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Unal Goktas

Publications and source records attributed to Unal Goktas.

10 recordsLinked to original sources
arXiv2023

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

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.

nlin.SI↗
arXiv2011

Scaling Invariant Lax Pairs of Nonlinear Evolution Equations

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 paper 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 do 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 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.

nlin.SI↗
arXiv2003

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Parallels are drawn through discussion and example to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.

nlin.SI↗
arXiv1998

Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations

Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlevé test, which is applicable to polynomial systems of ordinary and partial differential equations. The second and third algorithms allow one to explicitly compute polynomial conserved densities and higher-order symmetries of nonlinear evolution and lattice equations. The first algorithm is implemented in the symbolic syntax of both Macsyma and Mathematica. The second and third algorithms are available in Mathematica. The codes can be used for computer-aided integrability testing of nonlinear differential and lattice equations as they occur in various branches of the sciences and engineering. Applied to systems with parameters, the codes can determine the conditions on the parameters so that the systems pass the Painlevé test, or admit a sequence of conserved densities or higher-order symmetries.

solv-int↗
arXiv1998

Computation of Higher-order Symmetries for Nonlinear Evolution and Lattice Equations

A straightforward algorithm for the symbolic computation of higher-order symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the higher-order symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first-order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi-discrete lattice equations. With our Integrability Package, higher-order symmetries are obtained for several well-known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of higher-order symmetries exist. The existence of a sequence of such symmetries is a predictor for integrability.

solv-int↗
arXiv1998

Invariants and Symmetries for Partial Differential Equations and Lattices

Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as Mathematica. Invariants and symmetries are shown for several well-known equations. Our Mathematica package allows one to automatically compute invariants and symmetries. Applied to systems with parameters, the package determines the conditions on these parameters so that a sequence of invariants or symmetries exists. The software can thus be used to test the integrability of model equations for wave phenomena.

solv-int↗
arXiv1998

Computation of conservation laws for nonlinear lattices

An algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented. The algorithm is implemented in Mathematica and can be used as an automated integrability test. With the code diffdens.m, conserved densities are obtained for several well-known lattice equations. For systems with parameters, the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist.

solv-int↗
arXiv1997

Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations

A new algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented. The algorithm is implemented in Mathematica. The program condens.m automatically carries out the lengthy symbolic computations for the construction of conserved densities. The code is tested on several well-known partial differential equations from soliton theory. For systems with parameters, condens.m can be used to determine the conditions on these parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for integrability of the system.

solv-int↗