Nonlinear Differential-difference Equations Research Articles

A SIMPLE JUSTIFICATION OF TANH METHOD FOR DISCRETE SOLITARY WAVE SOLUTIONS

The tanh method is known as an effective method for solving partial differential equations and obtaining traveling wave solutions. While well-used in continuous systems, its application to discrete systems (nonlinear difference-differential equations, DDEs) is still less explored. This paper revisits the tanh method for discrete systems, providing a simple justification for the method to produce discrete solitary wave solutions, including pulse-type and (anti)kink-type. By transforming DDEs into nonlinear ordinary differential equations (ODEs) and using polynomial forms in terms of the tanh function, we outline steps for deriving and verifying exact solutions

Simplifications of Lax pairs for differential–difference equations by gauge transformations and (doubly) modified integrable equations

Matrix differential–difference Lax pairs play an essential role in the theory of integrable nonlinear differential–difference equations. We present sufficient conditions which allow one to simplify such a Lax pair by matrix gauge transformations. Furthermore, we describe a procedure for such a simplification and present applications of it to constructing new integrable equations connected by (non-invertible) discrete substitutions of Miura type to known equations with Lax pairs.Suppose that one has three (possibly multicomponent) equations E, E1, E2, a (Miura-type) discrete substitution from E1 to E, and a discrete substitution from E2 to E1. Then E1 and E2 can be called a modified version of E and a doubly modified version of E, respectively. We demonstrate how the above-mentioned procedure helps (in the considered examples) to construct modified and doubly modified versions of a given equation possessing a Lax pair satisfying certain conditions.The considered examples include scalar equations of Itoh–Narita–Bogoyavlensky type and 2-component equations related to the Toda lattice. We present several new integrable equations connected by new discrete substitutions of Miura type to known equations.

On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

This paper has involved the use of a variety of variations of the Fermat-type equation $f^n(z)+g^n(z)=1$, where $n(\geq 2)\in\mathbb{N}$. Many researchers have demonstrated a keen interest to investigate the Fermat-type equations for entire and meromorphic solutions of several complex variables over the past two decades. Researchers utilize the Nevanlinna theory as the key tool for their investigations. Throughout the paper, we call the pair $(f,g)$ as a finite order entire solution for the Fermat-type compatible system $\begin{cases} f^{m_1}+g^{n_1}=1;\\ f^{m_2}+g^{n_2}=1,\end{cases}$\!\! if $f$, $g$ are finite order entire functions satisfying the system, where $m_1,m_2,n_1,n_2\in\mathbb{N}\setminus\{1\} .$\ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: $\begin{cases} f^{m_1}+2\alpha f g+g^{n_1}=1;\\ f^{m_2}+2\alpha f g+g^{n_2}=1,\end{cases}$ \!\! where $\alpha\in\mathbb{C}\setminus\{0,\pm1\}.$ In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on $\mathbb{C}^2$, especially, those of Xu {\it{et al.}} (Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483(2), 2020) and then construct some systems of certain quadratic trinomial partial differential-difference equations with arbitrary coefficients. Our objective is to investigate the forms of the finite order transcendental entire functions of several complex variables satisfying the systems of certain quadratic trinomial partial differential-difference equations on $\mathbb{C}^n$. These results will extend the further study of this direction.

Meromorphic Solutions of Nonlinear Differential-Difference Equations Involving Periodic Functions

Meromorphic Solutions of Nonlinear Differential-Difference Equations Involving Periodic Functions

Existence of entire solutions for k-th derivative of certain type of non-linear differential-difference equations

The paper discusses the existence of entire solutions for k th derivative of certain type of non-linear differential-difference equations. In addition to, the paper evaluates some previous results for existence of finite order transcendental entiresolutions and extends to the higher order. Particularly, we obtain the conditions for occurrence and attainable forms of entire solutions.

The study of solutions of several systems of the product type nonlinear partial differential difference equations

The study of solutions of several systems of the product type nonlinear partial differential difference equations

Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂ n#

Abstract In this article, we investigate the existence and the precise form of finite-order transcendental entire solutions of some system of Fermat-type quadratic binomial and trinomial shift equations in C n {{\mathbb{C}}}^{n} . Our results are the generalizations of the results of [H. Y. Xu, S. Y. Liu, and Q. P. Li, Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483 (2020), 123641, 1–22, DOI: https://doi.org/10.1016/j.jmaa.2019.123641.] and [H. Y. Xu and Y. Y. Jiang, Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables, RACSAM 116 (2022), 8, DOI: https://doi.org/10.1007/s13398-021-01154-9.] to a large extent. Most interestingly, as a consequence of our main result, we have shown that the system of quadratic trinomial shift equation has no solution when it reduces to a system of quadratic trinomial difference equation. In addition, some examples relevant to the content of the article have been exhibited.

