Nonclassical symmetry reductions of the Boussinesq equation

Chaos, Solitons and Fractals

Volumn 5, Issue 12, 1995, Pages 2261-2301

  Clarkson, Peter A ^a  

^a UNIVERSITY OF EXETER   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646984319     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/0960-0779(94)E0099-B     Document Type: Article

References (274)

1. Similarity methods for differential equations
2. Symmetries and differential equations
3. Applications of Lie Groups to Differential Equations
4. The computer calculation of Lie point symmetries of large systems of differential equations
5. PDELIE: A partial differential equation solver: I
6. PDELIE: A partial differential equation solver: II
7. Isogroups of differential equations using algebraic computing
8. A triangulation algorithm which determines the Lie symmetry algebra of any system of PDEs
9. Algorithms for reducing a system of PDEs to standard form determining the dimension of its solution space and calculating its Taylor series solution
10. Finding abstract Lie symmetry algebras of differential equations without integrating determining equations
11. Lie symmetries of differential equations: A Mathematica program to determine Lie symmetries
12. MathSym: A Mathematica program for computing Lie symmetries
13. Lie: a PC program for the calculation of the Lie algebra of differential equations
14. Infinitesimal Symmetries: a Computational Approach
15. Interactive REDUCE programs for calculating classical, non-classical and Lie-Bäcklund symmetries of differential equations
16. Automatically determining symmetries of partial differential equations
17. Symmetries of differential equations: from Sophus Lie to computer algebra
18. An algorithm for determining the size of symmetry groups
19. Investigating symmetries and other analytical properties of ODEs with the computer algebra package CRACK
20. An efficiency improved program LIEPDE for determining Lie-symmetries of PDEs
21. Review of symbolic software for the computation of Lie symmetries of differential equations
22. The general similarity of the heat equation
23. Geometric approach to invariance groups and solution of partial differential equations
24. On the conditional symmetries of Levi and Winternitz
25. Nonclassical symmetry reduction: example of the Boussinesq equation
26. Symmetries of compatibility conditions for systems of differential equations
27. Method for solving the KdV equation
28. The construction of special solutions to partial differential equations
29. Group-invariant solutions of differential equations
30. On similarity solutions of the Boussinesq equation
31. Linearization of the Boussinesq equation and the modified Boussinesq equation
32. On similarity solutions of Boussinesq type equations
33. New similarity solutions of the Boussinesq equation
34. New similarity solutions for the modified Boussinesq equation
35. New similarity solutions and Painlevé analysis for the symmetric regularized long wave and the modified Benjamin-Bona-Mahoney equations
36. New exact solutions for the Boussinesq equation
37. Dimensional reductions and exact solutions of a generalized nonlinear Schrödinger equation
38. Nonclassical symmetry reductions of nonlinear partial differential equations
39. Nonclassical symmetry reductions and exact solutions of the Zabalotskaya-Khokhlov equation
40. Symmetry reductions of a generalized cylindrical nonlinear Schrödinger equation
41. New symmetry reductions and exact solutions of the Davey-Stewartson equation: I. Reductions to ordinary differential equations
42. New symmetry reductions and exact solutions of the Davey-Stewartson equation: II. Reductions to partial differential equations
43. Symmetry reductions, exact solutions and Painlevé analysis for a generalised Boussinesq equation
44. Nonclassical symmetry reductions and exact solutions of nonlinear reaction-diffusion equations
45. Symmetries of the nonlinear heat equation
46. Symmetry Reductions and Exact Solutions of a class of Nonlinear Heat Equations
47. Algorithms for the nonclassical method of symmetry reductions
48. On a shallow water wave equation
49. Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation
50. Conditional symmetry of the equations of Mathematical Physics
51. Symmetry and exact solutions of nonlinear spinor equations
52. Nonclassical Symmetry Reductions and Exact Solutions of Nonlinear Partial Differential Equations
53. Similarity solutions of the Kadomtsev-Petviashvili equation
54. A note on the new similarity reductions of the Boussinesq equation
55. Generalized Boussinesq equation and KdV equation—Painlevé properties, Bäcklund transformations and Lax pairs
56. Nonclassical symmetry reductions for the dispersive water waves
57. Some new solutions of the KP equation
58. Nonclassical analysis and Painlevé property for the Kuperschmidt equations
59. Similarity reductions of the KP equation by a direct method
60. Similarity reductions for variable coefficient coupled nonlinear Schrödinger equations
61. Nonclassical symmetries and Bäcklund transformations
62. Classical and nonclassical symmetries of the Helmholtz equation
63. The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation
64. Some symmetries of the nonlinear heat and wave equations
65. Similarity reductions of partial differential equations
66. On the weak symmetry groups of partial differential equations
67. The anisotropic Heisenberg spin chain and the nonlinear Schrödinger equation
68. Conditional symmetries and conditional integrability for nonlinear systems
69. Symmetry-based algorithms to relate partial differential equations I Local symmetries
70. When nonlinear differential equations are equivalent to linear differential equations
