Symbolic Computation

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Bo Tian - One of the best experts on this subject based on the ideXlab platform.

  • integrability aspects and soliton solutions for an inhomogeneous nonlinear system with Symbolic Computation
    Communications in Nonlinear Science and Numerical Simulation, 2012
    Co-Authors: Rui Guo, Bo Tian
    Abstract:

    Abstract Under investigation in this paper is an inhomogeneous nonlinear system, which describes the marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics under inhomogeneous media. Through Symbolic Computation, the Painleve integrable condition, Lax pair and conservation laws are derived for this system. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Parallel propagation of solitons: separation distance of the two parallel solitons depends on the value of | Im ( λ 1 ) | - | Im ( λ 2 ) | (where λ1 and λ2 are the spectrum parameters); (2) Periodic propagation of bound solitons: periodic bound solitons taking on contrary trends, and mutual attractions and repulsions of two bright bound solitons; (3) Elastic interactions of two one-peak bright solitons and of two one-peak dark solitons.

  • darboux transformation and soliton solutions for the generalized coupled variable coefficient nonlinear schrodinger maxwell bloch system with Symbolic Computation
    Computational Mathematics and Mathematical Physics, 2012
    Co-Authors: Bo Tian, Haiqiang Zhang, Rui Guo, Wenjun Liu
    Abstract:

    In an inhomogeneous nonlinear light guide doped with two-level resonant atoms, the generalized coupled variable-coefficient nonlinear Schrodinger-Maxwell-Bloch system can be used to describe the propagation of optical solitons. In this paper, the Lax pair and conservation laws of that model are derived via Symbolic Computation. Furthermore, based on the Lax pair obtained, the Darboux transformation is constructed and soliton solutions are presented. Figures are plotted to reveal the following dynamic features of the solitons: (1) Periodic mutual attractions and repulsions of four types of bound solitons: of two one-peak bright solitons; of two one-peak dark solitons; of two two-peak bright solitons and of two two-peak dark solitons; (2) Two types of elastic interactions of solitons: of two bright solitons and of two dark solitons; (3) Two types of parallel propagations of parabolic solitons: of two bright solitons and of two dark solitons. Those results might be useful in the study of optical solitons in some inhomogeneous nonlinear light guides.

  • types of solutions of the variable coefficient nonlinear schrodinger equation with Symbolic Computation
    Physical Review E, 2008
    Co-Authors: Wenjun Liu, Bo Tian, Haiqiang Zhang
    Abstract:

    By using Hirota's bilinear method and Symbolic Computation, solutions for a variable-coefficient nonlinear Schr\"odinger equation are obtained theoretically. It is found that the type of the solutions changes with the different choices of the group-velocity dispersion coefficient ${\ensuremath{\beta}}_{2}(z)$. According to those solutions, the relevant properties and features of physical and optical interest are illustrated. In addition, an effective technique for controlling the shape of the pulses is presented. The results of this paper will be valuable to the study of the future development of ultrahigh rate and long-distance optical communication systems.

  • integrable aspects and applications of a generalized inhomogeneous n coupled nonlinear schrodinger system in plasmas and optical fibers via Symbolic Computation
    Physics Letters A, 2008
    Co-Authors: Haiqiang Zhang, Bo Tian, Yitian Gao, Yaxing Zhang
    Abstract:

    Abstract For describing the general behavior of N fields propagating in inhomogeneous plasmas and optical fibers, a generalized N-coupled nonlinear Schrodinger system is investigated with Symbolic Computation in this Letter. When the coefficient functions obey the Painleve-integrable conditions, the ( N + 1 ) × ( N + 1 ) nonisospectral Lax pair associated with such a model is derived by means of the Ablowitz–Kaup–Newell–Segur formalism. Furthermore, the Darboux transformation is constructed so that it becomes exercisable to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some exact analytic one- and two-soliton solutions, our discussions are focused on the envelope soliton excitation in time-dependent inhomogeneous plasmas and the optical pulse propagation with the constant (or distance-related) fiber gain/loss and phase modulation.

