Symbolic Computation of Solitary Wave Solutions and Solitons Through Homogenization of Degree

Springer proceedings in mathematics & statistics · 2024 · 1 citations

• Mathematics
• Physics
• Mathematical analysis

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

Mathematical and Computational Applications · 2024 · 4 citations

• Mathematical physics
• Physics
• Classical mechanics

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

Preprints.org · 2024-07-30

Using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations thereby establishing their complete integrability. The Gardner equation is chosen as the key example for it comprises both the Korteweg-de Vries and modified Korteweg de Vries (mKdV) equations. The Gardner and Miura transformations which connect these equations are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case) whereas the focusing Gardner equation has the standard elastically colliding solitons. The paper’s aim is to provide a review of integrability properties and solutions of the Gardner equation and illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica but can be adapted for major computer algebra systems.

Symbolic computation of solitary wave solutions and solitons through homogenization of degree

arXiv (Cornell University) · 2023 · 1 citations

• Mathematics
• Mathematical analysis
• Applied mathematics

A simplified version of Hirota's method for the computation of solitary waves and solitons of nonlinear PDEs is presented. A change of dependent variable transforms the PDE into an equation that is homogeneous of degree. Solitons are then computed using a perturbation-like scheme involving linear and nonlinear operators in a finite number of steps. The method is applied to a class of fifth-order KdV equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt. The method works for non-quadratic homogeneous equations for which the bilinear form might not be known. Furthermore, homogenization of degree allows one to compute solitary wave solutions of nonlinear PDEs that do not have solitons. Examples include the Fisher and FitzHugh-Nagumo equations, and a combined KdV-Burgers equation. When applied to a wave equation with a cubic source term, one gets a bi-soliton solution describing the coalescence of two wavefronts. The method is largely algorithmic and is implemented in Mathematica.

Application of perturbation–iteration method to Lotka–Volterra equations

Alexandria Engineering Journal · 2016-03-05 · 8 citations

Perturbation–iteration method is generalized for systems of first order differential equations. Approximate solutions of Lotka–Volterra systems are obtained using the method. Comparisons of our results with each other and with numerical solutions are given. The method is implemented in Mathematica, a major computer algebra system. The package PerturbationIteration.m automatically carries out the tedious calculations of the method.

Application of Dempster-Schafer Method in Family-Based Association Studies

IEEE/ACM Transactions on Computational Biology and Bioinformatics · 2013-07-01

In experiments designed for family-based association studies, methods such as transmission disequilibrium test require large number of trios to identify single-nucleotide polymorphisms associated with the disease. However, unavailability of a large number of trios is the Achilles' heel of many complex diseases, especially for late-onset diseases. In this paper, we propose a novel approach to this problem by means of the Dempster-Shafer method. The simulation studies show that the Dempster-Shafer method has a promising overall performance, in identifying single-nucleotide polymorphisms in the correct association class, as it has 90 percent accuracy even with 60 trios.

Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential–Difference Equations

2011-09-24 · 16 citations

Scaling invariant Lax pairs of nonlinear evolution equations

Applicable Analysis · 2011-11-01 · 20 citations

Abstract A completely integrable nonlinear partial differential equation (PDE) can be associated with a system of linear PDEs in an auxiliary function whose compatibility requires that the original PDE is satisfied. This associated system is called a Lax pair. Two equivalent representations are presented. The first uses a pair of differential operators which leads to a higher order linear system for the auxiliary function. The second uses a pair of matrices which leads to a first-order linear system. In this article, we present a method, which is easily implemented in MAPLE or MATHEMATICA, to compute an operator Lax pair for a set of PDEs. In the operator representation, the determining equations for the Lax pair split into a set of kinematic constraints which are independent of the original equation and a set of dynamical equations which depend on it. The kinematic constraints can be solved generically. We assume that the operators have a scaling symmetry. The dynamical equations are then reduced to a set of nonlinear algebraic equations. This approach is illustrated with well-known examples from soliton theory. In particular, it is applied to a three parameter class of fifth-order Korteweg–de Vries (KdV)-like evolution equations which includes the Lax fifth-order KdV, Sawada-Kotera and Kaup–Kuperschmidt equations. A second Lax pair was found for the Sawada–Kotera equation. Keywords: Lax pairLax operatorscaling symmetrycomplete integrabilityfifth-order KdV-type equationsAMS Subject Classifications:: Primary: 37J3537K4035Q51Secondary: 68W3047J3570H06 Acknowledgements This material is based in part upon research supported by the National Science Foundation (NSF) under Grant No. CCF-0830783. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF. WH is grateful for the hospitality and support of the Department of Computer Engineering at Turgut Özal University (Keçiören, Ankara, Turkey) where code for Lax pair computations was further developed. MH thanks the Department of Applied Mathematics and Statistics, Colorado School of Mines for their hospitality while this work was completed. Undergraduate students Oscar Aguilar, Sara Clifton, William 'Tony' McCollom, and graduate student Jacob Rezac are thanked for their help with this project.

Symbolic Computation of Recursion Operators for Nonlinear Differential-Difference Equations

Mathematical and Computational Applications · 2011-04-01 · 2 citations

An algorithm for the symbolic computation of recursion operators for systems of nonlinear differential-difference equations (DDEs) is presented. Recursion operators allow one to generate an infinite sequence of generalized symmetries. The existence of a recursion operator therefore guarantees the complete integrability of the DDE. The algorithm is based in part on the concept of dilation invariance and uses our earlier algorithms for the symbolic computation of conservation laws and generalized symmetries. The algorithm has been applied to a number of well-known DDEs, including the Kacvan Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which recursion operators are shown. The algorithm has been implemented in Mathematica, a leading computer algebra system. The package DDERecursionOperator.m is briefly discussed.

Symbolic Computation of Recursion Operators for Nonlinear\n Differential-Difference equations

arXiv (Cornell University) · 2011-04-20

An algorithm for the symbolic computation of recursion operators for systems\nof nonlinear differential-difference equations (DDEs) is presented. Recursion\noperators allow one to generate an infinite sequence of generalized symmetries.\nThe existence of a recursion operator therefore guarantees the complete\nintegrability of the DDE. The algo-rithm is based in part on the concept of\ndilation invariance and uses our earlier algorithms for the symbolic\ncomputation of conservation laws and generalized symmetries.\n The algorithm has been applied to a number of well-known DDEs, including the\nKac-van Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which\nrecursion opera-tors are shown. The algorithm has been implemented in\nMathematica, a leading com-puter algebra system. The package\nDDERecursionOperator.m is briefly discussed.\n