Algorithm for solving a generalized Hirota-Satsuma Coupled KdV equation using homotopy perturbation transform method

In this paper, we merge homotopy perturbation method with He’s polynomials and Laplace transformation method to produce a highly effective algorithm for finding approximate solutions for generalized Hirota-Satsuma Coupled KdV equations. This technique is called the Homotopy Perturbation Transform Method (HPTM). With this technique, the solutions are obtained without any discretization or restrictive assumptions, and devoid of roundoff errors. This technique solved a generalized Hirota-Satsuma Coupled KdV equation without using Adomian’s polynomials which can be considered as a clear advantage over the decomposition method. MAPLE software was used to calculate the series generated from the algorithm. The results reveal that the homotopy perturbation transform method (HPTM) is very efficient, simple and can be applied to other nonlinear problems.Keywords: Coupled KdV equations, Homotopy perturbation transform method, Laplace transform method, Maple software, He’s polynomia

Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations

This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equation

Higher Order Non-Planar Electrostatic Solitary Potential in a Streaming Electron-Ion Magnetoplasma: Phase Plane Analysis

We investigate cylindrical and spherical solitons in electron-ion (EI) plasma that contains hot (cold) electrons with stationary ions. The magneto-hydrodynamic equations are solved with the aid of the reductive perturbation (RP) technique, leading to the modified Korteweg–De Vries (mKdV) equation for the non-linear behaviour of the solitary waves in EI plasma. By employing the reduced differential transform method (RDTM), an approximate solution of the mKdV is obtained for solitary waves. Phase plane analysis reveals that these excitations exhibit periodic oscillations. The phase plane and periodic behaviour of the obtained model are studied. It is observed that the amplitude and width of the electron acoustic waves (EAWs) are affected by a slight change in the cold to hot electron temperature ratio (σc) and the number density of the cold to hot electron ratio (α). The effect of the streaming speed u(0) and superthermality index κe are investigated. This study is important for understanding the symmetric properties of cylindrical and spherical plasma, relying on the bifurcation analysis, impacted by the streaming effect in the EI plasma.Basque Government: Grant IT1555-22 Basque Government: Grant KK-2022/00090 MCIN/ AEI 269.10.13039/501100011033: Grant PID2021-1235430B-C21/C22

On a deformation of the nonlinear Schr\"odinger equation

We study a deformation of the nonlinear Schr\"odinger equation recently derived in the context of deformation of hierarchies of integrable systems. This systematic method also led to known integrable equations such as the Camassa-Holm equation. Although this new equation has not been shown to be completely integrable, its solitary wave solutions exhibit typical soliton behaviour, including near elastic collisions. We will first focus on standing wave solutions, which can be smooth or peaked, then, with the help of numerical simulations, we will study solitary waves, their interactions and finally rogue waves in the modulational instability regime. Interestingly the structure of the solution during the collision of solitary waves or during the rogue wave events are sharper and have larger amplitudes than in the classical NLS equation

Particle and particle-like solitary wave dynamics in fluid media

This research deals with the study of various nonlinear wave processes in dispersive media by means of asymptotic methods developed upon existing exact methods in application to non-integrable systems. The aim of the research is to analyse wave models possessing solitary solutions and establish common features in the description of such solutions and classical particles. The new model equations have been derived for the description of long transverse waves propagating in the generalized atomic chain. The mathematical analogy between the model equations describing internal waves in stratified fluid (the Korteweg–de Vries and Gardner–Ostrovsky equations) and waves in discrete chain models (the generalized sine-Gordon–Toda model or Frenkel–Kontorova model) have been established. Chain models are described by sets of ODEs which can be readily solved with a high accuracy by existing well-developed solvers in mathematical software. The research includes solutions to important wave problems by means of approximate asymptotic and numerical methods. Results obtained provide an insight in understanding of details of nonlinear wave propagation in continuous and discrete media. An effective numerical code has been developed for the modeling of nonlinear phenomena both in continuous media and in the discrete models of interacting oscillators

Helical solitons in vector modified Korteweg-de Vries equations

We study existence of helical solitons in the vector modified Korteweg-de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is confirmed by direct numerical simulations.Comment: 20 pages, 10 figure

Exp-function Method for Wick-type Stochastic Combined KdV-mKdV Equations

Exp-function method is proposed to present soliton and periodic wave solutions for variable coefficients combined KdV- mKdV equation. By means of Hermite transform and white noise analysis, we consider the variable coefficients and Wick-type stochastic combined KdV-mKdV equations. As a result, we can construct new and more general formal solutions. These solutions include exact stochastic soliton and periodic wave solutions.Keywords: combined KdV-mKdV equation, Exp-function method, Wick product, Hermite transform, White noise

Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations

Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes

This work presents the application of the differential transform method (DTM) to the model of pollution for a system of three lakes interconnected by channels. Three input models (periodic, exponentially decaying, and linear) are solved to show that DTM can provide analytical solutions of pollution model in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend of a perturbation parameter