Hybrid soliton solutions in the (2+1)-dimensional nonlinear Schrödinger equation

2017 ◽  
Vol 31 (32) ◽  
pp. 1750298 ◽  
Author(s):  
Meidan Chen ◽  
Biao Li
Keyword(s):  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Bilinear Method ◽  
Nonlinear Schrödinger ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit

Rational solutions and hybrid solutions from N-solitons are obtained by using the bilinear method and a long wave limit method. Line rogue waves and lumps in the (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger (NLS) equation are derived from two-solitons. Then from three-solitons, hybrid solutions between kink soliton with breathers, periodic line waves and lumps are derived. Interestingly, after the collision, the breathers are kept invariant, but the amplitudes of the periodic line waves and lumps change greatly. For the four-solitons, the solutions describe as breathers with breathers, line rogue waves or lumps. After the collision, breathers and lumps are kept invariant, but the line rogue wave has a great change.

Rogue Waves and Hybrid Solutions of the Boussinesq Equation

2017 ◽  
Vol 72 (4) ◽  
pp. 307-314 ◽  
Author(s):  
Ji-Guang Rao ◽  
Yao-Bin Liu ◽  
Chao Qian ◽  
Jing-Song He
Keyword(s):  
Boussinesq Equation ◽  
Rogue Wave ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Hirota Bilinear

AbstractThe rational and semirational solutions in the Boussinesq equation are obtained by the Hirota bilinear method and long wave limit. It is shown that the rational solutions contain dark and bright rogue waves, and their typical dynamics are analysed and illustrated. The semirational solutions possess a range of hybrid solutions, and the hybrid of rogue wave and solitons are demonstrated in detail by the three-dimensional figures. Under certain parameter conditions, a new kind of semirational solutions consisted of rogue waves, breathers and solitons is discovered, which describes the dynamics of the rogue waves interacting with the breathers and solitons at the same time.

scholarly journals Hybrid Solutions of (3 + 1)-Dimensional Jimbo-Miwa Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Zhang ◽  
Shili Sun ◽  
Huanhe Dong
Keyword(s):  
Three Dimensional ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solution ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Hirota Bilinear

The rational solutions, semirational solutions, and their interactions to the (3+1)-dimensional Jimbo-Miwa equation are obtained by the Hirota bilinear method and long wave limit. The hybrid solutions contain rogue wave, lump solution, and the breather solution, in which the breathers which are manifested as growing and decaying periodic line waves show different dynamics in different planes. Rogue waves are localized in time and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods; they arise from a constant background at t≪0 and then disappear in the constant background when time goes on. More importantly, the interactions between some hybrid solutions are demonstrated in detail by the three-dimensional figures, such as hybrid solution between the stripe soliton and breather and hybrid solution between stripe soliton and lump solution.

Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation

2018 ◽  
Vol 32 (29) ◽  
pp. 1850359 ◽  
Author(s):  
Wenhao Liu ◽  
Yufeng Zhang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
The One ◽  
Wave Limit

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.

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.

Soliton molecules and some novel hybrid solutions for (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2021 ◽  
pp. 2150388
Author(s):  
Hongcai Ma ◽  
Huaiyu Huang ◽  
Aiping Deng
Keyword(s):  
Soliton Solutions ◽  
Dynamic Graphs ◽  
Bilinear Method ◽  
Resonance Mechanism ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Bkp Equation

In recent years, soliton molecules have received reinvigorating scientific interests in physics and other fields. Soliton molecules have been successfully found in optical experiments. In this paper, we attribute the solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation by employing the bilinear method. Based on the [Formula: see text]-soliton solutions, we establish the soliton molecules, asymmetric solitons and some novel hybrid solutions of this equation by means of the velocity resonance mechanism and the long wave limit method. Finally, we give dynamic graphs of soliton molecules, asymmetric solitons and some novel hybrid solutions.