  • integrability of an n coupled nonlinear schrodinger system for polarized optical waves in an isotropic medium via Symbolic Computation
    Physical Review E, 2008
    Co-Authors: Haiqiang Zhang, Bo Tian
    Abstract:

    Considering the simultaneous propagation of multicomponent fields in an isotropic medium, an $N$-coupled nonlinear Schr\"odinger system with the self-phase modulation, cross-phase modulation, and energy exchange terms is investigated in this paper. First, via Symbolic Computation, the Painlev\'e singularity structure analysis shows that such a system admits the Painlev\'e property. Then, with the Ablowitz-Kaup-Newell-Segur scheme, the linear eigenvalue problem (Lax pair) associated with this model is constructed in the frame of the block matrices. With the Hirota bilinear method, the bright one- and two-soliton solutions of this system are presented. In addition, the bright multisoliton solutions of the system for $N=2$ are straightforwardly derived by the linear superposition of soliton solutions of two independent scalar nonlinear Schr\"odinger equations. Furthermore, through the analysis for the soliton solutions, the corresponding propagation behavior and applications for soliton pulses in nonlinear optical fibers are considered. Finally, three significant conserved quantities, i.e., energy, momentum, and Hamiltonian, are also given.

Yitian Gao - One of the best experts on this subject based on the ideXlab platform.

  • integrable aspects and applications of a generalized inhomogeneous n coupled nonlinear schrodinger system in plasmas and optical fibers via Symbolic Computation
    Physics Letters A, 2008
    Co-Authors: Haiqiang Zhang, Bo Tian, Yitian Gao, Yaxing Zhang
    Abstract:

    Abstract For describing the general behavior of N fields propagating in inhomogeneous plasmas and optical fibers, a generalized N-coupled nonlinear Schrodinger system is investigated with Symbolic Computation in this Letter. When the coefficient functions obey the Painleve-integrable conditions, the ( N + 1 ) × ( N + 1 ) nonisospectral Lax pair associated with such a model is derived by means of the Ablowitz–Kaup–Newell–Segur formalism. Furthermore, the Darboux transformation is constructed so that it becomes exercisable to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some exact analytic one- and two-soliton solutions, our discussions are focused on the envelope soliton excitation in time-dependent inhomogeneous plasmas and the optical pulse propagation with the constant (or distance-related) fiber gain/loss and phase modulation.

  • Symbolic Computation on cylindrical modified dust ion acoustic nebulons in dusty plasmas
    Physics Letters A, 2007
    Co-Authors: Bo Tian, Yitian Gao
    Abstract:

    Abstract In this Letter, for the dust-ion-acoustic waves with azimuthal perturbation in a dusty plasma, a cylindrical modified Kadomtsev–Petviashvili (CMKP) model is constructed by virtue of Symbolic Computation, with three families of exact analytic solutions obtained as well. Dark and bright CMKP nebulons are investigated with pictures and related to such dusty-plasma environments as the supernova shells and Saturn's F-ring. Difference of the CMKP nebulons from other known nebulons is also analyzed, and possibly-observable CMKP-nebulonic effects for the future plasma experiments are proposed, especially those on the possible notch/slot and dark-bright bi-existence.

  • 3 1 dimensional generalized johnson model for cosmic dust ion acoustic nebulons with Symbolic Computation
    Physics of Plasmas, 2006
    Co-Authors: Yitian Gao, Bo Tian
    Abstract:

    In a cosmic dusty plasma, both azimuthal and height perturbations of a nonplanar cylindrical geometry are considered. For dust-ion-acoustic waves and with Symbolic Computation, (3+1)-dimensional generalized Johnson [(3+1)DGJ] model is derived and analytic solutions are constructed. Supernova-shell-typed expanding bright (3+1)DGJ nebulons and Saturn-F-ring-type expanding dark (3+1)DGJ nebulons are both pictured and discussed. Essential difference of this letter from the existing literature is pointed out, with the relevant, possibly observable (3+1)DGJ-nebulonic structures for the future cosmic experiments proposed.

