539.3
534.222

солітон
багатосолітонний розв’язок
відокремлена хвиля
рівняння КдВ
soliton
multi-soliton solutions
Korteweg de-Vries equations
separated waves

Запропоновано новий підхід до знаходження часткових розв’язків диференціальних рівнянь у частинних похідних, які моделюють одиночну хвилю. Відповідний підхід грунтується на використанні Т-представлень розв’язків. На цій основі знайдено новий клас солітонних розв’язків рівняння Кортевега де-Вріза та доведено, що відомий солітон рівняння КдВ є частковим випадком запропонованого нами представлення. Запропонований метод може бути застосований і до інших диференціальних рівнянь. Підхід, що грунтується на основі Т-представлень розв’язків, також може бути використаний для знаходження багатосолітонних розв’язків. У роботі запропоновано метод дослідження взаємодії одиночних хвиль. Відповідний метод продемонстровано на прикладі рівняння Кортевега де-Вріза та отримано двосолітонний розв’язок цього рівняння. Методи, запропоновані в даній роботі, можуть стати основою для досліджень одиночних хвиль та солітонів у рамках інших моделей, що описуються системами диференціальних рівнянь у частинних похідних.
In recent years the investigation of separated waves plays an important role in many applied scientific fields. Travelling wave solutions can describe various phenomena in fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology, meteorology and other fields. There are many models proposed to describe the physical phenomena of separated waves existence and variety of methods has been proposed to construct the exact and approximate solutions to nonlinear equations. In this paper, we propose a new technique of finding the PDE’s traveling wave solutions, which are based on the T- representation. This generalized representation of solutions has the advantages of classical δ-solitons in terms of their independence from the shape and profile, however, allowed to obtain limited smooth solution that simulates solitary waves.This technique guarantees isolation of a solution and allows to investigate the infinitesimal properties. Using T-representation method we found a new class of KdV solution, which simulate solitary wave and proved that well known KdV solution is a special case of our generalizations. The proposed method can be applied to finding solutions of a wide class of differential equations in partial derivatives in the form of solitary waves . T-representations can be useful for the investigation of multi-soliton solutions. Method for studying multi-soliton solutions is demonstrated on the example of KdV solutions. Multi-soliton solutions are represented as the sum of the T-representations with variable amplitudes. In this case, there are exact solutions for the amplitudes of the perturbations in special areas, which are determined by the maximum of perturbations. Problem of searching the perturbation amplitude is reduced to the Cauchy problem for some initial conditions. Changing conditions for maximum localization of perturbation we cover the area of solitones intersection by a system of curves, where exact solutions for the amplitude functions are found. For an approximate description of the waves interaction quite a few curves are needed. This approach allows to consider different laws of waves motion and simulate their amplitude in the region of intersection. An example of computer simulation is presented in this paper.

1. Asaraii, A. Infinite Series Method for Solving the Improved Modified KdV Equation [Тext] / A. Asaraii // Studies in Mathematical Sciences, 2012 – V.4, N.2. – Р.25–31.
2. Miura, M.R. Backlund Transformation [Тext] / M.R. Miura. – Berlin: Springer-Verlag, 1978.
3. Matveev, V.B. Darboux Transformations and Solitons [Тext] / V.B. Matveev, M.A. Salle. – Berlin: Springer, 1991.
4. Parkes E.J. An Automated Tanh-Function Method for Finding Solitary Wave Solution to Nonlinear Evolution Equations [Тext] / E.J. Parkes, B.R. Duffy // Comput. Phys. Commun, 1996. – No98. – Р.288–300.
5. Evans, D.J. The Tanh Function Method for Solving Some Important Non-Linear Partial Differential Equation [Тext] / D.J. Evans, K.R. Raslan // Int. J. comput. Math, 2005. – No82. – Р.897–905.
6. Fan, E. Extended Tanh-Function Method and Its Applications to Nonlinear Equations. [Тext] / E. Fan // Phys. Lett. A., 2000. – No277. – Р.212–218.
7. Elwakil, S.A. Modified Extended Tanh-Function Method For Solving Nonlinear Partial Differential Equations [Тext] / S.A. Elwakil, E.L. Labany, S.K. Zahran, M.A. Sabry // Phys. Lett, 2002. –A., 299. – Р.179– 188.
8. Gao, Y.T. Generalized Hyperbolic-Function Method with Computerized Symbolic Computation to Construct the Solitonic Solutions to Nonlinear Equations of Mathematical Physics [Text] / Y.T. Gao, B. Tian. // Comput. Phys. Commun, 2001. – No133. – P.158–164.
9. Kim, J.M. New Exact Solutons to the Kdv-Burgers-Kuramoto Equation whith the Exp-Function Method [Text] / J.M. Kim, C. Chun // Abstract and Applied Analysis, 2012. – Volume 2012. – Article ID 892420.
10. Krishnan, E.V. Higher-order KdV-Type Equations and their stability [Text] / E.V. Krishnan, Q.J.A. Khan // IJMMS, 2001. – 27:4. – P.215–220.
11. Adomian, G. The fifth-order korteveg-de vries equation [Text] / G. Adomian // Internet J. Math. & Math. Sci., 1996. – Vol.19, No.2. – P.415.
12. Huda, O. Bakodah Modified Adomain Decomposition Method for the Generelized Fifth order KdV Equation [Text] / O. Huda // American journal of computational mathematics, 2013. No3. – P.53–58.
13. Maomao, C. The Coupled Kuramoto Sivashinsky-KDV Equations for Surface Wave in Multilayed Liquid Films [Text] / C. Maomao, L. Dening, C. Rattanakul // Mathematical Physics, 2013. – Volume 2013. – Article ID 673546.
14. Doronin, G. Well and ill-posed problems for the KdV and Kawahara equations [Text] / G. Doronin, N.A. Larkin // Bol. Soc. Paran. Mat., 2008 – (3s.) v.26. – P.133–137.
15. Salas, A.H. Exact Solutions for a Third-Order KdV Equation whith Variable Coefficients and Forcing Term [Text] / A.H. Salas, C.A. Gomez // Mathematical Problems in Engineering, 2009. – Volume 2009. – Article ID 737928.
16. Turbal, Y. The trajectories of self-reinforsing solitary wave in the gas disc of galaxies [Text] / Y. Turbal // Proceedings of the 3-rd International Conference on Nonlinear Dynamic. Kharkov, 2010. – P.112–118.
17. Рыскин, Н.М. Нелинейные волны: учеб. пособие для вузов [Текст] / Н.М. Рыскин, Д.И. Трубецков. – М.: Наука. Физматлит, 2000. – 272 с.

