Symbolic calculation in chemistry: Selected examples

Barnett, Mike; Capitani, Joseph F.; Gathen, Joachim von zur; Gerhard, Jürgen

doi:10.1002/qua.20097

Cited by 247 publications

(177 citation statements)

References 294 publications

(308 reference statements)

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“…Note that the second and third density-flux pairs follow from the first pair via the cyclic permutations in (8). If we select W (u) = W (v) = W (w) = a = 1 4 , then the first three densities have rank 1 4 and ρ (4) , ρ (5) , and ρ (6) have ranks 1 2 , 1, 5 2 , respectively.…”

Section: Conservation Lawsmentioning

confidence: 99%

Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions

Poole

Hereman

2005

Int J of Quantum Chemistry

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A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead, it is solely based on calculus, variational calculus, and linear algebra.Densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative (Euler operator) is used to compute the undetermined coefficients. The homotopy operator is used to compute the fluxes.The method is illustrated with nonlinear PDEs describing wave phenomena in fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters, the method determines the conditions on the parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for complete integrability. The method is algorithmic, applicable to a variety of PDEs, and can be implemented in computer algebra systems such as Mathematica, Maple, and REDUCE.

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Section: Conservation Lawsmentioning

confidence: 99%

Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions

Poole

Hereman

2005

Int J of Quantum Chemistry

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“…Additionally, (1) can be also used to characterize other physical phenomena, such as the nonlinear lattice waves [6], ion sound waves in plasmas [7], vibrations in a nonlinear string [8], and percolation of water in the porous subsurface of a horizontal layer of material [9]. With diverse dispersion and nonlinear effects taken into account, several generalized and variant versions of (1) have also been derived [10,11].…”

Section: Introductionmentioning

confidence: 99%

Odd-Soliton-Like Solutions for the Variable-Coefficient Variant Boussinesq Model in the Long Gravity Waves

Wang

Gao

Gai

2010

Zeitschrift Für Naturforschung A

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Under investigation in this paper, with symbolic computation, is a variable-coefficient variant Boussinesq (vcvB) model for the nonlinear and dispersive long gravity waves travelling in two horizontal directions with varying depth. Connection between the vcvB model and a variable-coefficient Broer-Kaup (vcBK) system is revealed under certain constraints. By means of the N-fold Darboux transformation for the vcBK system, odd-soliton-like solutions in terms of the Vandermonde-like determinant for the vcvB model are derived. Dynamics of those solutions is analyzed graphically, on the three-parallel solitonic waves, head-on collisions, double structures, and inelastic interactions. It is reported that the shapes of the soliton-like waves and separation distance between them depend on the spectrum parameters and the variable coefficients affect the velocities of the waves. Our results could be helpful in interpreting certain nonlinear wave phenomena in fluid dynamics.

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“…In system (2), because the ratio between the parameters of self-phase modulation (SPM) and crossphase modulation (XPM) is considered to be equal, this type of integrable coupled NLS system is physically valid only for the pulse propagation in a special kind of optical fiber system [19]. But for wavelength division multiplexing in single-model fibers and for the propagations of linearly polarized pulses in birefringent fibers, the ratio between SPM and XPM in the integrable system, e. g., system (4), has to be 1/2 [14].…”

Section: Introductionmentioning

confidence: 99%

Soliton Solutions, Pairwise Collisions and Partially Coherent Interactions for A Generalized (1+1)-Dimensional Coupled Nonlinear Schrödinger System via Symbolic Computation

Tian

Zhang

et al. 2008

Zeitschrift Für Naturforschung A

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With the aid of symbolic computation, we investigate a generalized (1+1)-dimensional coupled nonlinear Schrödinger system with mixed nonlinear interactions, which has potential applications in nonlinear optics and elastic solids. The exact analytical one-, two-, and three-soliton solutions are firstly obtained by employing the bilinear method under two constraints. Some main propagation and interaction properties of the solitons are discussed simultaneously. Moreover, some figures are plotted to graphically analyze the pairwise collisions and partially coherent interactions of three solitons.

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Symbolic calculation in chemistry: Selected examples

Abstract: A selective classified bibliography of symbolic computation in some areas of chemistry is provided together with some examples of computer algebra algorithms and techniques to facilitate future joint work of chemists and computer scientists.

Cited by 247 publications

References 294 publications

Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions

Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions

Odd-Soliton-Like Solutions for the Variable-Coefficient Variant Boussinesq Model in the Long Gravity Waves

Soliton Solutions, Pairwise Collisions and Partially Coherent Interactions for A Generalized (1+1)-Dimensional Coupled Nonlinear Schrödinger System via Symbolic Computation

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