Self-consistent Source Research Articles

On the Hirota equation with a self-consistent source

On the Hirota equation with a self-consistent source

On the Integration of the Hierarchy of the Kaup–Boussinesq System with a Self-Consistent Source

On the Integration of the Hierarchy of the Kaup–Boussinesq System with a Self-Consistent Source

On the Integration of the Higher Order Toda Lattice with a Self-Consistent Integral Type Source

This work presents an algorithm that uses the inverse scattering method to find a solution for the higher-order Toda lattice with a self-consistent source. The higher-order Toda lattice with an integral-type source is also a significant theoretical model belonging to very integrable systems. The problem is solved by applying the direct and inverse scattering methods to the discrete Sturm-Liouville operator, and the time dependence of the scattering data for this operator is attained. The solution to the problem is set up using the inverse scattering transform (IST) approach.

Integration of the mKdV Equation with nonstationary coefficients and additional terms in the case of moving eigenvalues

In this paper, we consider the Cauchy problem for the non-stationary modified Korteweg–de Vries equation with an additional term and a self-consistent source in the case of moving eigenvalues. Also, the evolution of the scattering data of the Dirac operator is obtained, the potential of which is the solution of the loaded modified Korteweg–de Vries equation with a self-consistent source in the class of rapidly decreasing functions. Specific examples are given to illustrate the application of the obtained results.

Integration of the finite complex Toda lattice with a self-consistent source

In the paper, we derive a finite complex Toda lattice with a self-consistent source. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated finite Jacobi matrix. We show that the finite complex Toda lattice with a self-consistent source is also an important theoretical model as it is a completely integrable system. Namely, we have determined the time evolution of the spectral data for the Jacobi matrix associated with the solution of a finite complex Toda chain with a self-consistent source. Then, using the solution of the inverse spectral problem with respect to the time-dependent spectral data, we recover the time-dependent Jacobi matrix and hence the desired solution of the finite complex Toda chain with a self-consistent source. For the case N=2, explicit formulas for the solution of the problem under consideration are obtained.

Integration of the negative order Korteweg-de Vries equation with self-consistent source

Integration of the negative order Korteweg-de Vries equation with self-consistent source

Integration of the Korteweg-de Vries equation with loaded terms and a self-consistent source in the class of rapidly decreasing functions

In this paper, we solve the Cauchy problem for the Korteweg-de Vries equation with loaded terms and a self-consistent source in the class of rapidly decreasing functions. To solve this problem, the method of the inverse scattering problem is used. The evolution of the scattering data of the self-adjoint Sturm-Liouville operator, whose coefficient is a solution of the Korteweg-de Vries equation with loaded terms and a self-consistent source, is obtained. Examples are given to illustrate the application of the obtained results.

Интегрирование уравнения Кортевега – де Фриза отрицательного порядка с источником специального вида

In this paper, we consider the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem. It is shown that the considered equation can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm-Liouville operator with a periodic potential associated with the solution of the considered equation is determined. The results obtained make it possible to apply the inverse problem method for solving the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem.

Integration of the sine-Gordon equation with a source and an additional term

The work is devoted to solving the Cauchy problem for the sine-Gordon equation with an additional term and a self-consistent source in the class of rapidly decreasing functions. The problem is solved by the inverse scattering method. Several special cases of the sine-Gordon equation with an additional term are given, which can be integrated using the inverse scattering method, for example, the loaded sine-Gordon equation. Several examples are given to illustrate the application of the obtained results.

Integration of the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions

In this paper, we consider the negative order Korteweg–de Vries equation with a self-consistent integral source. It is shown that the negative-order Korteweg–de Vries equation with a self-consistent integral source can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm–Liouville operator with a periodic potential associated with the solution of the negative order Korteweg–de Vries equation with a self-consistent integral source is determined. The obtained results make it possible to apply the inverse problem method to solve the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions.

Integration of the Kaup–Boussinesq system with a self-consistent source via inverse scattering method

In this study we consider the Kaup–Boussinesq system with a self-consistent source. We show that the Kaup–Boussinesq system with a self-consistent source can be integrated by the method of inverse scattering theory. For a solving the problem under consideration, we use the direct and inverse scattering problem of the Sturm–Liouville equation with an energy-dependent potential. The time evolution of the scattering data for the Sturm–Liouville equation with an energy-dependent potentials associated with the solution of the Kaup–Boussinesq system with a self-consistent source is determined. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Kaup–Boussinesq system with a self-consistent source.

