraints, lumps, two- and three-lump and interaction solutions between dark soliton and dark lump are constructed, respectively. Finally, dark rogue waves, dark two-order rogue waves and related interaction solutions between dark soliton and dark rogue waves or dark lump are also demonstrated. Moreover, dynamical characteristics of these localized waves and interaction solutions are further vividly demonstrated through lots of three-dimensional graphs.

Are There Different Kinds of Rogue Waves?

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Paul C. Liu ◽  
Keith R. MacHutchon
Keyword(s):  
Rogue Wave ◽  
Rogue Waves ◽  
Rayleigh Distribution ◽  
The Other ◽  
South Indian Ocean ◽  
Gas Drilling ◽  
Wave Measurements ◽  
South Indian ◽  
Theoretical Considerations

There is clearly no immediate answer to the question posted by the title of this paper. Inasmuch as that there are not much definitively known about rogue waves and that there is still no universally accepted definition for rogue waves in the ocean, we think there might just be even more than one kind of rogue waves to contend with. While the conventional approach has generally designated waves with Hmax∕Hs greater than 2.2 as possible rogue waves, based on Rayleigh distribution considerations, there is conspicuously no provision as to how high the ratio of Hmax∕Hs can be and thus not known how high can a rogue wave be. In our analysis of wave measurements made from a gas-drilling platform in South Indian Ocean, offshore from Mossel Bay, South Africa, we found a number of cases that indicated Hmax∕Hs could be valued in the range between 4 and 10. If this were to be the case, then these records could be considered to be “uncommon” rogue waves, whereas a record of Hmax∕Hs in the range between 2 and 4 could be considered to comprise “typical” rogue waves. On the other hand, the spikes in the Hmax data could have been caused by equipment malfunction or some other phenomenon. Clearly, the question of whether or not there are different kinds of rogue waves cannot be readily answered by theoretical considerations alone and there is a crucial need for long-term wave time-series measurements for studying rogue waves.

Breather wave, rogue wave and lump wave solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

2018 ◽  
Vol 32 (20) ◽  
pp. 1850223 ◽  
Author(s):  
Ming-Zhen Li ◽  
Bo Tian ◽  
Yan Sun ◽  
Xiao-Yu Wu ◽  
Chen-Rong Zhang
Keyword(s):  
Wave Velocity ◽  
Water Waves ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Dispersion Effect ◽  
Wave Solutions ◽  
Weak Perturbation ◽  
Petviashvili Equation ◽  
Nonlinearity Effect ◽  
Lump Wave

Under investigation in this paper is a (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Via the Hirota method and symbolic computation, the lump wave, breather wave and rogue wave solutions are obtained. We graphically present the lump waves under the influence of the dispersion effect, nonlinearity effect, disturbed wave velocity effects and perturbed effects: Decreasing value of the dispersion effect can lead to the range of the lump wave decreases, but has no effect on the amplitude. When the value of the nonlinearity effect or disturbed wave velocity effects increases respectively, lump wave’s amplitude decreases but lump wave’s location keeps unchanged. Amplitudes of the lump waves are independent of the perturbed effects. Breather waves and rogue waves are displayed: Rogue waves emerge when the periods of the breather waves go to the infinity. When the value of the dispersion effect decreases, range of the rogue wave increases. When the value of the nonlinearity effect or disturbed wave velocity effects decreases respectively, rogue wave’s amplitude decreases. Value changes of the perturbed effects cannot influence the rogue wave.

A variable coefficient nonlinear Schrödinger equation: localized waves on the plane wave background and their dynamics

2019 ◽  
Vol 33 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Variable Coefficient ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Plane Wave Background ◽  
Localized Waves ◽  
Similarity Reductions

In this work, a variable coefficient nonlinear Schrödinger (vc-NLS) equation is under investigation, which can describe the amplification or absorption of pulses propagating in an optical fiber with distributed dispersion and nonlinearity. By means of similarity reductions, a similar transformation helps us to relate certain class of solutions of the standard NLS equation to the solutions of integrable vc-NLS equation. Furthermore, we analytically consider nonautonomous breather wave, rogue wave solutions and their interactions in the vc-NLS equation, which possess complicated wave propagation in time and differ from the usual breather waves and rogue waves. Finally, the main characteristics of the rational solutions are graphically discussed. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves.

Bilinear forms and soliton solutions for a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation in an optical fiber

2020 ◽  
Vol 34 (30) ◽  
pp. 2050336
Author(s):  
Dong Wang ◽  
Yi-Tian Gao ◽  
Jing-Jing Su ◽  
Cui-Cui Ding
Keyword(s):  
Schrödinger Equation ◽  
Optical Fiber ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Bilinear Forms ◽  
Dimensional Variable

In this paper, under investigation is a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation, which is introduced to the study of an optical fiber, where [Formula: see text] is the temporal variable, variable coefficients [Formula: see text] and [Formula: see text] are related to the group velocity dispersion, [Formula: see text] and [Formula: see text] represent the Kerr nonlinearity and linear term, respectively. Via the Hirota bilinear method, bilinear forms are obtained, and bright one-, two-, three- and N-soliton solutions as well as dark one- and two-soliton solutions are derived, where [Formula: see text] is a positive integer. Velocities and amplitudes of the bright/dark one solitons are obtained via the characteristic-line equations. With the graphical analysis, we investigate the influence of the variable coefficients on the propagation and interaction of the solitons. It is found that [Formula: see text] can only affect the phase shifts of the solitons, while [Formula: see text], [Formula: see text] and [Formula: see text] determine the amplitudes and velocities of the bright/dark solitons.

Exact Soliton Solutions of the Variable Coefficient KdV Equation with Forced Term

2014 ◽  
Vol 548-549 ◽  
pp. 1196-1200
Author(s):  
Yong Mei Bao ◽  
Siqintana Bao
Keyword(s):  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Nonlinear Ordinary Differential Equation ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Forced Term ◽  
Exact Soliton Solutions

In order to construct exact soliton solutions of nonlinear evolution equations with variable coefficients. By using a transformation, the variable coefficient KdV equation with forced Term is reduced to nonlinear ordinary differential equation (NLODE), after that, a number of exact solitons solutions of variable coefficient KdV equation with forced Term are obtained by using the equation shorted in NLODE. As it showed above, this kind of method can be applied in solving a large number of nonlinear evolution equations.

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev–Petviashvili equation

2017 ◽  
Vol 31 (36) ◽  
pp. 1750350 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Li Zou
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Variable Coefficient ◽  
Wave Equations ◽  
Direct Method ◽  
Dynamical Behavior ◽  
Rogue Waves ◽  
Nonlinear Wave Equations ◽  
Quasiperiodic Solutions ◽  
Petviashvili Equation

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

Sign in / Sign up

Export Citation Format

Share Document