High-order breathers, lumps and hybrid solutions to the (2+1)-dimensional fifth-order KdV equation

2019 ◽  
Vol 33 (22) ◽  
pp. 1950255 ◽  
Author(s):  
Wen-Tao Li ◽  
Zhao Zhang ◽  
Xiang-Yu Yang ◽  
Biao Li
Keyword(s):  
Kdv Equation ◽  
Perturbation Expansion ◽  
Soliton Solutions ◽  
High Order ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Fifth Order ◽  
Parameter Constraints

In this paper, the (2+1)-dimensional fifth-order KdV equation is analytically investigated. By using Hirota’s bilinear method combined with perturbation expansion, the high-order breather solutions of the fifth-order KdV equation are generated. Then, the high-order lump solutions are also derived from the soliton solutions by a long-wave limit method and some suitable parameter constraints. Furthermore, we extend this method to obtain hybrid solutions by taking long-wave limit for partial soliton solutions. Finally, the dynamic behavior of these solutions is presented in the figures.

Rational and semi-rational solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

2019 ◽  
Vol 33 (25) ◽  
pp. 1950296 ◽  
Author(s):  
Ya-Si Deng ◽  
Bo Tian ◽  
Yan Sun ◽  
Chen-Rong Zhang ◽  
Cong-Cong Hu
Keyword(s):  
Water Waves ◽  
Boussinesq Equation ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Different Shapes ◽  
Lump Wave ◽  
Kink Soliton

Nonlinear waves are seen in nature, such as the water waves and plasma waves. Investigated in this paper is a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Based on the bilinear method, we get the rational solutions, which are different from the published ones, semi-rational solutions and breather-type kink soliton solutions. Through the rational solutions, we observe two types of waves: the lump waves and line rogue waves. The semi-rational solutions depict two types of interactions: (1) The fusion or fission between the lump wave and soliton; (2) The interaction between the line rogue wave and soliton. During the interaction between the line rogue wave and soliton, the line rogue wave evolves with three different shapes: the bright rogue waves, bright–dark rogue waves and dark rogue waves. Via the breather-type kink soliton solutions, we observe the breather-soliton mixture.

Lumps, breathers, rogue waves and interaction solutions to a (3+1)-dimensional Kudryashov–Sinelshchikov equation

2020 ◽  
Vol 34 (12) ◽  
pp. 2050117 ◽  
Author(s):  
Xianglong Tang ◽  
Yong Chen
Keyword(s):  
Rogue Waves ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Long Wave ◽  
Wave Limit ◽  
Hirota Bilinear

Utilizing the Hirota bilinear method, the lump solutions, the interaction solutions with the lump and the stripe solitons, the breathers and the rogue waves for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kudryashov–Sinelshchikov equation are constructed. Two types of interaction solutions between the lumps and the stripe solitons are exhibited. Some different breathers are given by choosing special parameters in the expressions of the solitons. Through a long wave limit of breathers, the lumps and rogue waves are derived.

General high-order breather, lump, and semi-rational solutions to the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation

2020 ◽  
pp. 2150057
Author(s):  
Xin-Mei Zhou ◽  
Shou-Fu Tian ◽  
Ling-Di Zhang ◽  
Tian-Tian Zhang
Keyword(s):  
Bilinear Form ◽  
Soliton Solutions ◽  
High Order ◽  
Rational Solutions ◽  
Lump Solutions ◽  
Long Wave ◽  
Wave Limit

In this work, we investigate the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation. Based on its bilinear form, the [Formula: see text]th-order breather solutions of the gBK equation are successful given by taking appropriate parameters. Furthermore, the [Formula: see text]th-order lump solutions of the gBK equation are obtained via the long-wave limit method. In addition, the semi-rational solutions are generated to reveal the interaction between lump solutions, soliton solutions, and breather solutions.

scholarly journals Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

2022 ◽  
Author(s):  
Ren Bo ◽  
Shi Kai-Zhong ◽  
Shou-Feng Shen ◽  
Wang Guo-Fang ◽  
Peng Jun-Da ◽  
...  
Keyword(s):  
Bilinear Form ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Ultrashort Pulses ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Femtosecond Regime ◽  
Wave Limit

Abstract In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

Sign in / Sign up

Export Citation Format

Share Document