By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-br...By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-breather molecules,T-breather–L-soliton molecules and some interaction solutions when N≤6.Dynamical behaviors of these solutions are discussed analytically and graphically.The method adopted can be effectively used to construct soliton molecules and T-breather molecules of other nonlinear evolution equations.The results obtained may be helpful for experts to study the related phenomenon in oceanography and atmospheric science.展开更多
Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinan...Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent "phase shifts" of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.展开更多
We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation.What is more,each soliton molecule can be transformed into an asymmetr...We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation.What is more,each soliton molecule can be transformed into an asymmetric soliton by changing the parameterφ.In addition,the collision between soliton molecules(or asymmetric soliton)and several soliton solutions is observed.Finally,some related pictures are presented.展开更多
Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the ...Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the velocity resonance mechanism.The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper.With particular selections of the involved arbitrary parameters,especially the wave numbers,it is confirmed that,besides the usual multi-bright soliton molecules,the multi-dark soliton molecules and the mixed multibright-dark soliton molecules can also be obtained.In addition,we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions.The results demonstrate that when N≥4,the combined mKdVtype bilinear equation no longer admits soliton molecules comprising more than four solitons.展开更多
The main aim of this paper is to investigate the different types of soliton molecule solutions of the second extend(3+1)-dimensional Jimbo-Miwa equation in a fluid.Four different localized waves:line solitons,breather...The main aim of this paper is to investigate the different types of soliton molecule solutions of the second extend(3+1)-dimensional Jimbo-Miwa equation in a fluid.Four different localized waves:line solitons,breather waves,lump solutions and resonance Y-type solutions are obtained by the Hirota bilinear method directly.Furthermore,the molecule solutions consisting of only line waves,breathers or lump waves are generated by combining velocity resonance condition and long wave limit method.Also,the molecule solutions such as line-breather molecule,lump-line molecule,lump-breather molecule,etc.consisting of different waves are derived.Meanwhile,higher-order molecule solutions composed of only line waves are acquired.展开更多
The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition,which is called velocity resonance.One soliton molecule and interaction between ...The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition,which is called velocity resonance.One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters.The soliton molecules including the bright and bright soliton,the dark and bright soliton,and the dark and dark soliton are exhibited in figures 1–3,respectively.Meanwhile,the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevémethod.The consistent Riccati expansion(CRE)method is applied to the extended mKdV equation.It demonstrates that the extended mKdV equation is a CRE solvable system.A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.展开更多
We discuss a fifth order KdV(FOKdV)equation via a novel travelling wave method by introducing a background term.Results show that the background term plays an essential role in finding new abundant travelling wave str...We discuss a fifth order KdV(FOKdV)equation via a novel travelling wave method by introducing a background term.Results show that the background term plays an essential role in finding new abundant travelling wave structures,such as the soliton induced by negative background,the periodic travelling wave excited by the positive background,the few-cycle-pulse(FCP)solitons with and without background,the soliton molecules excited by the background.The FCP solitons are first obuained for the FOKdV equation.展开更多
For a one(2+1)-dimensional combined Kadomtsev-Petviashvili with its hierarchy equation, the missing D’Alembert type solution is derived first through the traveling wave transformation which contains several special k...For a one(2+1)-dimensional combined Kadomtsev-Petviashvili with its hierarchy equation, the missing D’Alembert type solution is derived first through the traveling wave transformation which contains several special kink molecule structures. Further, after introducing the B?cklund transformation and an auxiliary variable, the N-soliton solution which contains some soliton molecules for this equation, is presented through its Hirota bilinear form. The concrete molecules including line solitons, breathers and a lump as well as several interactions of their hybrid are shown with the aid of special conditions and parameters. All these dynamical features are demonstrated through the different figures.展开更多
Two nonlocal Alice–Bob Sawada–Kotera(ABSK) systems, accompanied by the parity and time reversal invariance are studied. The Lax pairs of two systems are uniformly written out in matrix form. The periodic waves, mult...Two nonlocal Alice–Bob Sawada–Kotera(ABSK) systems, accompanied by the parity and time reversal invariance are studied. The Lax pairs of two systems are uniformly written out in matrix form. The periodic waves, multiple solitons, and soliton molecules of the ABSK systems are obtained via the bilinear method and the velocity resonant mechanism. Though the interactions among solitons are elastic, the interactions between soliton and soliton molecules are not elastic.In particular, the shapes of the soliton molecules are changed explicitly after interactions.展开更多
The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial different...The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.展开更多
Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kau...Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps,breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.展开更多
