KdV soliton solutions to a model of hepatitis C virus evolution

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1 KdV soliton solutions to a model of hepatitis C virus evolution T. Telksnys, Z. Navickas, M. Ragulskis Kaunas University of Technology Differential Equations and Applications, Brno 2017 September 6th, 2017 T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

2 Introduction Solitons Historical Overview Russell and the Wave of Translation <...> the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation <...> which continued its course along the channel apparently without change of form or diminution of speed. <...> was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation. (1834) Figure: John Scott Russell ( ) T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

de Vries. The KdV equation u t 3 u u + 6u x3 x = 0. It was shown that there exist solutions of the form: KdV soliton Figure: D.

3 Introduction Solitons Historical Overview First Mathematical Model: KdV equation The first actual mathematical model of solitary waves (solitons) was discovered by Boussinesq (1877) and later rediscovered and studied in detail by Diederik Korteweg and Gustav de Vries. The KdV equation u t 3 u u + 6u x3 x = 0. It was shown that there exist solutions of the form: KdV soliton Figure: D. Korteweg (top), G. de Vries ( c ) u(t, x) = c 2 sech2 (x ct a). 2 T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

Introduction Solitons Historical Overview Zabusky and Kruskal: Rediscovery The works of Russell, Boussinesq, Korteweg and de Vries fell into obscurity until 1965,

The word soliton was coined and extensive studies into the nature of soliton (solitary) processes were launched that are still continuing today.

4 Introduction Solitons Historical Overview Zabusky and Kruskal: Rediscovery The works of Russell, Boussinesq, Korteweg and de Vries fell into obscurity until 1965, when Norman J. Zabusky and Martin Kruskal made connections between the KdV equation and the Fermi- Pasta-Ulam experiment. The word soliton was coined and extensive studies into the nature of soliton (solitary) processes were launched that are still continuing today. It has had a broad and far-reaching impact in myriad fields ranging from the purest mathematics to experimental science. Figure: N. Zabusky (top), M. Kruskal T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

Introduction Solitons Historical Overview Rise to fame: the Schrödinger equation A particular surge in interest of the analysis of solitary processes in physics came when Vladimir Zakharov and

5 Introduction Solitons Historical Overview Rise to fame: the Schrödinger equation A particular surge in interest of the analysis of solitary processes in physics came when Vladimir Zakharov and Aleksei Shabat demonstrated in 1972 that the nonlinear Schrödinger (NLS) equation has soliton solutions. NLS equation i u x + 2 u t 2 ± 2 u 2 u = 0. This discovery had enormous repercussions in physics, especially in nonlinear optics, Bose- Einstein condensates, where the NLS equation plays a very important role. Figure: V. Zakharov (top), A. Shabat T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

6 Introduction Applications Applications of Soliton Theory Soliton theory has had a great impact on myriad fields of science, including: Nonlinear optics; Bose-Einstein condensates; Hydrodynamics; Biophysics; MEMs and NEMs; Plasmas; Population dynamics. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

7 KdV Type Soliton Solutions Physical Properties Physical Properties of Solitary Solutions A nonlinear wave is called a soliton if: It maintains its shape as it propagates at a constant speed; If it collides with another soliton, it emerges from the collision unaltered, except for a phase shift. The definition is not universally accepted there are a few ways to define solitons in the physical sense (for example allowing a small loss of energy after collision ( light bullets )), however, from a mathematical perspective the definition given above is almost ubiquitous. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

8 KdV Type Soliton Solutions Analytic Expression Solitary solutions analytic expression Solitary solution The m-th order solitary solution reads: x(t) = m α k exp ( ηk(t c) ) k=0 m k=1 where η, α k, t k, c C are fixed parameters. ( exp ( η(t c) ) t k ), The shape is based on the KdV soliton, which is a special case of the above solution (hyperbolic secant): ϕ (ξ) = c 2 sech2 ( c 2 (ξ ξ 0) ). T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

9 KdV Type Soliton Solutions Examples Solitary solutions (a) Kink solitary solution (b) Bright solitary solution T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

10 KdV Type Soliton Solutions Examples Solitary solutions (c) Solitary solution with one singularity (d) Solitary solution with two singularities T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

