Nonlinear Evolution Equations Research Articles

Rogue waves for the (2+1)-dimensional Myrzakulov–Lakshmanan-IV equation on a periodic background

In this paper, the rogue wave solutions of the (2+1)-dimensional Myrzakulov–Lakshmanan (ML)-IV equation, which is described by five component nonlinear evolution equations, are studied on a periodic background. By using the Jacobian elliptic function expansion method, the Darboux transformation (DT) method and the nonlinearization of the Lax pair, two kinds of rogue wave solutions which are expressed by Jacobian elliptic functions dn and cn, are obtained. The relationship between these five kinds of potential is summarized systematically. Firstly, the periodic rogue wave solution of one potential is obtained, and then the periodic rogue wave solutions of the other four potentials are obtained directly. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u=be−ax2, a<0, a,b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u=be−ax22, respectively, for the third-order PDE, u=AeiΦ, where the amplitude b and the phase function a are complex-valued functions, A>0, and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.

Lump, periodic, travelling, semi-analytical solutions and stability analysis for the Ito integro-differential equation arising in shallow water waves

Nonlinear evolution equations are typically used to model and analyse the majority of natural processes. These equations are extremely important for comprehending many intricate situations involving wave propagation, including heat transfer, fluid dynamics, optical fibres, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other nonlinear science fields. In this study, the anticipation of a bilinear Kdv model known as the (1+1)-dimensional integro-differential Ito equation is considered which depicts the oceanic shallow water wave dynamics. Firstly, we determine the bilinear form of the model and then we develop lump waves and analyse their interactions with periodic waves using symbolic computation approaches. Later, the interactions between single, double-kink and lump waves solutions, as well as interactions between periodic, lump and soliton solutions is considered. Furthermore, the (G′/G2) expansion approach and the Adomian technique are implemented to acquire a variety of novel configurations for the governing dynamical model. Later, the corresponding absolute accuracy is evaluated through a comparison of several types of observed solutions. Additionally, the stability of the solitary travelling wave solutions to the specified model is also established. The 3D, 2D, and contour plots have been employed to demonstrate the characteristics of the nonlinear model and aid in determine the proper set of parameters. As a result, a collection of options for bright, dark, periodic, rational, and elliptic functions is established. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the integro-differential equation appearing in shallow water waves. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

Similarity Analysis of Lax Pairs for a Class of Nonlinear Evolution Equations.

Similarity Analysis of Lax Pairs for a Class of Nonlinear Evolution Equations.

Optical soliton solutions to the coupled Kaup-Newell equation in birefringent fibers

One of the fastest growing fields of study in nonlinear optics is optical solitons. This article investigates the optical soliton solutions of the coupled Kaup-Newell equation in birefringent fibers without four wave mixing. This model helps explain how modified structures, particularly Alfven waves, propagate in optical fiber and plasma physics. The results presented in this research clearly show how well the new mapping method handle nonlinear Kaup-Newell equation in coupled vector form, in birefringent fiber without four-wave mixing, to extract the analytical soliton solutions. Four-wave mixing expresses a nonlinear optical effect in which four waves interact each other as a result of the third order nonlinearity. As a valuable outcome of this article, a variety of solitons are discovered including bright, dark, periodic, singular bell-shaped, gap, flat-kink, dark-singular, and w-shaped. The existence criteria for these solitons are also obtained that are presented as constraint conditions. The constraint relations guarantee the existence of such solitons. Furthermore, graphs in 2D, 3D and contour are created to show the physical characteristics of the solutions that are produced. The new mapping method is clear and effective, and the solutions provide a wealth of avenues for additional research. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. The outcomes presented in this work are new versions which have never been observed before in existing literature.

New traveling‐wave solutions for a nonlinear Dodd–Bullough–Mikhailov equation

In this article, we will present new and more general traveling‐wave solutions for the Dodd–Bullough–Mikhailov equation based on the extended Fan sub‐equation method. The two‐dimensional affine sphere's intrinsic geometry is determined by Dodd–Bullough in the three‐dimensional unimodular affine space. The most significant accomplishment of our method is that we are able to provide all forms of the previously obtained solutions in one action, using at least four different approaches. As a result, the general elliptic equation with five parameters can be connected to other sub‐equations with three parameters that already exist, such as the Riccati equation, the first sort of elliptic equation, the auxiliary ordinary equation, and the modified Riccati equation. The extended Fan sub‐equation method is utilized to construct traveling‐wave solutions, solitary wave solutions, dark soliton solutions, and bell soliton solutions for the nonlinear Dodd–Bullough–Mikhailov equation. Mathematica is utilized to depict the dynamics of various wave structures in 3D, 2D, and contour formats, utilizing a specific set of parameters. Our graphical comparative analysis suggests that the employed method is both reliable and powerful for obtaining exact solutions of nonlinear evolution equations. The approach employed in this study may also be applicable in enhancing the understanding of numerous other complex physical phenomena.

