Journal ArticleDOI
Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg-de Vries equation
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TLDR
In this paper, an implicit cusp solitary-wave solution is constructed via a simple direct method, and the existing connections between the Harry Dym and the Korteweg-de Vries equations are uniformised and simplified, and transformations between their respective solutions are carried out explicitly.Abstract:
The Harry Dym equation, which is related to the classical string problem, is derived in three different ways. An implicit cusp solitary-wave solution is constructed via a simple direct method. The existing connections between the Harry Dym and the Korteweg-de Vries equations are uniformised and simplified, and transformations between their respective solutions are carried out explicitly. Whenever possible, physical insights are provided.read more
Citations
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Journal ArticleDOI
Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA
Willy Hereman,M Takaoka +1 more
TL;DR: The direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized in this paper, where the class of solitary wave solutions is extended to analytic functions of the real exponential solutions of the linearized equation Expanding the solutions in an infinite series in these real exponentials, an exact solution of the nonlinear PDE is obtained, whenever the series can be summed.
Journal ArticleDOI
Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann-Liouville derivative
Qing Huang,Renat Zhdanov +1 more
TL;DR: In this paper, group analysis of the time fractional Harry-Dym equation with Riemann-Liouville derivative is performed and the maximal symmetry group in Lie's sense and the corresponding optimal system of subgroups are determined.
Journal ArticleDOI
delta -dressing and exact solutions for the (2+1)-dimensional Harry Dym equation
TL;DR: In this paper, the delta-dressing method was used to construct exact implicit solutions with functional parameters of the (2+1)-dimensional Harry Dym (HD) equation.
Journal ArticleDOI
Symbolic Software for the Painleve Test of Nonlinear Ordinary and Partial Differential Equations
Douglas Baldwin,Willy Hereman +1 more
TL;DR: The PainleveTest.m as mentioned in this paper is a Mathematica package that allows for the testing of polynomial systems of nonlinear ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants).
Journal ArticleDOI
Finite-gap solutions of the Harry Dym equation
TL;DR: In this article, the approach for multisoliton integration of the Harry Dym equation is generalized to the finite-gap case and N-gap solutions of the system are constructed and their degeneracy to the N-soliton case is considered.
References
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Linear and Nonlinear Waves
TL;DR: In this paper, a general overview of the nonlinear theory of water wave dynamics is presented, including the Wave Equation, the Wave Hierarchies, and the Variational Method of Wave Dispersion.
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Applications of Lie Groups to Differential Equations
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
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Linear and Nonlinear Waves
G. B. Whitham,T. C. T. Ting +1 more
TL;DR: In this paper, a reference record was created on 2005-11-18, modified on 2016-08-08 and used for the purpose of ondes ; chocs ; onde de : choc reference record.
Journal ArticleDOI
A Simple model of the integrable Hamiltonian equation
TL;DR: In this paper, a method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested, based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equation with their symmetries by using symplectic operators.
Book
Transformation Groups Applied to Mathematical Physics
TL;DR: In this paper, the authors define the notion of invariant majorants and define an invariant transformation of a polytropic gas with respect to a group of motions, and show how to compute invariant transformations in the presence of invariants.