  • variable coefficient higher order nonlinear schrodinger model in optical fibers new transformation with burstons brightons and Symbolic Computation
    Physics Letters A, 2006
    Co-Authors: Bo Tian, Yitian Gao
    Abstract:

    In a realistic fiber of weakly dispersive and nonlinear dielectrics with distributed parameters, a variable-coefficient higher-order nonlinear Schrodinger (vcHNLS) model can be used to describe the femtosecond pulse propagation, applicable to, e.g., the design of ultrafast signal-routing and dispersion-managed fiber-transmission systems. In this Letter, new transformation is proposed, by virtue of Symbolic Computation, from a vcHNLS model to its known constant-coefficient counterpart without amplification/absorption. Features of the transformation are analyzed, and constraints on the variable coefficients are presented. Such physically/optically interesting examples as the variable-coefficient burstons and brightons are constructed in explicit forms with their properties discussed. Burstons and brightons are potentially observable with future optical-fiber experiments.

  • transformations for a generalized variable coefficient korteweg de vries model from blood vessels bose einstein condensates rods and positons with Symbolic Computation
    Physics Letters A, 2006
    Co-Authors: Bo Tian, Chunyi Zhang, Yitian Gao, Wenrui Shan, Guangmei Wei
    Abstract:

    Abstract The variable-coefficient Korteweg–de Vries (KdV)-typed models, although often hard to be studied, are of current interest in describing various real situations. Under investigation hereby is a large class of the generalized variable-coefficient KdV models with external-force and perturbed/dissipative terms. Recent examples of this class include those in blood vessels and circulatory system, arterial dynamics, trapped Bose–Einstein condensates related to matter waves and nonlinear atom optics, Bose gas of impenetrable bosons with longitudinal confinement, rods of compressible hyperelastic material and semiconductor heterostructures with positonic phenomena. In this Letter, based on Symbolic Computation, four transformations are proposed from this class either to the cylindrical or standard KdV equation when the respective constraint holds. The constraints have nothing to do with the external-force term. Under those transformations, such analytic solutions as those with the Airy, Hermit and Jacobian elliptic functions can be obtained, including the solitonic profiles. The roles for the perturbed and external-force terms to play are observed and discussed. Investigations on this class can be performed through the properties of solutions of cylindrical and standard KdV equations.

Haiqiang Zhang - One of the best experts on this subject based on the ideXlab platform.

Chunyi Zhang - One of the best experts on this subject based on the ideXlab platform.

  • multi soliton solutions and a backlund transformation for a generalized variable coefficient higher order nonlinear schrodinger equation with Symbolic Computation
    Physica A-statistical Mechanics and Its Applications, 2008
    Co-Authors: Xiang-hua Meng, Wenjun Liu, Hongwu Zhu, Chunyi Zhang, Bo Tian
    Abstract:

    In this paper, by virtue of Symbolic Computation, the investigation is made on a generalized variable-coefficient higher-order nonlinear Schrodinger equation with varying higher-order effects and gain or loss, which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide. A modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable-coefficient bilinear form. Based on the formal parameter expansion technique, the multi-soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints. A Backlund transformation in bilinear form is also obtained for the first time in this paper. Finally, discussions on the analytic soliton solutions are given and various propagation situations are illustrated.

  • Symbolic Computation construction of transformations for a more generalized nonlinear schrodinger equation with applications in inhomogeneous plasmas optical fibers viscous fluids and bose einstein condensates
    European Physical Journal B, 2007
    Co-Authors: Tao Xu, Chunyi Zhang, Xiang-hua Meng, Juan Li, Bo Tian
    Abstract:

    Currently, the variable-coefficient nonlinear Schrodinger (NLS)-typed models have attracted considerable attention in such fields as plasma physics, nonlinear optics, arterial mechanics and Bose-Einstein condensates. Motivated by the recent work of Tian et al. [Eur. Phys. J. B 47, 329 (2005)], this paper is devoted to finding all the cases for a more generalized NLS equation with time- and space-dependent coefficients to be mapped onto the standard one. With the computerized Symbolic Computation, three transformations and relevant constraint conditions on the coefficient functions are obtained, which turn out to be more general than those previously published in the literature. Via these transformations, the Lax pairs are also derived under the corresponding conditions. For physical applications, our transformations provide the feasibility for more currently-important inhomogeneous NLS models to be transformed into the homogeneous one. Applications of those transformations to several example models are illustrated and some soliton-like solutions are also graphically discussed. Copyright EDP Sciences/Societa Italiana di Fisica/Springer-Verlag 2007

