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Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear Spinor Equations
(Springer, 2023) Pashaev, Oktay
A new relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions, gauge equivalent to Papanicolau spin model, defined on the one sheet hyperboloid is introduced. By using the double numbers, the model is represented as hyperbolic complex valued relativistic massive Thirring type model. By Hirota’s bilinear method, an exact one and two dissipative soliton solutions of this model are constructed. Calculation of first three integrals of motion for one dissipation solution shows that the last one represents a particle-like nonlinear excitation, with relativistic dispersion and highly nonlinear mass. A nontrivial solution of the system of algebraic equations, showing fusion and fission of relativistic dissipations is found. Asymptotic analysis of exact two dissipaton solution confirms resonant character of our dissipaton interactions. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Citation - WoS: 17
Citation - Scopus: 18
Linear Wave Interaction With a Vertical Cylinder of Arbitrary Cross Section: an Asymptotic Approach
(American Society of Civil Engineers (ASCE), 2017) Dişibüyük, Nazile Buğurcan; Korobkin, A. A.; Yılmaz, Oğuz
An asymptotic approach to the linear problem of regular water waves interacting with a vertical cylinder of an arbitrary cross section is presented. The incident regular wave was one-dimensional, water was of finite depth, and the rigid cylinder extended from the bottom to the water surface. The nondimensional maximum deviation of the cylinder cross section from a circular one plays the role of a small parameter of the problem. A fifth-order asymptotic solution of the problem was obtained. The problems at each order were solved by the Fourier method. It is shown that the first-order velocity potential is a linear function of the Fourier coefficients of the shape function of the cylinder, the second-order velocity potential is a quadratic function of these coefficients, and so on. The hydrodynamic forces acting on the cylinder and the water surface elevations on the cylinder are presented. The present asymptotic results show good agreement with numerical and experimental results of previous investigations. Long-wave approximation of the hydrodynamic forces was derived and used for validation of the asymptotic solutions. The obtained values of the forces are exact in the limit of zero wave numbers within the linear wave theory. An advantage of the present approach compared with the numerical solution of the problem by an integral equation method is that it provides the forces and the diffracted wave field in terms of the coefficients of the Fourier series of the deviation of the cylinder shape from the circular one. The resulting asymptotic formula can be used for optimization of the cylinder shape in terms of the wave loads and diffracted wave fields.
Citation - WoS: 30
Citation - Scopus: 33
The Initial Stage of Dam-Break Flow
(Springer Verlag, 2009) Korobkin, Alexandre; Yılmaz, Oğuz
The liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy-Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.

Mathematics / Matematik

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