We investigate the alpha helical protein structure characterized by fourth-order interspine coupling,focusing on a three-coupled fourth-order nonlinear Schr??dinger system.We introduce a generalized Darboux transforma...We investigate the alpha helical protein structure characterized by fourth-order interspine coupling,focusing on a three-coupled fourth-order nonlinear Schr??dinger system.We introduce a generalized Darboux transformation,departing from the classical Darboux transformation.Based on this,we construct the two-and three-degenerate soliton solutions and four-degenerate asymptotic soliton solutions.Based on the asymptotic analysis,we find that the amplitudes of interacting solitons are retained upon the interactions.Elastic interactions between two degenerate solitons exhibiting four curve-type asymptotic solitons are depicted.When the lattice parameterβchanges,the velocities of the two degenerate solitons also change.Elastic interaction among three degenerate solitons comprising four curve-type asymptotic solitons and two line-type solitons is presented.Interaction among one soliton and two degenerate solitons with different velocities is shown.Elastic interaction among four degenerate solitons comprising eight curve-type asymptotic solitons is also presented.Interaction among two two-degenerate solitons with two spectral parameters is shown.The relative distance between two asymptotic solitons exhibits logarithmic growth with|t|,where t represents the retarded time.Acceleration of soliton separation decays exponentially with relative distance,and eventually approaches zero.Phase shifts depend on t.展开更多
The coupled nonlocal nonlinear Schrödinger equations with variable coefficients are researched using the nonstandard Hirota bilinear method.The two-soliton and double-hump one-soliton solutions for the equations ...The coupled nonlocal nonlinear Schrödinger equations with variable coefficients are researched using the nonstandard Hirota bilinear method.The two-soliton and double-hump one-soliton solutions for the equations are first obtained.By assigning different functions to the variable coefficients,we obtain V-shaped,Y-shaped,wave-type,exponential solitons,and so on.Next,we reveal the influence of the real and imaginary parts of the wave numbers on the double-hump structure based on the soliton solutions.Finally,by setting different wave numbers,we can change the distance and transmission direction of the solitons to analyze their dynamic behavior during collisions.This study establishes a theoretical framework for controlling the dynamics of optical fiber in nonlocal nonlinear systems.展开更多
本文研究了半耗散格点非线性Schrödinger方程组解的拉回渐近行为及其概率分布。该方程组描述带有杂质的Bose-Einstein浓缩模型,模型中的Bose波函数具有耗散性,杂质波函数的能量守恒。作者首先证明该问题的整体适定性,然后研究Bose...本文研究了半耗散格点非线性Schrödinger方程组解的拉回渐近行为及其概率分布。该方程组描述带有杂质的Bose-Einstein浓缩模型,模型中的Bose波函数具有耗散性,杂质波函数的能量守恒。作者首先证明该问题的整体适定性,然后研究Bose波函数在适当意义下拉回吸引子的存在性,接着应用该拉回吸引子和广义Banach极限构造统计解,并证明统计解满足Liouville型定理。In this paper, the pullback asymptotic behavior of solutions to the nonlinear Schrödinger system of equations with semi-dissipative lattices and their probability distributions are studied. The equations describe the Bose-Einstein condensation model with impurities, in which the Bose wave function is dissipative, and the energy of the impurity wave function is conserved. The authors first prove the global well-posed of the problem and then investigate the existence of a pullback attractor for the Bose wave function in a suitable sense. The authors then apply the pullback attractor and the generalized Banach limit to construct a statistical solution and show that the statistical solution satisfies the Liouville-type theorem.展开更多
The heavy quarks present in the quark-gluon plasma(QGP)can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history.In a previous study,a stochastic Schrödinger e...The heavy quarks present in the quark-gluon plasma(QGP)can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history.In a previous study,a stochastic Schrödinger equation(SSE)has been applied to describe the evolution of heavy quarks,where an external field with random phases is used to simulate the thermal medium.In this work,we study the connection between the SSE and the Boltzmann transport equation(BE)approach in the Keldysh Green’s function formalism.By comparing the Green’s function of the heavy quark from the SSE and the Keldysh Green’s functions leading to the Boltzmann equation,we demonstrate that the SSE is consistent with the Boltzmann equation in the weak coupling limit.We subsequently confirm their consistency through numerical calculations.展开更多
In this paper,we consider the following logarithmic Schrödinger equation:−Δu+ωu=u log|u|^(2),u∈H^(1)(R^(N)),where N≥3,and ω>0 is a constant.With an auxiliary equation,we obtain the existence of normalized...In this paper,we consider the following logarithmic Schrödinger equation:−Δu+ωu=u log|u|^(2),u∈H^(1)(R^(N)),where N≥3,and ω>0 is a constant.With an auxiliary equation,we obtain the existence of normalized solutions by using the constrained variational method.展开更多
