Introduction to Soliton Theory: Applications to Mechanics (Fundamental Theories of Physics Book 143)

This monograph provides the application of soliton theory to solve certain problems selected from the fields of mechanics. The work is based of the authors’ research, and on some specified, significant results existing in the literature. The present monograph is not a simple translation of its predecessor appeared in Publishing House of the Romanian Academy in 2002. Improvements outline the way in which the soliton theory is applied to solve some engineering problems. The book addresses concrete resolution methods of certain problems such as the motion of thin elastic rod, vibrations of initial deformed thin elastic rod, the coupled pendulum oscillations, dynamics of left ventricle, transient flow of blood in arteries, the subharmonic waves generation in a piezoelectric plate with Cantor-like structure, and some problems related to Tzitzeica surfaces. This comprehensive study enables the readers to make connections between the soliton physical phenomenon and some partical, engineering problems. Contents......Page 6 Preface......Page 10 1.2 Scattering theory......Page 14 1.3 Inverse scattering theory......Page 25 1.4 Cnoidal method......Page 30 1.5 Hirota method......Page 38 1.6 Linear equivalence method (LEM)......Page 44 1.7 Bäcklund transformation......Page 52 1.8 Painlevé analysis......Page 59 2.2 General properties of the linear waves......Page 66 2.3 Some properties of nonlinear equations......Page 72 2.4 Symmetry groups of nonlinear equations......Page 75 2.5 Noether theorem......Page 79 2.6 Inverse Lagrange problem......Page 82 2.7 Recursion operators......Page 86 3.2 Korteweg and de Vries equation (KdV)......Page 91 3.3 Derivation of the KdV equation......Page 99 3.4 Scattering problem for the KdV equation......Page 103 3.5 Inverse scattering problem for the KdV equation......Page 108 3.6 Multi-soliton solutions of the KdV equation......Page 114 3.7 Boussinesq, modified KdV and Burgers equations......Page 120 3.8 The sine-Gordon and Schrödinger equations......Page 125 3.9 Tricomi system and the simple pendulum......Page 128 4.1 Scope of the chapter......Page 134 4.2 Fundamental equations......Page 135 4.3 The equivalence theorem......Page 145 4.4 Exact solutions of the equilibrium equations......Page 147 4.5 Exact solutions of the motion equations......Page 159 5.2 Linear and nonlinear vibrations......Page 162 5.3 Transverse vibrations of the helical rod......Page 168 5.4 A special class of DRIP media......Page 172 5.5 Interaction of waves......Page 176 5.6 Vibrations of a heterogeneous string......Page 179 6.2 Motion equations. Problem E1......Page 186 6.3 Problem E2......Page 190 6.4 LEM solutions of the system E2......Page 193 6.5 Cnoidal solutions......Page 198 6.6 Modal interaction in periodic structures......Page 204 7.1 Scope of the chapter......Page 210 7.2 The mathematical model......Page 211 7.3 Cnoidal solutions......Page 219 7.4 Numerical results......Page 222 7.5 A nonlinear system with essential energy influx......Page 226 8.1 Scope of the chapter......Page 233 8.2 A nonlinear model of blood flow in arteries......Page 234 8.3 Two-soliton solutions......Page 241 8.4 A micropolar model of blood flow in arteries......Page 248 9.1 Scope of the chapter......Page 255 9.2 A plate with Cantor-like structure......Page 256 9.3 The eigenvalue problem......Page 261 9.4 Subharmonic waves generation......Page 262 9.5 Internal solitary waves in a stratified fluid......Page 268 9.6 The motion of a micropolar fluid in inclined open channels......Page 272 9.7 Cnoidal solutions......Page 278 9.8 The effect of surface tension on the solitary waves......Page 282 10.2 Tzitzeica surfaces......Page 286 10.3 Symmetry group theory applied to Tzitzeica equations......Page 289 10.4 The relation between the forced oscillator and a Tzitzeica curve......Page 296 10.5 Sound propagation in a nonlinear medium......Page 298 10.6 The pseudospherical reduction of a nonlinear problem......Page 304 References......Page 311 D......Page 318 K......Page 319 R......Page 320 Y......Page 321 This monograph is planned to provide the application of the soliton theory to solve certain practical problems selected from the fields of solid mechanics, fluid mechanics and biomechanics. The work is based mainly on the authors’ research carried out at their home institutes, and on some specified, significant results existing in the published literature. The methodology to study a given evolution equation is to seek the waves of permanent form, to test whether it possesses any symmetry properties, and whether it is stable and solitonic in nature. Students of physics, applied mathematics, and engineering are usually exposed to various branches of nonlinear mechanics, especially to the soliton theory. The soliton is regarded as an entity, a quasi-particle, which conserves its character and interacts with the surroundings and other solitons as a particle. It is related to a strange phenomenon, which consists in the propagation of certain waves without attenuation in dissipative media. This phenomenon has been known for about 200 years (it was described, for example, by the Joule Verne's novel Les histoires de Jean Marie Cabidoulin, Éd. Hetzel), but its detailed quantitative description became possible only in the last 30 years due to the exceptional development of computers. The discovery of the physical soliton is attributed to John Scott Russell. In 1834, Russell was observing a boat being drawn along a narrow channel by a pair of horses. "This comprehensive study enables the readers to make connections between the soliton physical phenomenon and some partical, engineering problems."--Jacket