Integration of the Loaded KdV Equation with a Self-Consistent Source of Integral Type in the Class of Rapidly Decreasing Complex-Valued Functions

Integration of the Loaded KdV Equation with a Self-Consistent Source of Integral Type in the Class of Rapidly Decreasing Complex-Valued Functions

Integration of the differential–difference sine-Gordon equation with a self-consistent source

Integration of the differential–difference sine-Gordon equation with a self-consistent source

Integration of the differential-difference sine-Gordon equation with a self-consistent source

Метод обратной задачи рассеяния применяется для интегрирования дифференциально-разностного уравнения синус-Гордон с самосогласованным источником.

Integration of Camassa-Holm equation with a self-consistent source of integral type

Integration of Camassa-Holm equation with a self-consistent source of integral type

Predictive multi-channel integrated modeling of a reversed magnetic shear H-mode discharge with internal transport barrier in EAST

A reversed magnetic shear high plasma performance H-mode discharge with internal transport barrier (ITB) has been realized and modeled on experimental advanced superconducting tokamak (EAST) in deuterium (D) plasmas with an ITER-like tungsten (W) upper divertor. In this shot (#71326), the dimensionless parameter G () can reach to 0.25 (H 89 ⩽ 2, β N ⩽ 2, q 95 ∼ 4, I p ⩽ 450 kA, B T ⩽ 1.6 T). The ITB has been observed in the channels of particle (electron density n e), electron and ion temperature (T e and T i) and toroidal rotation velocity (V t). However, in the formation process of V t and n e ITBs, the heavy impurities accumulate in the core plasmas, which may be one of the most important reasons for the limitation of the improvement of plasma performance. A time slice (t = 4.65 s) of the high plasma performance phase with n e, T i and V t ITBs has been analyzed. It is found that the position of minimum safety factor (q min) is at about normalized radius ρ = 0.4 where may be the location of ion ITB foot, during equilibrium reconstruction from EFIT using external magnetic and internal polarimeter-INTferometer (POINT) measurement constraints. Based on this equilibrium, the source is self-consistently calculated by ONETWO and NUBEAM. Then the n e, T e, T i and V t have been modeled predictively and simultaneously by TGYRO utilizing the reconstructed equilibrium, the self-consistent source and the experimental radiated power for the first time in EAST. This modeling well reproduces the experimental n e and T i profiles in the reversed magnetic shear plasmas. In addition, it is shown that reversed magnetic shear rather than shear of radial electric field (E r) mainly produced by V t plays a key role to the formation of n e and T i ITBs. The reasons for central accumulation of W impurity are also clarified by computing both neoclassical and turbulent transport components of W. It turns out that the neoclassical transport dominates over turbulent transport for W and neoclassical pinch of W causes its central accumulation.

Integrating the loaded KdV equation with a self-consistent source of integral type in the class of rapidly decreasing complex-valued functions

Integrating the loaded KdV equation with a self-consistent source of integral type in the class of rapidly decreasing complex-valued functions

Integration of the Harry Dym equation with an integral type source

In the work, we deduce the evolution of scattering data for a spectral problem associated with the nonlinear evolutionary equation of Harry Dym with a self-consistent source of integral type. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Harry Dym equation with an integral type source.

On the Construction and Integration of a Hierarchy for the Kaup System with a Self-Consistent Source in the Class of Periodic Functions

On the Construction and Integration of a Hierarchy for the Kaup System with a Self-Consistent Source in the Class of Periodic Functions

Интегрирование матричного нелинейного уравнения Шредингера с источником

This paper is concerned with studying the matrix nonlinear Schr¨odinger equation with a self-consistent source. The source consists of the combination of the eigenfunctions of the corresponding spectral problem for the matrix Zakharov-Shabat system which has not spectral singularities. The theorem about the evolution of the scattering data of a non-self-adjoint matrix Zakharov-Shabat system which potential is a solution of the matrix nonlinear Schr¨odinger equation with the self-consistent source is proved. The obtained results allow us to solve the Cauchy problem for the matrix nonlinear Schr¨odinger equation with a self-consistent source in the class of the rapidly decreasing functions via the inverse scattering method. A one-to-one correspondence between the potential of the matrix Zakharov-Shabat system and scattering data provide the uniqueness of the solution of the considering problem. A step-by-step algorithm for finding a solution to the problem under consideration is presented.