Exclusive soliton solutions arise in mono-mode optical fibre connecting to nonlinear Fokas system

Nonlinear partial evolution equations are mostly significant to illustrate critical phenomena in wave theory concerning real-world problems. The current study deals with the (2 + 1)-dimensional nonlinear Fokas model depicting the nonlinear pulse through the mono-mode optical fibers. Improved auxiliary equation and improved tanh schemes are executed on the considering governing system. Subsequently, a variety of optical soliton solutions with the nature of dynamic nonlinear waves are made accessible throughout the present exploration. Some of constructed solutions are figured out in 3D, 2D and contour sense for the visualization to the readers for making them understand of the characteristics of dynamic waves. The solitons are visible to be bright, dark, kink, anti-kink, singular kink, periodic, compacton, anti-compacton etc. in the current exploration. Involved free parameters are assigned with various numerical values and brought out the effects of nonlinear pulses in wave propagation along mono-mode optical fibers. The entire work might claim to be recorded in the literature as new aspects of research.

Wave propagation analysis in the modified nonlinear time fractional Harry Dym equation: Insights from Khater II method and B-spline schemes

This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation.

Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation

We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns using analytical or semi-analytical approach. The interaction of quasiparticles stems from nonlocal competitive losses in the FKPP model. We developed the formalism of our approach relying on ideas of the Maslov method. The construction of the asymptotic expansion of a solution to the original nonlinear evolution equation is based on solutions to an auxiliary dynamical system of ODEs. The asymptotic solutions for various specific cases corresponding to various spatial profiles of the reproduction rate and nonlocal competitive losses are studied within the framework of the approach proposed.

On new computations of the time-fractional nonlinear KdV-Burgers equation with exponential memory

This paper examines the Korteweg–de Vries-Burgers (KdV-Burgers) equation with nonlocal operators using the exponential decay and Mittag-Leffler kernels. The Caputo-Fabrizio and Atangana-Baleanu operators are used in the natural transform decomposition method (NTDM). By coupling a decomposition technique with the natural transform methodology, the method provides an effective analytical solution. When the fractional order is equal to unity, the proposed approach computes a series form solution that converges to the exact values. By comparing the approximate solution to the precise values, the efficacy and trustworthiness of the proposed method are confirmed. Graphs are also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both operators outcome is examined using diagrams and numerical data. These graphs show how the approximated solution’s graph and the precise solution’s graph eventually converge as the non-integer order gets closer to 1. The outcomes demonstrate the method’s high degree of accuracy and its wide applicability to fractional nonlinear evolution equations. In order to further explain these concepts, simulations are run using a computationally packed software that helps interpret the implications of solutions. NTDM is considered the best analytical method for solving fractional-order phenomena, especially KdV-Burgers equations.

Non‐singular complexiton, singular complexiton and complex multiple soliton solutions to the (3 + 1)‐dimensional nonlinear evolution equation

This research mainly concerned with some new solutions of the (3 + 1)‐dimensional nonlinear evolution equation (NEE). First, we extract the resonant multiple soliton solutions (RMSSs) by taking advantage of the linear superposition principle (LSP) and weight algorithm (WA). Then the non‐singular complexiton and singular complexiton solutions are developed by introducing pairs of the conjugate parameters. Besides, the complex multiple‐soliton solutions are also explored with the aid of the bilinear approach. The graph descriptions of the attained solutions are paraded to show the dynamical properties. The outcomes of this work are all new and are desirous to bring some new perspective to the investigation of the complexiton solutions to the other partial differential equations (PDEs).

