arXiv subjects

Willy Hereman

Publications and source records attributed to Willy Hereman.

At least 19 recordsLinked to original sources

arXiv2025

Conservation laws of nonlinear PDEs arising in elasticity and acoustics in Cartesian, cylindrical, and spherical geometries

Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave propagation in a circular cylinder and a cylindrical annulus. Next, using the multiplier method, conservation laws are derived for a parameterized system of constitutive equations in cylindrical coordinates involving a general expression for the Cauchy stress. Conservation laws for the Khokhlov-Zabolotskaya-Kuznetsov equation and Westervelt-type equations in various coordinate systems are also presented.

math.AP↗

arXiv2024

Lie symmetries, closed-form solutions, and conservation laws of a constitutive equation modeling stress in elastic materials

The Lie-point symmetry method is used to find some closed-form solutions for a constitutive equation modeling stress in elastic materials. The partial differential equation (PDE), which involves a power law with arbitrary exponent n, was investigated by Mason and his collaborators (Magan et al., Wave Motion, 77, 156-185, 2018). The Lie algebra for the model is five-dimensional for the shearing exponent n > 0, and it includes translations in time, space, and displacement, as well as time-dependent changes in displacement and a scaling symmetry. Applying Lie's symmetry method, we compute the optimal system of one-dimensional subalgebras. Using the subalgebras, several reductions and closed-form solutions for the model are obtained both for general exponent n and special case n = 1. Furthermore, it is shown that for general n > 0 the model has interesting conservation laws which are computed with symbolic software using the scaling symmetry of the given PDE.

nlin.SI↗

arXiv2024

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

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.

nlin.SI↗

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↗

arXiv2019

Lax Pairs for Edge-constrained Boussinesq Systems of Partial Difference Equations

The method due to Nijhoff and Bobenko & Suris to derive Lax pairs for partial difference equations (PDeltaEs) is applied to edge constrained Boussinesq systems. These systems are defined on a quadrilateral. They are consistent around the cube but they contain equations defined on the edges of the quadrilateral. By properly incorporating the edge equations into the algorithm, it is straightforward to derive Lax matrices of minimal size. The 3 by 3 Lax matrices thus obtained are not unique but shown to be gauge-equivalent. The gauge matrices connecting the various Lax matrices are presented. It is also shown that each of the Boussinesq systems admits a 4 by 4 Lax matrix. For each system, the gauge-like transformations between Lax matrices of different sizes are explicitly given. To illustrate the analogy between continuous and lattice systems, the concept of gauge-equivalence of Lax pairs of nonlinear partial differential equations is briefly discussed. The method to find Lax pairs of PDeltaEs is algorithmic and is being implemented in Mathematica. The Lax pair computations for this chapter helped further improve and extend the capabilities of the software under development.

nlin.SI↗

arXiv2013

Shallow Water Waves and Solitary Waves

Encyclopedic article covering shallow water wave models used in oceanography and atmospheric science. Sections: Definition of the Subject; Introduction and Historical Perspective; Completely Integrable Shallow Water Wave Equations; Shallow Water Wave Equations of Geophysical Fluid Dynamics; Computation of Solitary Wave Solutions; Numerical Methods; Water Wave Experiments and Observations; Future Directions, and Bibliography.

nlin.SI↗

arXiv2013

The Korteweg-de Vries Equation

Two page encyclopedic article about the Korteweg-de Vries equation covering historical perspective, solitary wave and periodic solutions, modern developments, properties and applications, and further reading.