The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear...The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear method,we systematically construct single-and double-periodic lump solutions.To provide a detailed insight into the dynamic behavior of the nonlinear waves,we explore diverse mixed solutions,including bright-dark,W-shaped,multi-peak,and bright soliton solutions.Building upon single-periodic lump solutions,we analyze the dynamics of lump waves on both plane-wave and periodic backgrounds using the long-wave limit method.Moreover,we obtain the interaction solutions involving lumps,periodic lumps,and solitons.The interactions among two solitons,multiple lumps,and mixed waves are illustrated and analyzed.Comparative analysis reveals that these multi-lump solutions exhibit richer dynamical properties than conventional single-lump ones.These results contribute to a deeper understanding of nonlinear systems and may facilitate solving nonlinear problems in nature.展开更多
We devote ourselves to finding exact solutions(including perturbed soliton solutions)to a class of semi-linear Schrödinger equations incorporating Kudryashov's self-phase modulation subject to stochastic pert...We devote ourselves to finding exact solutions(including perturbed soliton solutions)to a class of semi-linear Schrödinger equations incorporating Kudryashov's self-phase modulation subject to stochastic perturbations described by multiplicative white noise based on Stratonvich's calculus.By borrowing ideas of the sub-equation method and utilizing a series of changes of variables,we transform the problem of identifying exact solutions into the task of analyzing the dynamical behaviors of an auxiliary planar Hamiltonian dynamical system.We determine the equilibrium points of the introduced auxiliary Hamiltonian system and analyze their Lyapunov stability.Additionally,we conduct a brief bifurcation analysis and a preliminary chaos analysis of the auxiliary Hamiltonian system,assessing their impact on the Lyapunov stability.Based on the insights gained from investigating the dynamics of the introduced auxiliary Hamiltonian system,we discover‘all'of the exact solutions to the stochastic semi-linear Schrödinger equations under consideration.We obtain explicit formulas for exact solutions by examining the phase portrait of the introduced auxiliary Hamiltonian system.The obtained exact solutions include singular and periodic solutions,as well as perturbed bright and dark solitons.For each type of obtained exact solution,we pick one representative to plot its graph,so as to visually display our theoretical results.Compared with other methods for finding exact solutions to deterministic or stochastic partial differential equations,the dynamical system approach has the merit of yielding all possible exact solutions.The stochastic semi-linear Schrödinger equation under consideration can be used to portray the propagation of pulses in an optical fiber,so our study therefore lays the foundation for discovering new solitons optimized for optical communication and contributes to the improvement of optical technologies.展开更多
Under investigation is the n-component nonlinear Schrödinger equation with higher-order effects,which describes the ultrashort pulses in the birefringent fiber.Based on the Lax pair,the eigenfunction and generali...Under investigation is the n-component nonlinear Schrödinger equation with higher-order effects,which describes the ultrashort pulses in the birefringent fiber.Based on the Lax pair,the eigenfunction and generalized Darboux transformation are derived.Next,we construct several novel higher-order localized waves and classified them into three categories:(i)higher-order rogue waves interacting with bright/antidark breathers,(ii)higher-order breather fission/fusion,(iii)higherorder breather interacting with soliton.Moreover,we explore the effects of parameters on the structure,collision process and energy distribution of localized waves and these characteristics are significantly different from previous ones.Finally,the dynamical properties of these solutions are discussed in detail.展开更多
In this paper,we present some properties of scattering data for the derivative nonlinear Schrödinger equation in H^(S)(R)(s≥1/2)starting from the Lax pair.We show that the reciprocal of the transmission coeffici...In this paper,we present some properties of scattering data for the derivative nonlinear Schrödinger equation in H^(S)(R)(s≥1/2)starting from the Lax pair.We show that the reciprocal of the transmission coefficient can be expressed as the sum of some iterative integrals,and its logarithm can be written as the sum of some connected iterative integrals.We provide the asymptotic properties of the first few iterative integrals of the reciprocal of the transmission coefficient.Moreover,we provide some regularity properties of the reciprocal of the transmission coefficient related to scattering data in H^(S)(R).展开更多