  • Symbolic Computation construction of transformations for a more generalized nonlinear schrodinger equation with applications in inhomogeneous plasmas optical fibers viscous fluids and bose einstein condensates
    European Physical Journal B, 2007
    Co-Authors: Chunyi Zhang, Xiang-hua Meng, Guangmei Wei, Bo Tian
    Abstract:

    Currently, the variable-coefficient nonlinear Schrodinger (NLS)-typed models have attracted considerable attention in such fields as plasma physics, nonlinear optics, arterial mechanics and Bose-Einstein condensates. Motivated by the recent work of Tian et al. [Eur. Phys. J. B 47, 329 (2005)], this paper is devoted to finding all the cases for a more generalized NLS equation with time- and space-dependent coefficients to be mapped onto the standard one. With the computerized Symbolic Computation, three transformations and relevant constraint conditions on the coefficient functions are obtained, which turn out to be more general than those previously published in the literature. Via these transformations, the Lax pairs are also derived under the corresponding conditions. For physical applications, our transformations provide the feasibility for more currently-important inhomogeneous NLS models to be transformed into the homogeneous one. Applications of those transformations to several example models are illustrated and some soliton-like solutions are also graphically discussed.

  • Symbolic Computation on generalized hopf cole transformation for a forced burgers model with variable coefficients from fluid dynamics
    Wave Motion, 2007
    Co-Authors: Chunyi Zhang, Hongwu Zhu, Xiang-hua Meng, Bo Tian
    Abstract:

    Abstract Considering the inhomogeneities of media, nonuniformities of boundaries and external forces, a forced Burgers model with space- and time-dependent coefficients is hereby investigated. In this paper, we perform Symbolic Computation and construct the generalized Hopf–Cole transformation from such a model to the standard heat equation with the relevant constraint conditions on the variable-coefficient and external-force functions. Physically speaking, this transformation provides the feasibility of linearizing many forced and/or variable-coefficient Burgers models from various branches of physics. Specially, we present the N -shock-wave-like solution, based on which the coalescence structures of shock waves with inhomogeneous and forcing effects are discussed, and possible applications in some fields are also pointed out. In like manner, we can also generalize the Hopf–Cole transformation to bilinearize many other variable-coefficient nonlinear evolution equations.

  • inelastic interaction and non traveling wave effects for two multi dimensional burgers models from fluid dynamics and astrophysics with Symbolic Computation
    Zeitschrift für Naturforschung A, 2006
    Co-Authors: Tao Xu, Chunyi Zhang, Haiqiang Zhang, Juan Li, Lili Li, Bo Tian
    Abstract:

    Describing the surface perturbations of a shallow viscous fluid, cosmic-ray-modified shock structures and electromagnetic waves in a saturated ferrite, the (2+1)- and (3+1)-dimensional Burgers equations are investigated in this paper. In view of the higher space dimensionality, the transformations from such two models to a (1+1)-dimensional Burgers equation are constructed with Symbolic Computation. Via the obtained transformations, three families of multi-dimensional N-shock-wave-like solutions are specially presented, which recover some previously published solutions. The inelastically interacting properties and some non-traveling-wave effects of shock waves are discussed through the figures for several sample solutions. Additionally, possible applications for those solutions and effects in some fields are also pointed out.

Wenrui Shan - One of the best experts on this subject based on the ideXlab platform.