71. Equivalence Transformations for Classes of Differential Equations
72. Group theory and exact solutions of partially integrable differential systems
73. Lie groups and solutions of nonlinear partial differential equations
74. On new exact blow-up solutions for nonlinear heat conduction equations with source and applications
75. Quasilinear heat equations with first-order sign-invariants and new explicit solutions
76. Quasilinear heat equations with source: blow-up, localization, symmetry, exact solutions, asymptotics, structures
77. Optimal numerical algorithms
78. Transformation groups in net spaces
79. Invariante Variationsprobleme
80. Invariant variation problems
81. Bifurcation and symmetry breaking
82. Reduction and equivariant branching lemma: dynamical systems, evolution PDEs, and Gauge theories
83. Group-theoretic Methods in Bifurcation Theory
84. Symmetries in optical control
85. Controlled invariant distribuations and group invariance
86. Hamiltonian structure, symmetries and conservation laws for water waves
87. Nonlinear evolution equations and ordinary differential equations of Painlevé type
88. A connection between nonlinear evolution equations and ordinary differential equations of P-type I
89. A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation
90. The connection between partial differential equations soluble by inverse scattering and ordinary differential equations of Painlevé type
91. Solitons, Nonlinear Evolution Equations and Inverse Scattering
92. Lie transformations nonlinear evolution equations and Painlevé forms
93. The Painlevé property and singularity analysis of integrable and non-integrable systems
94. Resonantly coupled nonlinear evolution equations
95. 3u
96. The inverse problem for a general N × N spectral equation
97. Inverse scattering and the Boussinesq equation
98. On stochastization of one-dimensional chains of nonlinear oscillations
99. Théorie de l'intumescence appelée onde solitaire ou de translation se propagente dans un canal rectangulaire
100. Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblemant parielles de la surface au fond
101. The long-wave paradox in the theory of gravity waves
102. Studies of a nonlinear lattice
103. A synergetic approach to problems of nonlinear dispersive wave propagation and interaction
104. The application of Bäcklund transforms to physical problems
105. On a unified approach to transformations and elementary solutions of Painlevé equations
106. Rational solutions of Painlevé equations
107. Integral equations and exact solutions for the fourth Painlevé equation
108. Numerical studies of the fourth Painlevé equation
109. Lax pairs, Bäcklund transformations and special solutions for ordinary differential equations
110. One-parameter systems of solutions of Painlevé's equations
111. Theory of the fourth Painlevé equation
112. Elementary solutions of certain Painlevé equations
113. Theory of the fourth Painlevé equation
114. The second Painlevé equation
115. Rational solutions of the second and fourth Painlevé equations
116. iv
117. Direct reduction and differential constraints
118. Nonclassical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal
119. Nichtklassische Symmetrierduktionen hydrodynamischer Plasmagleichungen
120. Conditional symmetries and the direct reduction of partial differential equations
121. Similarity methods with reference to a high-order nonlinear diffusion equation
122. Nonlinear diffusion in population genetics, combustion and nerve propagation
123. Multidimensional nonlinear diffusion arising in population genetics
124. Impulses and physiological states in theoretical models of nerve membrane
125. An active pulse transmission line simulating nerve axon
126. Reducing systems of linear-differential equations to passive form
127. A survey on the method of Gröbner bases for solving problems in connection with systems of multi-variate polynomials
128. Applications of Gröbner bases in nonlinear computational geometry
129. Computations in differential and difference modules
130. On the evolution of packets of water waves
131. An analytical model of periodic waves in shallow water
132. The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev-Petviashvili equation
133. On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili (I) equation
134. On the inverse scattering transform for the Kadomtsev-Petviashvili equation
135. Interactions of traveling fronts—an exact solution of a nonlinear diffusion equation
136. Direct methods in soliton theory
137. Application of a MACSYMA program for the Painlevé test to the Fitzhugh-Nagumo equation
138. The Painlevé property for partial differential equations
139. The Painlevé property for partial differential equations II Backluand transformation Lax pairs and the Schwarzian derivative
140. Nonclassical symmetries and the singular manifold method for the Fitzhugh-Nagumo equation
141. Multiphase and rational solutions of a family of nonlinear equations
142. Painlevé expansions for nonintegrable evolution equations
143. Similarity reductions from extended Painlevé expansions for nonintegrable evolution equations
144. Universal invariance properties of Painlevé analysis and Backhand transformation in nonlinear partial differential equations
145. Invariant Painlevé analysis of partial differential equations
146. Painlevé analysis and Backhand transformation in the Kuramoto-Shivshinsky equation
147. Painlevé analysis of the generalized Burgers-Huxley equation
148. On types of nonlinear nonintegrable equations with exact solutions
149. Partial differential equations with solutions having movable first-order singularities
150. Truncated expansions and nonlinear integrable partial differential equations