基金supported by the Natural Science Foundation of Shandong Province(Grant No.ZR2025QC30)。
文摘We investigate the alpha helical protein structure characterized by fourth-order interspine coupling,focusing on a three-coupled fourth-order nonlinear Schr??dinger system.We introduce a generalized Darboux transformation,departing from the classical Darboux transformation.Based on this,we construct the two-and three-degenerate soliton solutions and four-degenerate asymptotic soliton solutions.Based on the asymptotic analysis,we find that the amplitudes of interacting solitons are retained upon the interactions.Elastic interactions between two degenerate solitons exhibiting four curve-type asymptotic solitons are depicted.When the lattice parameterβchanges,the velocities of the two degenerate solitons also change.Elastic interaction among three degenerate solitons comprising four curve-type asymptotic solitons and two line-type solitons is presented.Interaction among one soliton and two degenerate solitons with different velocities is shown.Elastic interaction among four degenerate solitons comprising eight curve-type asymptotic solitons is also presented.Interaction among two two-degenerate solitons with two spectral parameters is shown.The relative distance between two asymptotic solitons exhibits logarithmic growth with|t|,where t represents the retarded time.Acceleration of soliton separation decays exponentially with relative distance,and eventually approaches zero.Phase shifts depend on t.
基金supported by the National Key R&D Program of China(Grant No.2022YFA1604200)the National Natural Science Foundation of China(Grant No.12261131495)Institute of Systems Science,Beijing Wuzi University(Grant No.BWUISS21).
文摘The coupled nonlocal nonlinear Schrödinger equations with variable coefficients are researched using the nonstandard Hirota bilinear method.The two-soliton and double-hump one-soliton solutions for the equations are first obtained.By assigning different functions to the variable coefficients,we obtain V-shaped,Y-shaped,wave-type,exponential solitons,and so on.Next,we reveal the influence of the real and imaginary parts of the wave numbers on the double-hump structure based on the soliton solutions.Finally,by setting different wave numbers,we can change the distance and transmission direction of the solitons to analyze their dynamic behavior during collisions.This study establishes a theoretical framework for controlling the dynamics of optical fiber in nonlocal nonlinear systems.
文摘本文研究了半耗散格点非线性Schrödinger方程组解的拉回渐近行为及其概率分布。该方程组描述带有杂质的Bose-Einstein浓缩模型,模型中的Bose波函数具有耗散性,杂质波函数的能量守恒。作者首先证明该问题的整体适定性,然后研究Bose波函数在适当意义下拉回吸引子的存在性,接着应用该拉回吸引子和广义Banach极限构造统计解,并证明统计解满足Liouville型定理。In this paper, the pullback asymptotic behavior of solutions to the nonlinear Schrödinger system of equations with semi-dissipative lattices and their probability distributions are studied. The equations describe the Bose-Einstein condensation model with impurities, in which the Bose wave function is dissipative, and the energy of the impurity wave function is conserved. The authors first prove the global well-posed of the problem and then investigate the existence of a pullback attractor for the Bose wave function in a suitable sense. The authors then apply the pullback attractor and the generalized Banach limit to construct a statistical solution and show that the statistical solution satisfies the Liouville-type theorem.
文摘The heavy quarks present in the quark-gluon plasma(QGP)can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history.In a previous study,a stochastic Schrödinger equation(SSE)has been applied to describe the evolution of heavy quarks,where an external field with random phases is used to simulate the thermal medium.In this work,we study the connection between the SSE and the Boltzmann transport equation(BE)approach in the Keldysh Green’s function formalism.By comparing the Green’s function of the heavy quark from the SSE and the Keldysh Green’s functions leading to the Boltzmann equation,we demonstrate that the SSE is consistent with the Boltzmann equation in the weak coupling limit.We subsequently confirm their consistency through numerical calculations.
基金supported by the Natural Science Research Project of Department of Education of Guizhou Province(No.QJJ2023062)the National Natural Science Foundation of China(No.52174184)。
文摘In this paper,we consider the following logarithmic Schrödinger equation:−Δu+ωu=u log|u|^(2),u∈H^(1)(R^(N)),where N≥3,and ω>0 is a constant.With an auxiliary equation,we obtain the existence of normalized solutions by using the constrained variational method.