Erratum to: Some Results on Nonlinear Difference-Differential Equations

Erratum to: Some Results on Nonlinear Difference-Differential Equations

On Meromorphic Solutions of Non-linear Differential-Difference Equations

In this paper, we investigate the non-existence of transcendental entire solutions for non-linear differential-difference equations of the forms fn(z)+Q(z,f)=β1eα1z+β2eα2z+⋯+βseαsz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f^{n}(z)+Q(z,f)=\\beta _{1}e^{\\alpha _{1}z}+\\beta _{2}e^{\\alpha _{2}z}+\\cdots +\\beta _{s}e^{\\alpha _{s}z} \\end{aligned}$$\\end{document}and fn(z)f(k)(z)+Ld(z,f)=∑i=1spi(z)eαi(z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f^{n}(z)f^{(k)}(z)+L_d(z,f)=\\sum ^{s}_{i=1}p_i(z)e^{\\alpha _i{(z)}}, \\end{aligned}$$\\end{document}where n, s are positive integers, nge s+2,Q(z, f) is a differential-difference polynomial in f of degree d.

Method of separation of variables and exact solution of time fractional nonlinear partial differential and differential-difference equations

Method of separation of variables and exact solution of time fractional nonlinear partial differential and differential-difference equations

Some Results on Nonlinear Difference-Differential Equations

Some Results on Nonlinear Difference-Differential Equations

Construction of abundant solitons in a coupled nonlinear pendulum lattice through two discrete distinct techniques

In this paper, we deal with two distinct discrete techniques named the discrete Tanh method and the differential-difference Jacobi elliptic functions sub-ODE method to extract abundant soliton solutions for a coupled nonlinear pendulum chains. According to the two straightforward schemes’, we construct various solutions likewise, kink, anti-kink, bright, dark, singular-bright solitary wave, traveling compacton, traveling gray soliton-like, periodic soliton-like have not reported on the most recently studied mechanical model. We extract the sought solutions with less approximation (introduction of a discrete wave transformation) and display some graphical representations showing well known, novel shapes discovered in a coupled nonlinear pendulum lattice. More often, the tediousness computations of discrete techniques lead some researchers to changing a nonlinear differential-difference equation into a nonlinear ordinary differential equation throughout the semi-discrete approximation before introducing a wave transformation. Nonetheless, the semi-discrete approximation involve a Taylor bounded expansion’s such as the threshold order depends on the ability and skills of researchers to get solutions through available methods and those that they can perform. Hence, while an algorithm can be lead us to avoiding approximation (eventual source of errors), tediousness analytical computations, we perform by using it, in addition of software availability. And therefore, we establish as is the case in the present work various soliton-like solutions by means of two direct discrete techniques. The effect of some physical parameters are displayed on the shaped through 2D and 3D graphical simulations. The used methods (tanh, sn, cn, dn Jacobi elliptic functions approaches) previously applied on other physical models (discrete electrical transmission lattices) are presented here in manner to be replicable for exploring more soliton solutions in different nonlinear discrete physical models.

Toda lattice with constraint of type B

We introduce a new integrable hierarchy of nonlinear differential–difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the constraint can be regarded as a discretization of the BKP hierarchy. We construct its algebraic–geometrical solutions in terms of Riemann and Prym theta-functions.

Soliton solutions for the time-fractional nonlinear differential-difference equation with conformable derivatives in the ferroelectric materials

Soliton solutions for the time-fractional nonlinear differential-difference equation with conformable derivatives in the ferroelectric materials

Quasi‐periodic solutions to a hierarchy of integrable nonlinear differential–difference equations

Using the discrete zero‐curvature equation, we derive a hierarchy of new nonlinear differential–difference equations associated with a discrete matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix, we introduce a trigonal curve and the associated three‐sheeted Riemann surface, from which we derive the Baker–Akhiezer function and meromorphic function. By comparing the asymptotic expansions of the meromorphic function and its Riemann theta function representations, we obtain quasi‐periodic solutions for a hierarchy of nonlinear differential–difference equations.

