KdV soliton solutions to a model of hepatitis C virus evolution

Similar documents
Dark & Bright Solitons in Strongly Repulsive Bose-Einstein Condensate

Solitons : An Introduction

Solitons in optical fibers. by: Khanh Kieu

Shock Waves & Solitons

Nonlinear Optics (WiSe 2015/16) Lecture 7: November 27, 2015

Nonlinear Optics (WiSe 2017/18) Lecture 12: November 28, 2017

Nonlinear Optics (WiSe 2018/19) Lecture 7: November 30, 2018

Convergence of operator splitting for the KdV equation

Are Solitary Waves Color Blind to Noise?

Soliton Molecules. Fedor Mitschke Universität Rostock, Institut für Physik. Benasque, October

What Is a Soliton? by Peter S. Lomdahl. Solitons in Biology

Dynamics of Solitary Waves Induced by Shock Impulses in a Linear Atomic Chain*

What we do understand by the notion of soliton or rather solitary wave is a wave prole which is very stable in the following sense

12 Lecture 12. Lax pair and Toda lattice

Chapter 3. Head-on collision of ion acoustic solitary waves in electron-positron-ion plasma with superthermal electrons and positrons.

The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method

Exact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially Integrable Equations

Periodic, hyperbolic and rational function solutions of nonlinear wave equations

The integrating factor method (Sect. 1.1)

Direct and inverse relationships between Riccati systems coupled with multiplicative terms

Soliton Solutions of a General Rosenau-Kawahara-RLW Equation

Nonlinear Fourier Analysis

A note on the G /G - expansion method

A NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (1+2)-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION

Recurrence in the KdV Equation? A. David Trubatch United States Military Academy/Montclair State University

Basic Concepts and the Discovery of Solitons p. 1 A look at linear and nonlinear signatures p. 1 Discovery of the solitary wave p. 3 Discovery of the

TWO SOLITON SOLUTION OF TZITZEICA EQUATION. Bulgarian Academy of Sciences, Sofia, Bulgaria

Vector Schroedinger Equation. Generalized Wave Equation. Boussinesq Paradigm. Nonintegrability and Conservativity of Soliton Solutions

Exact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized ( G. )-Expansion Method

Solving Nonlinear Wave Equations and Lattices with Mathematica. Willy Hereman

A NEW VARIABLE-COEFFICIENT BERNOULLI EQUATION-BASED SUB-EQUATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS

New Approach of ( Ǵ/G ) Expansion Method. Applications to KdV Equation

Lecture 10: Whitham Modulation Theory

Propagation of Solitons Under Colored Noise

Compacton-like solutions in some nonlocal hydrodynamic-type models

Linear degeneracy in multidimensions

MACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

SOLITON SOLUTIONS OF SHALLOW WATER WAVE EQUATIONS BY MEANS OF G /G EXPANSION METHOD

INVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS

Exact Solutions of Space-time Fractional EW and modified EW equations

SPECIAL TYPES OF ELASTIC RESONANT SOLITON SOLUTIONS OF THE KADOMTSEV PETVIASHVILI II EQUATION

2. Examples of Integrable Equations

Available online at J. Math. Comput. Sci. 2 (2012), No. 1, ISSN:

exp Φ ξ -Expansion Method

NUMERICAL SOLITARY WAVE INTERACTION: THE ORDER OF THE INELASTIC EFFECT

Self-trapped optical beams: From solitons to vortices

New explicit solitary wave solutions for (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation

Simulating Solitons of the Sine-Gordon Equation using Variational Approximations and Hamiltonian Principles

Solitons. M.W.J. Verdult

Inverse scattering technique in gravity

Introduction to Inverse Scattering Transform

STUDY OF SOLITARY WAVE SOLUTIONS FOR A CLASS OF NON LINEAR EQUATIONS

IAN STRACHAN. 1. Introduction

Introduction to the Hirota bilinear method

Nongauge Bright Soliton of the Nonlinear Schrodinger (NLS) Equation and a Family of Generalized NLS Equations

Dispersion relations, stability and linearization

On the N-tuple Wave Solutions of the Korteweg-de Vnes Equation

Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using Trigonometric and Hyperbolic Function Methods.

The cosine-function method and the modified extended tanh method. to generalized Zakharov system

Kinetic equation for a dense soliton gas

Research Article Application of the G /G Expansion Method in Ultrashort Pulses in Nonlinear Optical Fibers

MATHEMATICAL STRUCTURES IN CONTINUOUS DYNAMICAL SYSTEMS

On Decompositions of KdV 2-Solitons

Soliton and Numerical Solutions of the Burgers Equation and Comparing them

Lecture Notes on the Fermi-Pasta-Ulam-Tsingou Problem

2.10 Saddles, Nodes, Foci and Centers

The first three (of infinitely many) conservation laws for (1) are (3) (4) D t (u) =D x (3u 2 + u 2x ); D t (u 2 )=D x (4u 3 u 2 x +2uu 2x ); D t (u 3

JACOBI ELLIPTIC FUNCTION EXPANSION METHOD FOR THE MODIFIED KORTEWEG-DE VRIES-ZAKHAROV-KUZNETSOV AND THE HIROTA EQUATIONS

NUMERICAL SOLUTIONS OF THE FORCED PERTURBED KORTEWEG-DE VRIES EQUATION WITH VARIABLE COEFFICIENTS

Research Article New Exact Solutions for the 2 1 -Dimensional Broer-Kaup-Kupershmidt Equations

KINK DEGENERACY AND ROGUE WAVE FOR POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION

A Short historical review Our goal The hierarchy and Lax... The Hamiltonian... The Dubrovin-type... Algebro-geometric... Home Page.