11 KdV Type Soliton Solutions Coupled Riccati Equations Coupled Riccati equations (1) Regula et al (2009) introduced the following system for modeling Hepatitis C virus (HCV) evolution: x t = x (1 x y) (1 θ) bxy + qy + s; y t = ry (1 x y) + (1 + θ) bxy (d + q)y, θ, b, q, s, r, d R. Simple generalization leads to: x t = a 0 + a 1 x + a 2 x 2 + a 3 xy + a 4 y; y t = b 0 + b 1 y + b 2 y 2 + b 3 xy + b 4 x, a j, b j R. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

12 KdV Type Soliton Solutions Coupled Riccati Equations Coupled Riccati equations (2) x t = a 0 + a 1 x + a 2 x 2 + a 3 xy + a 4 y; y t = b 0 + b 1 y + b 2 y 2 + b 3 xy + b 4 x, a j, b j R. Standard diffusive coupling terms a 4 y, b 4 x; Multiplicative coupling terms a 3 xy, b 3 xy; Conditions for existence of soliton solutions? Construction of soliton solutions? T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

13 Inverse Balancing Technique Main Idea Inverse balancing technique The inverse idea of common ansatz methods; Insert known solution into DE; If DE depends linearly on equation parameters, solve for them; Application of technique is not used to solve the DE, but to obtain necessary existence conditions for the solitary solutions. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

14 Inverse Balancing Technique Existence of solitary solutions in a class of PDEs PDEs with polynomial nonlinearity Class of PDEs m u t m +A m 1 u m 1,0 t m 1 +A m 1 u 0,m 1 z m 1 + +A u 10 t +A u 01 z = a nu n + +a 0. When do the considered PDEs have solitary solutions (of any order l)? u (t αz) = l α k exp ( ηk (t αz) ) k=0 l k=1 ( exp ( η (t αz) ) t k ). T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

15 Inverse Balancing Technique Existence of solitary solutions in a class of PDEs Necessary existence conditions (1) Condition #1: derivative and nonlinear term balance n = m + 1. Condition #2: equation and solution order balance (m + 1)l 2 l + m + 1, l, m N. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

16 Inverse Balancing Technique Existence of solitary solutions in a class of PDEs Necessary existence conditions (2) Table of necessary existence conditions of solitary solutions to the considered PDEs. denotes existence with all parameter values, denotes existence with additional constraints on parameters, denotes the nonexistence of solitary solutions. (n, m) l (2, 1) (3, 2) (4, 3) (5, 4) (6, 5) (7, 6) (8, 7) T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

17 Generalized Differential Operator Technique Operators Generalized differential operator The notation D α := α will be used. Generalized differential operator (GDO) D csu := R (c, s, u) D c + P (c, s, u) D s + Q (c, s, u) D u, where R, P, Q are analytic. Properties of GDO D csu (f 1 + f 2 ) = D csu f 1 + D csu f 2 ; D csu (f 1 f 2 ) = (D csu f 1 ) f 2 + f 1 D csu f 2. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

18 Generalized Differential Operator Technique Operators Multiplicative operator Suppose a GDO D csu is given. The multiplicative operator reads: G := + j=0 (t c) j j! D j csu. Main property Gf (c, s, u) = f (Gc, Gs, Gu). Each ODE or ODE system has unique generalized differential and multiplicative operators. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

19 Generalized Differential Operator Technique Operator Expression of Solutions First order ODE system x t = P (t, x, y) ; x = x (t; c, s, u), x (c; c, s, u) = s; y t = Q (t, x, y) ; y = y (t; c, s, u), y (c; c, s, u) = u. System operators D csu = D c + P (c, s, u) D s + Q (c, s, u) D u ; G = + j=0 (t c) j j! D j csu. General solution x(t) = + j=0 (t c) j j! ( ) + D j csus = Gs, y(t) = j=0 (t c) j j! ( ) D j csuu = Gu. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

20 Nonlinear System Transformation Solution Transformation Image of solution Suppose an ODE is given: x t = P (x), x = x (t; c, s) ; x (c; c, s) = s. Variable substitution t := exp (ηt), ĉ := exp (ηc) ; η R\{0} Image of solution ( ) 1 ) x = x(t) = x η ln t =: x ( t = x; x t = η t x t. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

21 Nonlinear System Transformation Equation/Operator Transformation Image of ODE Image of ODE η t x = P ( x) ; t ) x = x ( t; ĉ, s, x (ĉ; ĉ, s) = s. Operators of transformed ODE Dĉs := Dĉ + 1 ηĉ P (s) D s; Ĝ := + j=0 ( t ĉ) j j! D j ĉs. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