We present experimental observations of soliton pulsations in the net normal-dispersion fiber laser by using the dispersive Fourier transform(DFT) technique. According to the pulsating characteristics, the soliton pul...We present experimental observations of soliton pulsations in the net normal-dispersion fiber laser by using the dispersive Fourier transform(DFT) technique. According to the pulsating characteristics, the soliton pulsations are classified as visible and invisible soliton pulsations. The visible soliton pulsation is converted from single-into dual-soliton pulsation with the common characteristics of energy oscillation and bandwidth breathing. The invisible soliton pulsation undergoes periodic variation in the spectral profile and peak power but remains invariable in pulse energy. The reason for invisible soliton pulsation behavior is periodic oscillation of the pulse inside the soliton molecule. These results could be helpful in deepening our understanding of the soliton pulsation phenomena.展开更多
Soliton molecules(SMs)of the(2+1)-dimensional generalized KonopelchenkoDubrovsky-Kaup-Kupershmidt(gKDKK)equation are found by utilizing a velocity resonance ansatz to N-soliton solutions,which can transform to asymmet...Soliton molecules(SMs)of the(2+1)-dimensional generalized KonopelchenkoDubrovsky-Kaup-Kupershmidt(gKDKK)equation are found by utilizing a velocity resonance ansatz to N-soliton solutions,which can transform to asymmetric solitons upon assigning appropriate values to some parameters.Furthermore,a double-peaked lump solution can be constructed with breather degeneration approach.By applying a mixed technique of a resonance ansatz and conjugate complexes of partial parameters to multisoliton solutions,various kinds of interactional structures are constructed;There include the soliton molecule(SM),the breather molecule(BM)and the soliton-breather molecule(SBM).Graphical investigation and theoretical analysis show that the interactions composed of SM,BM and SBM are inelastic.展开更多
The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is exp...The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation.A novel(2+1)-dimensional KdV extension,the cKP3-4 equation,is obtained by combining the third member(KP3,the usual KP equation)and the fourth member(KP4)of the KP hierarchy.The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair.The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable.Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations.For instance,the soliton molecules and the missing D'Alembert type solutions(the arbitrary travelling waves moving in one direction with a fixed model dependent velocity)including periodic kink molecules,periodic kink-antikink molecules,few-cycle solitons,and envelope solitons exist for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.展开更多
基金Jiangsu Provincial Natural Science Foundation of China(Grant Nos.BK20221508,11775116,BK20210380,and JSSCBS20210277)SRT(Grant No.202210307165Y)Jiangsu Qinglan High-level Talent Project.
文摘By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-breather molecules,T-breather–L-soliton molecules and some interaction solutions when N≤6.Dynamical behaviors of these solutions are discussed analytically and graphically.The method adopted can be effectively used to construct soliton molecules and T-breather molecules of other nonlinear evolution equations.The results obtained may be helpful for experts to study the related phenomenon in oceanography and atmospheric science.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11775121 and 11435005)the K. C. Wong Magna Fund in Ningbo University.
文摘Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent "phase shifts" of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11371086, 11671258, and 11975145)the Fund of Science and Technology Commission of Shanghai Municipality, China (Grant No. 13ZR1400100)+1 种基金the Fund of Institute for Nonlinear Sciences, Donghua Universitythe Fundamental Research Funds for the Central Universities, China (Grant No. 2232021G-13)
文摘We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation.What is more,each soliton molecule can be transformed into an asymmetric soliton by changing the parameterφ.In addition,the collision between soliton molecules(or asymmetric soliton)and several soliton solutions is observed.Finally,some related pictures are presented.
基金the National Natural Science Foundation of China(Grant Nos.11975204 and 12075208)the Project of Zhoushan City Science and Technology Bureau(Grant No.2021C21015)the Training Program for Leading Talents in Universities of Zhejiang Province。
文摘Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the velocity resonance mechanism.The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper.With particular selections of the involved arbitrary parameters,especially the wave numbers,it is confirmed that,besides the usual multi-bright soliton molecules,the multi-dark soliton molecules and the mixed multibright-dark soliton molecules can also be obtained.In addition,we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions.The results demonstrate that when N≥4,the combined mKdVtype bilinear equation no longer admits soliton molecules comprising more than four solitons.
文摘The main aim of this paper is to investigate the different types of soliton molecule solutions of the second extend(3+1)-dimensional Jimbo-Miwa equation in a fluid.Four different localized waves:line solitons,breather waves,lump solutions and resonance Y-type solutions are obtained by the Hirota bilinear method directly.Furthermore,the molecule solutions consisting of only line waves,breathers or lump waves are generated by combining velocity resonance condition and long wave limit method.Also,the molecule solutions such as line-breather molecule,lump-line molecule,lump-breather molecule,etc.consisting of different waves are derived.Meanwhile,higher-order molecule solutions composed of only line waves are acquired.
基金supported by the National Natural Science Foundation of China Grant Nos.11775146,11305106,11975156,11775047 and 11835011。
文摘The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition,which is called velocity resonance.One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters.The soliton molecules including the bright and bright soliton,the dark and bright soliton,and the dark and dark soliton are exhibited in figures 1–3,respectively.Meanwhile,the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevémethod.The consistent Riccati expansion(CRE)method is applied to the extended mKdV equation.It demonstrates that the extended mKdV equation is a CRE solvable system.A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.