151. A unified approach to Painlevé expansions
152. Hirota's method and the singular manifold expansion
153. Nongeneric connections corresponding to front solutions
154. Painlevé analysis of a coupled Burgers' heat equation system, and nonclassical similarity solutions of the heat equation
155. Painlevé analysis of the three-dimensional Burgers equation
156. Painlevé analysis of the two-dimensional Burgers equation
157. On the Painlevé analysis of the 2-dimensional Burgers-equation
158. Wave diffraction in weak cosmic-ray-modified shocks
159. The Sine-Gordon equations: complete and partial integrability
160. Resonance of Solitons in one dimension
161. Initial boundary value-problems for the Korteweg-de Vries equation
162. The effect of conductivity on hotspots
163. Solitary wave interactions in elastic rods
164. Solitary waves on nonlinear rods
165. Solitary waves on nonlinear rods: II
166. Solitons in DNA
167. Solitons in DNA
168. Similarity solutions of the porous media equation
169. Conditional symmetries of the equations of Mathematical Physics
170. The symmetry and exact solutions of the nonlinear multi-dimensional Liouville, d'Alembert and eikonal equations
171. On some exact solutions of the three-dimensional nonlinear Schrödinger equation
172. On some exact solutions of the nonlinear Dirac equation
173. On some exact solutions of the nonlinear equations of quantum electrodynamics
174. Reduction and exact solutions of the Navier-Stokes equations
175. On reduction and solutions of nonlinear wave equations with broken symmetry
176. The symmetry and exact solutions of the nonlinear d'Alembert equations for complex fields
177. On the reduction and some exact solution of the nonlinear Dirac and Dirac-Klein-Gordon equations
178. On some new exact solutions of the nonlinear d'Alembert-Hamilton system
179. On the non-Lie reduction of the linear Dirac equation
180. Conditional symmetry and reduction of partial differential equations
181. On the reduction of the nonlinear multi-dimensional wave equations and compatibility of the d'Alembert-Hamilton system
182. A class of solutions of the boundary layer equations
183. A class of similarity flows for mixed convection along heated plates with suction
184. Closed-form solutions of some partial differential equations via quasi-solutions: I
185. Closed-form solutions of some partial differential equations via quasi-solutions: II
186. Some closed-form solutions Burgers' equations
187. Functional separation of variables for Laplace equations in two dimensions
188. Integrability of a two-dimensional generalization of the Toda chain
189. Integrable nonlinear Klein-Gordon equations and Toda Lattices
190. On the method of solution to the 2 + 1 Toda equation
191. On the inverse scattering transform of the 2 + 1 Toda equation
192. Continuous symmetries of discrete equations
193. Symmetries and conditional symmetries of differential-difference equations
194. Dimensional reduction for equations involving discreate and continuous variables
195. Continuous symmetries of differential-difference equations-the Kac-van Moerbeke equation and Painlevé reduction
196. Continuous symmetries and Painlevé reduction of the Kac-van Moerbeke equation
197. Finite difference models entirely inheriting symmetry of original differential equations
198. Canonical structure and symmetries for discrete systems
199. The similarity method for difference equations
200. Continuous symmetries of difference equations
201. Lie symmetries and the integration of difference equations
202. Hidden symmetries and linearization of the modified Painlevé-Ince equation
203. Hidden symmetries associated with the projective group of nonlinear ordinary differential equations
204. Hidden symmetries of energy-conserving differential equations
205. Hidden and nonlocal symmetries of nonlinear differential equations
206. Hidden symmetries of nonlinear ordinary differential equations
207. Hidden symmetries of energy-conserving differential equations
208. Two-dimensional Burgers equation
209. Transverse diffraction of nonlinear waves in radiative magnetogasdynamics
210. Weakly multi-dimensional cosmic-ray-modified MHD shocks
211. Diffracting nonlinear acoustic beams in 3 + 1-dimensions with applications to oceanic acoustics
212. Nonlinear diffraction and caustic formation
213. Model equations of nonlinear acoustics
214. Transverse diffraction of nonlinear waves and singular rays
215. Similarity solutions of the generalized Kadomtsev-Petviashvili-Burgers equations
216. Reduction of Burgers equations to Riccati equation
217. On the invariance properties of the wave equation
218. New classes of symmetries for partial differential equations
219. Potential symmetries and linearization
220. Use and construction of potential symmetries
221. An overview of potential symmetries
222. Potential symmetries and equivalent conservation laws
223. Potential symmetries and solutions by reduction of partial differential equations
224. Potential symmetries and of Fokker-Planck equations
225. Symmetry-based algorithms to relate partial differential equations II Linearization by nonlocal symmetries
226. Internal, external and generalized symmetries
227. Internal symmetries of differential equations
228. Ovsiannikov's method and the construction of partially invariant solutions
229. Partially invariant solutions of a class of nonlinear Schrödinger equations
230. Partially invariant solutions of nonlinear Klein-Gordon and Laplace equations
231. Symmetry and explicit solutions of partial differential equations
232. The reducibility of partially invariant solutions of systems of PDEs
233. Iterating the nonclassical symmetries method
234. Differential constraints and one-parameter Lie-Bäcklund groups