22 Closed Form Solutions Linear Recurring Sequences Linear recurring sequences Let p j := D j csus. Solutions can be written in the closed form if the sequence ( pj ; j Z 0 ) (or a sequence constructed from pj in a known way) is linearly recurring. p 0 p 1... p k 1 p 1 p 2... p k d k := det p k 1 p k... p 2k 2 Linear recurring sequence ( pj ; j Z 0 ) is an m-th order linear recurring sequence (LRS), if d m 0, d m+l = 0; l = 1, 2,.... It is denoted as order ( p j ; j Z 0 ) = m. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

23 Closed Form Solutions Linear Recurring Sequences Canonical expression of LRS The characteristic equation reads: p 0 p 1... p m 1 p m p 1 p 2... p m p m = 0. p m 1 p m... p 2m 2 p 2m 1 1 ρ... ρ m 1 ρ m Characteristic roots: ρ 1,..., ρ m ; ρ k ρ j, k j. Canonical expression p j = m k=1 λ k ρ j k, j = 0, 1,.... Coefficients λ k, k = 1,..., m are determined from a system of linear equations. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

24 Closed Form Solutions Application of LRS: Exponent Sum Solution Exponent sum solution Suppose that ) order (D j csus; j Z 0 = m; m D j csus = λ k ρ j k. k=1 Form of solution x (t; c, s, u) = Gs = = + j=0 (t c) j j! m λ k ρ j k k=1 m λ k exp ( ρ k (t c) ). k=1 T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

25 Closed Form Solutions Application of LRS: Soliton Solution Solitary (soliton) solution (1) Let ) order (D j csus; j Z 0 = +, but order ( ) 1 j! D j ĉsus; j Z 0 = m. Then 1 j! D m j ĉsus = λ k ρ j k. k=1 Theorem The soliton solution exists, if the following relations hold true: Dĉsu ρ k = ρ 2 k, Dĉsu λ k = λ k ρ k ; k = 1,..., m. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

26 Closed Form Solutions Application of LRS: Soliton Solution Solitary (soliton) solution (2) Form of solution ) x ( t; + ĉ, s, u = Ĝs = j ( t ĉ) m m λ k ρ j k = λ k ); 1 ρ k ( t ĉ j=0 x (t; c, s, u) = k=1 k=1 m α k exp ( ηk(t c) ) k=0 m k=1 ( exp ( η(t c) ) t k ). T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

27 Closed Form Solutions Application of LRS: General Case General case Suppose that f (ξ) = + j=0 j 1 q j j! ξj ; q 0 = 1, q j = (a + bk) 0, j = 1, 2,.... k=0 ( ) m 1 1 Let order q Dj j ĉsus; j Z 0 = m, q Dj j ĉsus = λ k ρ j k. Form of solution j=0 k=1 ) + ŷ ( t; ĉ, s, u = Ĝs = q j j ( t ĉ) m m λ k ρ j k j! = λ k f x (t; c, s, u) = m λ k f k=1 k=1 k=1 ( α k β k exp ( η(t c) )). ( ρ k ( t ĉ) ) ; T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

28 Solitary Solutions Hepatitis C Evolution Model Hepatitis C model (Coupled Riccati equations) x t = a 0 + a 1 x + a 2 x 2 + a 3 xy + a 4 y; x(c) = s; y t = b 0 + b 1 y + b 2 y 2 + b 3 xy + b 4 x; y(c) = u, a j, b j R. Kink solutions (order = 1); Bright/dark and singular solutions (order = 2); T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

29 Solitary Solutions Hepatitis C Evolution Model Bright/Dark Solutions Bright/dark solutions Analytical expression (exp ( η(t c) ) ) x 1 (exp ( η(t c) ) ) x 2 x (t) = σ (exp ( η(t c) ) ) t 1 (exp ( η(t c) ) ) ; (1) t 2 (exp ( η(t c) ) ) y 1 (exp ( η(t c) ) ) y 2 y (t) = γ (exp ( η(t c) ) ) t 1 (exp ( η(t c) ) ). (2) t 2 σ, γ, η are constants; x 1, x 2, y 1, y 2, t 1, t 2 depend on initial conditions s, u; Solutions which hold for all initial conditions are constructed. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