  • water wave Symbolic Computation for the earth enceladus and titan the higher order boussinesq burgers system auto and non auto backlund transformations
    Applied Mathematics Letters, 2020
    Co-Authors: Xinyi Gao, Yongjiang Guo, Wenrui Shan
    Abstract:

    Abstract In the Solar System, water and water waves are commonly seen: For the Earth, water is “at the core of sustainable development” and “at the heart of adaptation to climate change”; For the Enceladus, Cassini spacecraft discovers a possible global ocean of liquid water beneath an icy crust; For the Titan, Cassini spacecraft suggests an icy shell floating atop a global ocean. Shallow water waves near the ocean beaches or in the lakes can be described by the Boussinesq-Burgers-type equations. In this Letter, on the higher-order Boussinesq-Burgers system, Symbolic Computation helps us to go from the two-dimensional Bell polynomials to construct two non-auto-Backlund transformations and to proceed from the Painleve-Backlund format to obtain four auto-Backlund transformations with some soliton solutions. All of our results are shown to be dependent on the constant coefficient in the system.

  • application of exp function method to the whitham broer kaup shallow water model using Symbolic Computation
    Applied Mathematics and Computation, 2009
    Co-Authors: Zhi Zheng, Wenrui Shan
    Abstract:

    In this letter, the Exp-function method is applied to the Whitham-Broer-Kaup shallow water model. With the help of Symbolic Computation, several kinds of new solitary wave solutions are formally derived.

  • transformations for a generalized variable coefficient korteweg de vries model from blood vessels bose einstein condensates rods and positons with Symbolic Computation
    Physics Letters A, 2006
    Co-Authors: Bo Tian, Chunyi Zhang, Yitian Gao, Wenrui Shan, Guangmei Wei
    Abstract:

    Abstract The variable-coefficient Korteweg–de Vries (KdV)-typed models, although often hard to be studied, are of current interest in describing various real situations. Under investigation hereby is a large class of the generalized variable-coefficient KdV models with external-force and perturbed/dissipative terms. Recent examples of this class include those in blood vessels and circulatory system, arterial dynamics, trapped Bose–Einstein condensates related to matter waves and nonlinear atom optics, Bose gas of impenetrable bosons with longitudinal confinement, rods of compressible hyperelastic material and semiconductor heterostructures with positonic phenomena. In this Letter, based on Symbolic Computation, four transformations are proposed from this class either to the cylindrical or standard KdV equation when the respective constraint holds. The constraints have nothing to do with the external-force term. Under those transformations, such analytic solutions as those with the Airy, Hermit and Jacobian elliptic functions can be obtained, including the solitonic profiles. The roles for the perturbed and external-force terms to play are observed and discussed. Investigations on this class can be performed through the properties of solutions of cylindrical and standard KdV equations.

  • transformations for a generalized variable coefficient nonlinear schrodinger model from plasma physics arterial mechanics and optical fibers with Symbolic Computation
    European Physical Journal B, 2005
    Co-Authors: Bo Tian, Chunyi Zhang, Yitian Gao, Wenrui Shan, Guangmei Wei
    Abstract:

    Describing space and laboratory plasmas, arterial mechanics and optical fibers, a generalized variable-coefficient nonlinear Schrodinger model is hereby under investigation. Four transformations have been constructed from such a model to the known standard and cylindrical nonlinear Schrodinger equations with the relevant constraints on the variable coefficients presented. Symbolic Computation is performed. Specialities of those transformations are discussed. Analytic solutions of such a generalized variable-coefficient model can be obtained via those transformations from the analytic solutions of the standard and cylindrical ones.

  • transformations for a generalized variable coefficient nonlinear schrodinger model from plasma physics arterial mechanics and optical fibers with Symbolic Computation
    European Physical Journal B, 2005
    Co-Authors: Bo Tian, Chunyi Zhang, Yitian Gao, Wenrui Shan, Guangmei Wei
    Abstract:

    Describing space and laboratory plasmas, arterial mechanics and optical fibers, a generalized variable-coefficient nonlinear Schrodinger model is hereby under investigation. Four transformations have been constructed from such a model to the known standard and cylindrical nonlinear Schrodinger equations with the relevant constraints on the variable coefficients presented. Symbolic Computation is performed. Specialities of those transformations are discussed. Analytic solutions of such a generalized variable-coefficient model can be obtained via those transformations from the analytic solutions of the standard and cylindrical ones. Copyright EDP Sciences/Societa Italiana di Fisica/Springer-Verlag 2005

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