基金The authors also acknowledge the support of NNSFC(Grant No.11675084)K C Wong Magna Fund in Ningbo University。
文摘We discuss a fifth order KdV(FOKdV)equation via a novel travelling wave method by introducing a background term.Results show that the background term plays an essential role in finding new abundant travelling wave structures,such as the soliton induced by negative background,the periodic travelling wave excited by the positive background,the few-cycle-pulse(FCP)solitons with and without background,the soliton molecules excited by the background.The FCP solitons are first obuained for the FOKdV equation.
基金supported by the National Natural Science Foundation of China(Nos.11705077,11775104 and 11447017)。
文摘For a one(2+1)-dimensional combined Kadomtsev-Petviashvili with its hierarchy equation, the missing D’Alembert type solution is derived first through the traveling wave transformation which contains several special kink molecule structures. Further, after introducing the B?cklund transformation and an auxiliary variable, the N-soliton solution which contains some soliton molecules for this equation, is presented through its Hirota bilinear form. The concrete molecules including line solitons, breathers and a lump as well as several interactions of their hybrid are shown with the aid of special conditions and parameters. All these dynamical features are demonstrated through the different figures.
文摘Two nonlocal Alice–Bob Sawada–Kotera(ABSK) systems, accompanied by the parity and time reversal invariance are studied. The Lax pairs of two systems are uniformly written out in matrix form. The periodic waves, multiple solitons, and soliton molecules of the ABSK systems are obtained via the bilinear method and the velocity resonant mechanism. Though the interactions among solitons are elastic, the interactions between soliton and soliton molecules are not elastic.In particular, the shapes of the soliton molecules are changed explicitly after interactions.
基金supported by the National Natural Science meters restrain as the relation:Foundation of China Grant No.11775146.
文摘The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.
基金supported by the National Natural Science Foundation of China (project Nos. 11371086,11671258,11975145)the Fund of Science and Technology Commission of Shanghai Municipality (project No. 13ZR1400100)the Fund of Donghua University,Institute for Nonlinear Sciences and the Fundamental Research Funds for the Central Universities。
文摘Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps,breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.
基金supported by the National Natural Science Foundation of China (Grant No. 41875040)the Natural Science Foundation of Anhui Province, China (Grant No. 2008085MF211)+1 种基金the Foundation for Young Talents in College of Anhui Province, China (Grant No. gxyqZD2019034)the Innovation Fund for Postgraduates of Huaibei Normal University, China (Grant No. CX2022035)。
文摘We present experimental observations of soliton pulsations in the net normal-dispersion fiber laser by using the dispersive Fourier transform(DFT) technique. According to the pulsating characteristics, the soliton pulsations are classified as visible and invisible soliton pulsations. The visible soliton pulsation is converted from single-into dual-soliton pulsation with the common characteristics of energy oscillation and bandwidth breathing. The invisible soliton pulsation undergoes periodic variation in the spectral profile and peak power but remains invariable in pulse energy. The reason for invisible soliton pulsation behavior is periodic oscillation of the pulse inside the soliton molecule. These results could be helpful in deepening our understanding of the soliton pulsation phenomena.
基金Supported by the National Natural Science Foundation of China(12001424)the Natural Science Basic Research Program of Shaanxi Province(2021JZ-21)the Fundamental Research Funds for the Central Universities(2020CBLY013)。
文摘Soliton molecules(SMs)of the(2+1)-dimensional generalized KonopelchenkoDubrovsky-Kaup-Kupershmidt(gKDKK)equation are found by utilizing a velocity resonance ansatz to N-soliton solutions,which can transform to asymmetric solitons upon assigning appropriate values to some parameters.Furthermore,a double-peaked lump solution can be constructed with breather degeneration approach.By applying a mixed technique of a resonance ansatz and conjugate complexes of partial parameters to multisoliton solutions,various kinds of interactional structures are constructed;There include the soliton molecule(SM),the breather molecule(BM)and the soliton-breather molecule(SBM).Graphical investigation and theoretical analysis show that the interactions composed of SM,BM and SBM are inelastic.
基金the National Natural Science Foundation of China(Grant Nos.11975131 and 11435005)and K.C.Wong Magna Fund in Ningbo University.
文摘The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation.A novel(2+1)-dimensional KdV extension,the cKP3-4 equation,is obtained by combining the third member(KP3,the usual KP equation)and the fourth member(KP4)of the KP hierarchy.The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair.The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable.Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations.For instance,the soliton molecules and the missing D'Alembert type solutions(the arbitrary travelling waves moving in one direction with a fixed model dependent velocity)including periodic kink molecules,periodic kink-antikink molecules,few-cycle solitons,and envelope solitons exist for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.