30 Solitary Solutions Hepatitis C Evolution Model Bright/Dark Solutions Existence conditions Bright/dark solitary solutions exist if: and a 3 = b 2 ; a 2 = b 3, 9a 0 a 1 a 2 + 9b 0 b 1 b 2 18a 0 a 2 b 1 18b 0 b 2 a 1 + 3a 1 b b 1 a 2 1 2a 3 1 2b 3 1 9a 1 a 4 b 4 9b 1 b 4 a a 0 b 2 b b 0 a 2 a 4 = 0. In the phase plane, bright/dark solution trajectories are conic sections: Ax 2 (t) + By 2 (t) + Cx(t)y(t) + Ex(t) + F y(t) = G; A, B, C, E, F, G R. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

31 Solitary Solutions Hepatitis C Evolution Model Bright/Dark Solutions Time evolution of solutions (1) T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

32 Solitary Solutions Hepatitis C Evolution Model Bright/Dark Solutions Time evolution of solutions (2) T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

33 Solitary Solutions Hepatitis C Evolution Model Bright/Dark Solutions Phase portrait T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

34 Publications Publications T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

35 Publications Publications I Z. Navickas, M. Ragulskis, R. Marcinkevicius, and T.Telksnys. Kink solitary solutions to generalized riccati equations with polynomial coefficients. Journal of Mathematical Analysis and Applications, 448: , Z. Navickas, M.Ragulskis, N. Listopadskis, and T.Telksnys. Comments on soliton solutions to fractional-order nonlinear differential equations based on the exp-function method. Optik - International Journal for Light and Electron Optics, 132: , Z. Navickas, R. Marcinkevicius, T.Telksnys, and M.Ragulskis. Existence of second order solitary solutions to riccati differential equations coupled with amultiplicative term. IMA Journal of Applied Mathematics, 81: , R. Marcinkevicius, Z. Navickas, M. Ragulskis, and T. Telksnys. Solitary solutions to a relativistic two-body problem. Astrophysics and Space Science, 361: , Z. Navickas, R. Vilkas, T. Telksnys, and M. Ragulskis. Direct and inverse relationships between riccati systems coupled with multiplicative terms. Journal of Biological Dynamics, 10: , T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

36 Publications Publications II Z. Navickas, M. Ragulskis, and T. Telksnys. Existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity. Applied Mathematics and Computation, 283: , Z. Navickas, M. Ragulskis, R. Marcinkeviius, and T. Telksnys. Generalized solitary waves in nonintegrable kdv equations. Journal of Vibroengineering, 18: , Z. Navickas, T. Telksnys, and M. Ragulskis. Comments on The exp-function method and generalized solitary solutions. Computers and Mathematics with Applications, 69(8): , Z. Navickas and M. Ragulskis. Comments on A new algorithm for automatic computation of solitary wave solutions to nonlinear partial differential equations based on the exp-function method. Applied Mathematics and Computation, 243: , Z. Navickas and M. Ragulskis. Comments on Two exact solutions to the general relativistic Binets equation. Astrophysiscs Space Science, 344: , T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

37 Publications Publications III Z. Navickas, L. Bikulciene, M. Rahula, and M. Ragulskis. Algebraic operator method for the construction of solitary solutions to nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 18: , Z. Navickas, M. Ragulskis, and L. Bikulciene. Special solutions of Huxley differential equation. Mathematical Modeling and Analysis, 16: , Z. Navickas, L. Bikulciene, and M. Ragulskis. Generalization of Exp-function and other standard function methods. Applied Mathematics and Computation, 216: , Z. Navickas, M. Ragulskis, and L. Bikulciene. Be careful with the Exp-function method additional remarks. Communications in Nonlinear Science and Numerical Simulation, 15: , Z. Navickas and M. Ragulskis. How far can one go with the Exp-function method? Applied Mathematics and Compututation, 211: , Z. Navickas and L. Bikulciene. Expressions of solutions of ordinary differential equations by standard functions. Mathematical Modeling and Analysis, 11: , T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

38 The End Thank You For Your Attention T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, / 38

Dark & Bright Solitons in Strongly Repulsive Bose-Einstein Condensate

Dark & Bright Solitons in Strongly Repulsive Bose-Einstein Condensate Indu Satija, George Mason Univ & National Institute of Standard and Tech ( NIST) collaborator:radha Balakrishnan, Institute of Mathematical