Wave analysis in generalized fractional Tzitzéica-type nonlinear PDEs: Contributions to nonlinear sciences

In this paper, the extended direct algebraic approach with the general fractional derivative is employed to attain various novel wave structures of the non-linear fractional Tzitzéica type non-linear evolution equations in the form of kink, singular, periodic singular, dark, bright and dark-bright combine solitons. For the purpose to illustrate the physical behavior of the acquired solutions, some of the extracted results are sketched in the pattern of 3-D and 2-D plots which show the efficiency and authenticity of the proposed method. The under consideration method can also utilize to any other non-linear model appears in optics and engineering.

Mild solutions to a class of nonlinear second order evolution equations

The purpose of this paper is to study the existence of mild solutions to a class of second order nonlinear evolution equations of the form \begin{equation*} \begin{cases} u''(t)+A(u'(t))+B(u(t))\ni f(t), &t\in(0,T),\\ u(0)=u_0, \quad u'(0)=g(u') \end{cases} \end{equation*} where $A\colon D(A)\subseteq X\rightarrow 2^{X}$ is an $m$-accretive operator on a Banach space $X,$ $B: X\rightarrow X$ is a lipschitz mapping, $g\colon C([0,T];X)\to X$ is a function and $f\in L^1(0,T,X)$. We obtain sufficient conditions for this problem to have at least a mild solution.

Bifurcation analysis and exact wave solutions of the nano-ionic currents equation: Via two analytical techniques

In this work, numerous nonlinear evolution equations, namely the nano-ionic current equation, have wide applications in various fields of science and technology. To achieve this, we employ the generalized auxiliary equation and extended hyperbolic function techniques to investigate the exact wave solutions of the model under consideration. We obtained a range of exact wave solutions, including bright, dark, singular-periodic, periodic, and w-shape solutions, for the nonlinear model under examination, as well as rational and exponential function solutions with various structures, which are explored in our manuscript. In this paper, we compared the solutions obtained using the GAE and EHF approaches and also compared the obtained solutions using the GAE scheme with the solutions reported in Aljahdaly et al. [43]. Moreover, the three-dimensional and contour graphs are included to illustrate the structure of some obtained solutions by selecting specific parameter values. Bifurcation analysis is illustrated by phase plane analysis; phase plots are employed to depict the dynamic structure and are also analyzed for the existence of other solutions. These techniques serve as reliable, simple, and potent tools to analyze various nonlinear evolution equations found in physics, applied mathematics, and engineering.

Novel solitonic structure, Hamiltonian dynamics and lie symmetry algebra of biofilm

In this study, the Lie point symmetries and optimal system have been established. We discuss the biofilm model’s soliton solutions. To examine a nonlinear dynamical biofilm system, which is simply a bistable Allen–Cahn equation with quartic potential, to determine solitary wave profiles using the considered equation, a new auxiliary equation technique is used. A suitable variable transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The unified approach is utilized to evaluate periodic solutions, solitary and soliton solutions as well as several newly discovered exact solitary wave solutions, it can be accomplished via Mathematica (https://www.wolfram.com/mathematica/online/). The new auxiliary equation approach is an efficient method for creating unique wave profiles based on a variety of soliton families. Also, the results are graphically visualized, by using the appropriate parametric settings. The outcomes are shown graphically in two dimensions, three dimensions, and contour form. The Hamiltonian conditions are satisfied by the planer dynamical framework of equations to ensure as the system that was generated is a conservative Hamiltonian dynamical system which includes all traveling wave structures. A sensitivity evaluation is used to explore the governing model’s thoroughly dynamical properties. It is demonstrated that the model becomes more dependent on the beginning conditions than the variables. The strategy could be used to look for exact solutions to various nonlinear evolution equations. We believe that this study will have important applications in different areas of science.

Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this article, we consider a new (3 + 1)-dimensional evolution equation, which can be used to interpret the propagation of nonlinear waves in the oceans and seas. We effectively investigate the integrable properties of the considered nonlinear evolution equation through several aspects. First of all, we present some elementary properties of multi-dimensional Bell polynomial theory and its relation with Hirota bilinear form. Utilizing those relations, we derive a Hirota bilinear form and a bilinear Bäcklund transformation. By employing the Cole–Hopf transformation in the bilinear Bäcklund transformation, we present a Lax pair. Additionally, using the Bell polynomial theory, we compute an infinite number of conservation laws. Moreover, we obtain one-, two-, and three-soliton solutions explicitly from Hirota bilinear form and illustrate them graphically. Breather solutions are also derived by employing appropriate complex conjugate parameters in the two-soliton solution. Choosing the generalized algorithm for rogue waves derived from the N-soliton solution, we directly obtain a first-order center-controllable rogue wave. Lump solutions are formulated by employing a well-established quadratic test function as a solution to the Hirota bilinear form. Further taking the test function in a combined form of quadratic and exponential functions, we obtain lump-multi-stripe solutions. Furthermore, a combined form of quadratic and hyperbolic cosine functions produces lump-multi-soliton solutions. The fission and fusion effects in the evolution of lump-multi-stripe solutions and lump-soliton-solutions are demonstrated pictorially.