Dynamic Model of Economic Agents Including Interaction and Delay Effects

Abstract: The dynamic model of economic growth of several agents is presented. This is a system of non-linear differential-difference equations. The model takes into account the interaction of economic agents and the delay effect. Some simple analytic particular solutions are shown. These analytical solutions allow some clear economic interpretation. The comparative scenario quantitative calculations are given. The calculations are presented for time period of a decade for the typical global development rate of 3% per year. There are two main scenarios. The first scenario of cooperative or isolated agent development is considered. There is the scenario of the agent development due to use the other partners. The consequence of isolation is the backlog of 60% per decade. The second scenario is the time delay-lead effect to use the achievements of other agents. A lead of one - two years is considered. In this case, ceteris paribus, the dynamic growth of the agent is 1.5 times ahead of the others. More effective growth occurs with the faster and advance reaction rate to partner achievements. In this case there is the effect of the multiplicative accumulation of advantage. This scenario can be interpreted as the illustration of the accelerated economic growth of post-war Japan.

Lie symmetry analysis and exact solution of $$(2+1)$$-dimensional nonlinear time-fractional differential-difference equations

Lie symmetry analysis and exact solution of $$(2+1)$$-dimensional nonlinear time-fractional differential-difference equations

Nonlinear differential-difference hierarchy relevant to the Ablowitz-Ladik equation: Lax pair, conservation laws, N-fold Darboux transformation and explicit exact solutions

Nonlinear differential-difference equations appear in optics, condensed matter physics, plasma physics and other fields. In this paper, we investigate a nonlinear differential-difference hierarchy relevant, in the case of θ=0, to the Ablowitz-Ladik equation, where θ=0,1. That hierarchy is obtained via a discrete spectral problem and the associated discrete spectral problem. When θ=1, Lax pair of the first nonlinear differential-difference system in that hierarchy is obtained. When θ=1, conservation laws and N-fold Darboux transformation of the first nonlinear differential-difference system in that hierarchy are derived with the aid of that Lax pair, where N is a positive integer. When θ=1, explicit exact solutions of that system are determined via that N-fold Darboux transformation. Discrete one soliton and interaction between the discrete one soliton and one breather-like wave are graphically depicted.

Entire solutions of one certain type of nonlinear differential-difference equations

We describe the entire solutions for two kinds of nonlinear differential-difference equations of the form fn(z)+ωfn−1(z)f′(z)+q1(z)eQ1(z)fc1+q2(z)eQ2(z)fc2=u(z)ev(z),n≥3 and fn(z)+q(z)eQ(z)f(z+c)=u(z)ev(z),n≥2, where q,Q,u,v,qj(z),Qj(z), for j=1,2, are polynomials such that Q(z) and at least one of Qj(z) are not constants, q(z) and qj(z) are not identically zero, and ω,c1,c2,c are constants. Our results improve and generalize some previous results.

DISCRETE MATURITY AND DELAY DIFFERENTIAL-DIFFERENCE MODEL OF HEMATOPOIETIC CELL DYNAMICS WITH APPLICATIONS TO ACUTE MYELOGENOUS LEUKEMIA

In the last few years, many efforts were oriented towards describing the hematopoiesis phenomenon in normal and pathological situations. This complex biological process is organized as a hierarchical system that begins with primitive hematopoietic stem cells (HSCs) and ends with mature blood cells: red blood cells, white blood cells and platelets. Regarding acute myelogenous leukemia (AML), a cancer of the bone marrow and blood, characterized by a rapid proliferation of immature cells, which eventually invade the bloodstream, there is a consensus about the target cells during the HSCs development which are susceptible to leukemic transformation. We propose and analyze a mathematical model of HSC dynamics taking into account two phases in the cell cycle, a resting and a proliferating one, by allowing just after division a part of HSCs to enter the resting phase and the other part to come back to the proliferating phase to divide again. The resulting mathematical model is a system of nonlinear differential-difference equations. Due to the hierarchical organization of the hematopoiesis, we consider n stages of HSCs characterized by their maturity levels. We obtain a system of 2n nonlinear differential-difference equations. We study the existence, uniqueness, positivity, boundedness and unboundedness of the solutions. We then investigate the existence of positive and axial steady states for the system, and obtain conditions for their stability. Sufficient conditions for the global asymptotic stability of the trivial steady state as well as conditions for its instability are obtained. Using neutral differential equation associated to the differential-difference system, we also obtain results on the local asymptotic stability of the positive steady state. Numerical simulations are carried out to show the influence of variations of the differentiation rates and self-renewal coefficients of the HSCs on the behavior of the system. In particular, we show that a blocking of differentiation at an early stage of HSC development can lead in an overexpression of very immature cells. Such situation corresponds to the observation in the case of AML.