1. Asaraii, A. Infinite Series Method for Solving the Improved Modified KdV Equation [Text] / A. Asaraii // Studies in Mathematical Sciences, 2012 – V.4, N.2. – R.25–31.
2. Miura, M.R. Backlund Transformation [Text] / M.R. Miura. – Berlin: Springer-Verlag, 1978.
3. Matveev, V.B. Darboux Transformations and Solitons [Text] / V.B. Matveev, M.A. Salle. – Berlin: Springer, 1991.
4. Parkes E.J. An Automated Tanh-Function Method for Finding Solitary Wave Solution to Nonlinear Evolution Equations [Text] / E.J. Parkes, B.R. Duffy // Comput. Phys. Commun, 1996. – No98. – R.288–300.
5. Evans, D.J. The Tanh Function Method for Solving Some Important Non-Linear Partial Differential Equation [Text] / D.J. Evans, K.R. Raslan // Int. J. comput. Math, 2005. – No82. – R.897–905.
6. Fan, E. Extended Tanh-Function Method and Its Applications to Nonlinear Equations. [Text] / E. Fan // Phys. Lett. A., 2000. – No277. – R.212–218.
7. Elwakil, S.A. Modified Extended Tanh-Function Method For Solving Nonlinear Partial Differential Equations [Text] / S.A. Elwakil, E.L. Labany, S.K. Zahran, M.A. Sabry // Phys. Lett, 2002. –A., 299. – R.179– 188.
8. Gao, Y.T. Generalized Hyperbolic-Function Method with Computerized Symbolic Computation to Construct the Solitonic Solutions to Nonlinear Equations of Mathematical Physics [Text] / Y.T. Gao, B. Tian. // Comput. Phys. Commun, 2001. – No133. – P.158–164.
9. Kim, J.M. New Exact Solutons to the Kdv-Burgers-Kuramoto Equation whith the Exp-Function Method [Text] / J.M. Kim, C. Chun // Abstract and Applied Analysis, 2012. – Volume 2012. – Article ID 892420.
10. Krishnan, E.V. Higher-order KdV-Type Equations and their stability [Text] / E.V. Krishnan, Q.J.A. Khan // IJMMS, 2001. – 27:4. – P.215–220.
11. Adomian, G. The fifth-order korteveg-de vries equation [Text] / G. Adomian // Internet J. Math. & Math. Sci., 1996. – Vol.19, No.2. – P.415.
12. Huda, O. Bakodah Modified Adomain Decomposition Method for the Generelized Fifth order KdV Equation [Text] / O. Huda // American journal of computational mathematics, 2013. No3. – P.53–58.
13. Maomao, C. The Coupled Kuramoto Sivashinsky-KDV Equations for Surface Wave in Multilayed Liquid Films [Text] / C. Maomao, L. Dening, C. Rattanakul // Mathematical Physics, 2013. – Volume 2013. – Article ID 673546.
14. Doronin, G. Well and ill-posed problems for the KdV and Kawahara equations [Text] / G. Doronin, N.A. Larkin // Bol. Soc. Paran. Mat., 2008 – (3s.) v.26. – P.133–137.
15. Salas, A.H. Exact Solutions for a Third-Order KdV Equation whith Variable Coefficients and Forcing Term [Text] / A.H. Salas, C.A. Gomez // Mathematical Problems in Engineering, 2009. – Volume 2009. – Article ID 737928.
16. Turbal, Y. The trajectories of self-reinforsing solitary wave in the gas disc of galaxies [Text] / Y. Turbal // Proceedings of the 3-rd International Conference on Nonlinear Dynamic. Kharkov, 2010. – P.112–118.
17. Ryskin, N.M. Nelineinye volny: ucheb. posobie dlia vuzov [Text] / N.M. Ryskin, D.I. Trubetskov. – M.: Nauka. Fizmatlit, 2000. – 272 p.