基金supported by the Applied Basic Research Program of Shanxi Province,China(Grant Nos.202403021212253 and 202203021221217).
文摘The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear method,we systematically construct single-and double-periodic lump solutions.To provide a detailed insight into the dynamic behavior of the nonlinear waves,we explore diverse mixed solutions,including bright-dark,W-shaped,multi-peak,and bright soliton solutions.Building upon single-periodic lump solutions,we analyze the dynamics of lump waves on both plane-wave and periodic backgrounds using the long-wave limit method.Moreover,we obtain the interaction solutions involving lumps,periodic lumps,and solitons.The interactions among two solitons,multiple lumps,and mixed waves are illustrated and analyzed.Comparative analysis reveals that these multi-lump solutions exhibit richer dynamical properties than conventional single-lump ones.These results contribute to a deeper understanding of nonlinear systems and may facilitate solving nonlinear problems in nature.
基金partially supported by Qing Lan Project of Jiangsu,Suqian Sci.&Tech.Program(Grant Nos.Z2023131 and M202206)the Startup Foundation for Newly Recruited Employees,the Xichu Talents Foundation of Suqian University(Grant No.2022XRC033)the National Natural Science Foundation of China(Grant No.11701050)。
文摘We devote ourselves to finding exact solutions(including perturbed soliton solutions)to a class of semi-linear Schrödinger equations incorporating Kudryashov's self-phase modulation subject to stochastic perturbations described by multiplicative white noise based on Stratonvich's calculus.By borrowing ideas of the sub-equation method and utilizing a series of changes of variables,we transform the problem of identifying exact solutions into the task of analyzing the dynamical behaviors of an auxiliary planar Hamiltonian dynamical system.We determine the equilibrium points of the introduced auxiliary Hamiltonian system and analyze their Lyapunov stability.Additionally,we conduct a brief bifurcation analysis and a preliminary chaos analysis of the auxiliary Hamiltonian system,assessing their impact on the Lyapunov stability.Based on the insights gained from investigating the dynamics of the introduced auxiliary Hamiltonian system,we discover‘all'of the exact solutions to the stochastic semi-linear Schrödinger equations under consideration.We obtain explicit formulas for exact solutions by examining the phase portrait of the introduced auxiliary Hamiltonian system.The obtained exact solutions include singular and periodic solutions,as well as perturbed bright and dark solitons.For each type of obtained exact solution,we pick one representative to plot its graph,so as to visually display our theoretical results.Compared with other methods for finding exact solutions to deterministic or stochastic partial differential equations,the dynamical system approach has the merit of yielding all possible exact solutions.The stochastic semi-linear Schrödinger equation under consideration can be used to portray the propagation of pulses in an optical fiber,so our study therefore lays the foundation for discovering new solitons optimized for optical communication and contributes to the improvement of optical technologies.
基金Project supported by the National Natural Science Foundation of China(Grant No.12271096)the Natural Science Foundation of Fujian Province(Grant No.2021J01302)。
文摘Under investigation is the n-component nonlinear Schrödinger equation with higher-order effects,which describes the ultrashort pulses in the birefringent fiber.Based on the Lax pair,the eigenfunction and generalized Darboux transformation are derived.Next,we construct several novel higher-order localized waves and classified them into three categories:(i)higher-order rogue waves interacting with bright/antidark breathers,(ii)higher-order breather fission/fusion,(iii)higherorder breather interacting with soliton.Moreover,we explore the effects of parameters on the structure,collision process and energy distribution of localized waves and these characteristics are significantly different from previous ones.Finally,the dynamical properties of these solutions are discussed in detail.
基金W.W.was supported by the China Postdoctoral Science Foundation(Grant No.2023M741992)Z.Y.was supported by the National Natural Science Foundation of China(Grant No.11925108).
文摘In this paper,we present some properties of scattering data for the derivative nonlinear Schrödinger equation in H^(S)(R)(s≥1/2)starting from the Lax pair.We show that the reciprocal of the transmission coefficient can be expressed as the sum of some iterative integrals,and its logarithm can be written as the sum of some connected iterative integrals.We provide the asymptotic properties of the first few iterative integrals of the reciprocal of the transmission coefficient.Moreover,we provide some regularity properties of the reciprocal of the transmission coefficient related to scattering data in H^(S)(R).