Modeling the stability of thin liquid film flows on a uniformly heated slippery inclined substrate: A realistic approach

We investigate the stability of gravity-driven, Newtonian, thin liquid film falling down a uniformly heated slippery rigid inclined wall. The authors of previous research works considered specified temperature (ST) boundary condition to study the effects of slip length. However, the ST boundary condition does not include the effects of heat fluxes at wall–air and wall–liquid interfaces and so fails to incorporate the real situation. Consequently, we consider heat flux/mixed-type boundary condition as the thermal boundary condition on the rigid plate. This boundary condition involves the heat flux from the rigid plate to the surrounding liquid and the heat losses from the wall to the ambient air. Using long-wave expansion method, we construct a highly nonlinear evolution equation in terms of the film thickness at any instant. Using normal mode approach, the linear study reveals the stabilizing (destabilizing) behavior of the wall film Biot number (dimensionless slip length). It is found that the destabilizing tendency of the slip length is more in the absence of thermocapillary stress. The linear study reveals that the destabilizing role of MB may be controlled to some extent by increasing the wall film Biot number Bw. Using asymptotic expansions of the flow variables in terms of the small wave number k, the Orr–Sommerfeld boundary value problem gives an onset of instability in terms of critical Reynolds number. It slightly differs from that of the same as obtained by Benney's long-wave expansion method, due to the consideration of small free surface Biot number [B=O(ϵ)]. For arbitrary wave numbers, using Chebyshev spectral collocation method, the effect of Marangoni number (Ma), slip length (δ), and wall film Biot number (Bw) on the H, S, P, and shear modes of instability are discussed in detail. Near the threshold, both Ma and δ show the destabilizing effect on H mode of instability, whereas Bw gives the stabilizing effect. Interestingly, their roles on H mode of instability becomes diametrically opposite far from the onset of instability. For S mode, both Ma and Bw display the destabilizing effect, whereas δ plays the dual role. For P mode, both Ma and δ show the destabilizing effect, whereas Bw plays the stabilizing role. The slip length (δ) plays the stabilizing role, in the case of shear mode. In the absence of thermocapillary effect, the vorticity balance at the liquid–air interface explains that the amplitude of the vorticity perturbation amplifies the surface deformation due to the presence of inertia and the slip length. In the absence of the slip length, a weakly nonlinear study transforms the evolution equation to the famous Kuramoto–Sivashinsky (KS) equation.

Soliton equations: admitted solutions and invariances via B\"acklund transformations

A couple of applications of B\"acklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on proving a new invariance admitted by a third order nonlinear evolution equation and, on the other one, on the construction of solutions. Indeed, via B\"acklund transformations, a {\it B\"acklund chart}, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third order nonlinear evolution equations of KdV type. On the basis of the Abelian wide B\"acklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the {\it Korteweg-deVries interacting soliton} (int.sol.KdV) equation is obtained and a related new explicit solution is constructed. Then, the corresponding non-Abelian {\it B\"acklund chart}, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

A bivariate spectral linear partition method for solving nonlinear evolution equations

This work develops a method for solving nonlinear evolution equations. The method, termed a bivariate spectral linear partition method (BSLPM), combines the Chebyshev spectral collocation method, bivariate Lagrange interpolation, and a linear partition technique as an underlying linearization method. It is developed for an th‐order nonlinear differential equation and then used to solve three known evolution problems. The results are compared with known exact solutions from literature. The method's applicability, reliability, and accuracy are confirmed by the congruence between the numerical and exact solutions. Tables, error graphs, and convergence graphs were generated using MATLAB (R2015a), to confirm the order of accuracy of the method and verify its convergence. The performance of the method is also observed against other methods performing well in these types of differential equations and is found to be comparable in terms of accuracy. The proposed method is also efficient as it uses minimal computation time.