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↗

arXiv2011

Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov-Kuznetsov and Kadomtsev-Petviashvili equations as examples. The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations. The code ConservationLawsMD.m has been applied to (2+1)-dimensional versions of the Sawada-Kotera, Camassa-Holm, and Gardner equations, and the multi-dimensional Khokhlov-Zabolotskaya equation.

nlin.SI↗

arXiv2011

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

Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in {\em Mathematica}. The codes {\sc InvariantsSymmetries.m} and {\sc DDERecursionOperator.m} can aid researchers interested in properties of nonlinear DDEs.

math-ph↗

arXiv2011

Symbolic Computation of Recursion Operators for Nonlinear Differential-Difference equations

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 algo-rithm 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 Kac-van Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which recursion opera-tors are shown. The algorithm has been implemented in Mathematica, a leading com-puter algebra system. The package DDERecursionOperator.m is briefly discussed.

cs.SC↗

arXiv2009

The homotopy operator method for symbolic integration by parts and inversion of divergences with applications

Using standard calculus, explicit formulas for one-, two- and three-dimensional homotopy operators are presented. A derivation of the one-dimensional homotopy operator is given. A similar methodology can be used to derive the multi-dimensional versions. The calculus-based formulas for the homotopy operators are easy to implement in computer algebra systems such as Mathematica, Maple, and REDUCE. Several examples illustrate the use, scope, and limitations of the homotopy operators. The homotopy operator can be applied to the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs). Conservation laws provide insight into the physical and mathematical properties of the PDE. For instance, the existence of infinitely many conservation laws establishes the complete integrability of a nonlinear PDE.

nlin.SI↗

arXiv2008

Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations

We present direct methods and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).The methods are applied to nonlinear PDEs in (1+1) dimensions with polynomial nonlinearities which include the Korteweg-de Vries (KdV), Boussinesq, and Drinfel'd-Sokolov-Wilson equations. An adaptation of the methods is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville equations. With respect to nonlinear DDEs, our methods are applied to Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. To overcome the shortcomings of the undetermined coefficients method, we designed a new direct method which uses leading order analysis. That method is applied to discretizations of the KdV and modified KdV equations, and a combination thereof. Additional examples include lattices due to Bogoyavlenskii, Belov-Chaltikian, and Blaszak-Marciniak. The undetermined coefficient methods for PDEs and DDEs have been implemented in Mathematica. The code "TransPDEDensityFlux.m" computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code "DDEDensityFlux.m" does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library "discrete" computes densities and fluxes of nonlinear DDEs.

nlin.SI↗

arXiv2006

Symbolic Software for the Painleve Test of Nonlinear Ordinary and Partial Differential Equations

The automation of the traditional Painleve test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants). Except where limited by memory, there is no restriction on the number of independent or dependent variables. The package is quite robust in determining all the possible dominant behaviors of the Laurent series solutions of the differential equation. The omission of valid dominant behaviors is a common problem in many implementations of the Painleve test, and these omissions often lead to erroneous results. Finally, our package is compared with the other available implementations of the Painleve test.

nlin.SI↗

arXiv2005

Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multi-dimensions

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.

nlin.SI↗

arXiv2005

Continuous and Discrete Homotopy Operators with Applications in Integrability Testing

We introduce calculus-based formulas for the continuous Euler and homotopy operators. The 1D continuous homotopy operator automates integration by parts on the jet space. Its 3D generalization allows one to invert the total divergence operator. As a practical application, we show how the operators can be used to symbolically compute local conservation laws of nonlinear systems of partial differential equations in multi-dimensions. By analogy to the continuous case, we also present concrete formulas for the discrete Euler and homotopy operators. Essentially, the discrete homotopy operator carries out summation by parts. We use it to algorithmically invert the forward difference operator. We apply the discrete operator to compute fluxes of differential-difference equations in (1+1) dimensions. Our calculus-based approach allows for a straightforward implementation of the operators in major computer algebra system, such as Mathematica and Maple. The symbolic algorithms for integration and summation by parts are illustrated with elementary examples. The algorithms to compute conservation laws are illustrated with nonlinear PDEs and their discretizations arising in fluid dynamics and mathematical physics.

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↗

arXiv2003

Symbolic algorithms for the Painleve test, special solutions, and recursion operators for nonlinear PDEs

This paper discusses the algorithms and implementations of three Mathematica packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs. The first package, PainleveTest.m, symbolically performs the Painleve integrability test. The second package, PDESpecialSolutions.m, computes exact solutions expressible in hyperbolic or elliptic functions. The third package, PDERecursionOperator.m, generates and tests recursion operators.

nlin.SI↗