The Solitary Wave Solutions of Zoomeron Equation

From the SelectedWorks of Ji-Huan He. Soliton Perturbation. Ji-Huan He, Donghua University. Available at:

Model Equation, Stability and Dynamics for Wavepacket Solitary Waves

ADVANCES IN MECHAN ICS Aug125, ,, KdV. , KdV. KdV. (m KdV). (non - propagating soliton or standing soliton), (precursor soliton)

SOLITON SOLUTIONS OF THE CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION WITH VARIABLE COEFFICIENTS

Exact Solutions of the Generalized- Zakharov (GZ) Equation by the Infinite Series Method

The 3-wave PDEs for resonantly interac7ng triads

Kink-antikink collisions in the φ 4 equation: The n-bounce resonance and the separatrix map

TRAVELLING WAVE SOLUTIONS

Wronskian, Grammian and Pfaffian Solutions to Nonlinear Partial Differential Equations

The extended homogeneous balance method and exact 1-soliton solutions of the Maccari system

Solitons and Inverse Scattering Transform: a brief introduction

An Introduction to Nonlinear Waves. Erik Wahlén

SYMBOLIC SOFTWARE FOR SOLITON THEORY: INTEGRABILITY, SYMMETRIES CONSERVATION LAWS AND EXACT SOLUTIONS. Willy Hereman

NUMERICAL SIMULATIONS OF THE STOCHASTIC KDV EQUATION

arxiv: v1 [math-ph] 14 Jan 2019

Freedom in the Expansion and Obstacles to Integrability in Multiple-Soliton Solutions of the Perturbed KdV Equation

EXACT TRAVELLING WAVE SOLUTIONS FOR NONLINEAR SCHRÖDINGER EQUATION WITH VARIABLE COEFFICIENTS

Waves on deep water, I Lecture 13

New Solutions for Some Important Partial Differential Equations

Applied Asymptotic Analysis

1 Introduction. A. M. Abourabia 1, K. M. Hassan 2, E. S. Selima 3

Relation between Periodic Soliton Resonance and Instability

Dynamics of solitons of the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients

The effect of disturbances on the flows under a sluice gate and past an inclined plate

Solitons and Radiation in Nonintegrable Systems

A Generalized Method and Exact Solutions in Bose Einstein Condensates in an Expulsive Parabolic Potential

Computers and Mathematics with Applications. Stability analysis for Zakharov Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma

Transcription:

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, 2017 1 / 38

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 (1808 1882) T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, 2017 2 / 38

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, 2017 3 / 38

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, 2017 4 / 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 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, 2017 5 / 38

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, 2017 6 / 38

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, 2017 7 / 38

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, 2017 8 / 38

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, 2017 9 / 38

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, 2017 10 / 38

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, 2017 11 / 38

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, 2017 12 / 38

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, 2017 13 / 38

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, 2017 14 / 38

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, 2017 15 / 38

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) 1 2 3 4 5 T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, 2017 16 / 38

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, 2017 17 / 38

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, 2017 18 / 38

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, 2017 19 / 38

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, 2017 20 / 38

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, 2017 21 / 38

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, 2017 22 / 38

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+1....... = 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, 2017 23 / 38

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, 2017 24 / 38

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, 2017 25 / 38

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, 2017 26 / 38

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, 2017 27 / 38

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, 2017 28 / 38

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, 2017 29 / 38

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 2 1 + 3b 1 a 2 1 2a 3 1 2b 3 1 9a 1 a 4 b 4 9b 1 b 4 a 4 + 27a 0 b 2 b 4 + 27b 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, 2017 30 / 38

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

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

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

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

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, 448:156 170, 2017. 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, 132:223 231, 2017. 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, 81:1163 1190, 2016. R. Marcinkevicius, Z. Navickas, M. Ragulskis, and T. Telksnys. Solitary solutions to a relativistic two-body problem. Astrophysics and Space Science, 361:201 209, 2016. Z. Navickas, R. Vilkas, T. Telksnys, and M. Ragulskis. Direct and inverse relationships between riccati systems coupled with multiplicative terms. Journal of Biological Dynamics, 10:297 313, 2016. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, 2017 35 / 38

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, 283:333 338, 2016. Z. Navickas, M. Ragulskis, R. Marcinkeviius, and T. Telksnys. Generalized solitary waves in nonintegrable kdv equations. Journal of Vibroengineering, 18:1270 1279, 2016. Z. Navickas, T. Telksnys, and M. Ragulskis. Comments on The exp-function method and generalized solitary solutions. Computers and Mathematics with Applications, 69(8):798 803, 2015. 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, 243:419 425, 2014. Z. Navickas and M. Ragulskis. Comments on Two exact solutions to the general relativistic Binets equation. Astrophysiscs Space Science, 344:281 285, 2013. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, 2017 36 / 38

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, 18:1374 1389, 2013. Z. Navickas, M. Ragulskis, and L. Bikulciene. Special solutions of Huxley differential equation. Mathematical Modeling and Analysis, 16:248 259, 2011. Z. Navickas, L. Bikulciene, and M. Ragulskis. Generalization of Exp-function and other standard function methods. Applied Mathematics and Computation, 216:2380 2393, 2010. Z. Navickas, M. Ragulskis, and L. Bikulciene. Be careful with the Exp-function method additional remarks. Communications in Nonlinear Science and Numerical Simulation, 15:3874 3886, 2010. Z. Navickas and M. Ragulskis. How far can one go with the Exp-function method? Applied Mathematics and Compututation, 211:522 530, 2009. Z. Navickas and L. Bikulciene. Expressions of solutions of ordinary differential equations by standard functions. Mathematical Modeling and Analysis, 11:399 412, 2006. T. Telksnys (KUT FMNS) DiffEq & App 2017 Brno September 6th, 2017 37 / 38

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

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback