Nonlinear Evolution Partial Differential Equations Research Articles

Optimized technique and dynamical behaviors of fractional Lax and Caudrey–Dodd–Gibbon models modelized by the Caputo fractional derivative

The presented paper aims to investigate, examine, and analyze the nonlinear time-fractional evolution partial differential equations (TFNE-PDEs) in the sense of Caputo essential in numerous nonlinear wave propagation phenomena. To achieve this, the Laplace-residual power series method (L-RPSM) is used to construct approximate Laplace-residual power series solutions (AL-RPSSs) for fifth-order Korteweg–de Vries PDEs (KdV-PDEs). The L-RPSM provides analytical solutions of the dynamic wavefunction of several equations including time-fractional Lax PDE (TFL-PDE), and time-fractional Caudrey–Dodd–Gibbon PDEs (TFCDG-PDEs). The theoretical and numerical consequences of the models are discussed, and error’s analysis of the method are discussed. The outcomes are introduced in 2D and 3D figures, and dynamic behaviors of parameters are discussed for distinct values of α. Moreover, the accuracy of this method is demonstrated by comparing it with results obtained using other methods. The results show that the suggested iterative approach is a suitable tool with computational efficiency for long-wavelength solutions of nonlinear time-fractional PDEs in various phenomena.

Lie symmetry analysis, multiple exp-function method and conservation laws for the (2+1)-dimensional Boussinesq equation

In this study, we take into account the (2 + 1)-dimensional Boussinesq equation, a nonlinear evolution partial differential equation that describes how gravity waves move across the surface of the ocean. The symmetry reductions and group invariant precise solutions are systematically determined using the Lie symmetry analysis. We derive the precise multiple wave solutions using the multiple exp-function method, and then, using the multiplier method, we give the conservation laws. The dynamics of complicated waves and their interplay are faithfully recreated by the findings.

Analysis of soliton interactions of modified Korteweg-de Vries equation using conserved quantities

In this paper, the conservative quantities are used to develop an approximate method to calculate the merged waveform shape of the solitary waves described by modified Korteweg–de Vries (mKdV) equation. With this method, we can efficiently and effectively capture the physics of the complicated merging phenomena when two solitary waves described by the nonlinear evolution partial differential equation merge at the maximum without the need to solve the equation in detail. This offers a simple and robust tool to analyse the interactions between solitons and to benchmark the results obtained by the asymptotic and numerical methods. It is expected that the approximate analysis demonstrated in this paper can be applied to a series of nonlinear evolution equations to simulate various solitary wave interaction problems. In future, our goal is to extend this simple method to other nonlinear wave evolution phenomena.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of an integrable evolution equation in one spatial dimension. Do there exist integrable analogs of the modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Existence of classical solutions for a class of nonlinear impulsive evolution partial differential equations

This paper is devoted to the study of a class of impulsive nonlinear evolution partial differential equations. We give new results about existence and multiplicity of global classical solutions. The method used is based on the use of fixed points for the sum of two operators. Our main results will be illustrated by an application to an impulsive Burgers equation.

Investigation of Exact Solutions of Nonlinear Evolution Equations Using Unified Method

In this article, an analytical technique based on unified method is applied to investigate the exact solutions of non-linear homogeneous evolution partial differential equations. These partial differential equations are reduced to ordinary differential equations using different traveling wave transformations, and exact solutions in rational and polynomial forms are obtained. The obtained solutions are presented in the form of 2D and 3D graphics to study the behavior of the analytical solution by setting out the values of suitable parameters. The acquired results reveal that the unified method is a suitable approach for handling non-linear homogeneous evolution equations.

Pseudo-spectral approach for extracting optical solitons of the complex Ginzburg Landau equation with six nonlinearity forms

In this paper, a compelling pseudo-spectral approach namely split-step Fourier transform (SSFT) is utilized for the first time to report the complex Ginzburg Landau (CGL) equation. In this regard, six nonlinear forms of this notable equation are associated with this model, which are the kerr law nonlinearity, power law nonlinearity, quadratic-cubic nonlinearity, cubic-quintic nonlinearity, dual power nonlinearity, and polynomial law nonlinearity. Since opting for an analytical solution is complicated and only provided for a limited set of soliton solutions, the numerical solution might render a more generalized aspect in addressing versatile solutions for a wide range of arbitrary input pulses. Therefore, the SSFT scheme is efficiently employed to extract a plethora of optical solitons solutions such as kink waves, bright, dark, bright-dark, singular, singular periodic solitons. The obtained results, via MATLAB, are in total agreement with the findings explored form the exact analytical solutions. As a result, this numerical technique may provide a creditable mathematical tool to solve a wide range of other nonlinear evolution partial differential equations in the mathematical physics field.

Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized -symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized -symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized -symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized -symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized -symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized -symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized -symmetric Scarf-II potential in the deep learning.

Stochastic phase field α-Navier-Stokes vesicle-fluid interaction model

We consider a stochastic perturbation of the phase field alpha-Navier-Stokes model with vesicle-fluid interaction. It consists in a system of nonlinear evolution partial differential equations modeling the fluid-structure interaction associated to the dynamics of an elastic vesicle immersed in a moving incompressible viscous fluid. This system of equations couples a phase-field equation -for the interface between the fluid and the vesicle- to the alpha-Navier-Stokes equation -for the viscous fluid- with an extra nonlinear interaction term, namely the bending energy. The stochastic perturbation is an additive space-time noise of trace class on each equation of the system. We prove the existence and uniqueness of solution in classical spaces of L2 functions with estimates of non-linear terms and bending energy. It is based on a priori estimate about the regularity of solutions of finite dimensional systems, and tightness of the approximated solution.

Solving second-order nonlinear evolution partial differential equations using deep learning**The project is supported by the National Natural Science Foundation of China (No. 11675054), Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers’ equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions.

Embedded solitons in the $$(2+1)$$-dimensional sine-Gordon equation

An effective and simple method to solve nonlinear evolution partial differential equations is the self-similarity transformation, in which one utilizes solutions of the known equation to find solutions of the unknown. In this paper, we employ an improved similarity transformation to transform the $$(2+1)$$-dimensional (D) sine-Gordon (SG) equation into the $$(1+1)$$-D SG equation and obtain non-rational solutions of the $$(2+1)$$-D SG equation by utilizing the known solutions of the $$(1+1)$$-D SG equation. Based on the solutions obtained, and with the help of special choices of the involved solution parameters, several localized structures of the $$(2+1)$$-D SG model are analyzed on a finite background, such as the embedded hourglass, split silo, dumbbell, and pie solitons. Their spatiotemporal profiles are displayed, and their properties are discussed.

Solitary wave solutions of pZK equation using Lie point symmetries

The nonlinear propagation of dust-ion acoustic solitary waves and shocks can be represented by a nonlinear evolution partial differential equation, namely the perturbed $$(3+1)$$-dimensional Zakharov–Kuznetsov (pZK) equation. Based on some subalgebras of symmetries, several reductions and many group-invariant solutions are found for the pZK equation. One of the reduced partial differential equations is dealt using new generalized exponential rational function method which was proposed by Ghanbari and Inc (Eur. Phys. J. Plus 133: 142, 2018), to obtain closed-form analytical solutions. Obtained solutions are new solitary wave, multi-soliton and kink type which is significant in the field of plasma physics.

Analytic solutions for a new model of the(3+1)-Boussinesq equation

In this paper, we have applied the mapping method to solve the (3+1)-dimensional Boussinenesq equation where we have obtained exact solutions for evolution equation to construct exact periodic and soliton solutions of nonlinear partial differential evolution equation. Many have obtained new families of exact traveling wave solutions, but the Boussinesq equation is successfully. These solutions may be significantly important for the explanation of some practical physical problems. New exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions and elliptic functions. It is shown that the mapping method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

N-soliton and rogue wave solutions of (2+1)-dimensional integrable system with Lax pair

A new generalized (2[Formula: see text]+[Formula: see text]1)-dimensional model, obtained from the Kadomtsev–Petviashvili (KP) equation considered as a system of nonlinear evolution partial differential equations (PDEs), is introduced. With the usage of the Hirota bilinear method and the KP-hierarchy reduction method, N-soliton solutions of the integrable system are constructed. Considering the case of s[Formula: see text]=[Formula: see text]−1 in the linear differential operators, L1 and L2, and a specific set of parameters, two dark solitons, mixed solutions consisting of soliton-type and periodic waves solution are obtained. Based on the particular definition of the matrix elements, one and two rogue waves solutions expressed in terms of rational functions are derived. It is shown that the fundamental rogue waves are line rogue waves, which is different from the property of the moving line solitons of the soliton equations.

An upper order bound of the invariant manifold in Lax pairs of a nonlinear evolution partial differential equation

In Habibullin et al (2016 J. Phys. A: Math. Theor. 49 035202); Habibullin and Khakimova (2017 J. Phys. A: Math. Theor. 50 305206), Habibullin et al proposed an approach to construct Lax pairs of a nonlinear integrable partial differential equation (PDE), where one is the linearized equation of the studied PDE and the other is the invariant manifold of the linearized equation. In this paper, we show that the invariant manifold is the characteristic of a generalized conditional symmetry of the system composed of the studied PDE and its linearized PDE. Then we give an upper order bound of the invariant manifold which provides a theoretical basis for a complete classification of such type of invariant manifold. As an application, we give a complete classification of the given type invariant manifold for the KdV and mKdV equations and also construct several invariant manifolds and Lax pairs for the Sharma–Tasso–Olver equation.

Solutions and conservation laws of a generalized 3D Kawahara equation

We study a nonlinear evolution partial differential equation, namely, the generalized $(3+1)$ -dimensional (3D) Kawahara equation. The Lie symmetry method in conjunction with the Kudryashov method will be employed to derived solutions of the 3D generalized Kawahara equation. Furthermore, the multiplier method will be implemented to construct conservation laws of the generalized 3D Kawahara equation. Thereafter, a brief physical meaning of the derived conserved quantities is discussed.

Complexification and integrability in multidimensions

The complexification of the independent variables of nonlinear integrable evolution partial differential equations (PDEs) in two space dimensions, like the celebrated Kadomtsev-Petviashvili and Davey-Stewartson (DS) equations, yields nonlinear integrable equations in genuine 4 + 2, namely, in four real space dimensions (x1, x2, y1, y2) and two real time dimensions (t1, t2), as opposed to two complex space dimensions and one complex time dimension. The associated initial value problem for such equations, namely, the problem where the dependent variables are specified for all space variables at t1 = t2 = 0, can be solved via a non-local d-bar formalism. Here, the details of this formalism for the 4 + 2 DS system are presented. Furthermore, the linearised version of the 3 + 1 reduction of the 4 + 2 DS system is discussed. The construction of the nonlinear 3 + 1 reduction remains open, in spite of the fact that multi-soliton solutions for the 3 + 1 DS system already exist.

Parametric excitation of an axisymmetric flow of a thin liquid film down a vertical fiber

We investigate axisymmetric flow of thin liquid films falling down the outer side of a vertical cylindrical surface oscillating harmonically in the axial direction. A set of two coupled nonlinear partial differential evolution equations in terms of the local instantaneous film thickness and volumetric flow rate is derived using the Galerkin method. A linear stability analysis of the time-periodic thickness-uniform base state is performed using the Floquet theory. We carry out numerical investigation of the nonlinear dynamics of the films in the framework of the set of evolution equations derived here. We find that the film evolution exhibits strange attractor dynamics ranging from coherent to fully irregular regimes. We show that harmonic parametric excitation affects the spatial topological structure of the interfacial wave and may modify its type from depression wave to elevation wave. The harmonic parametric excitation may also result in a significant decrease in the interfacial wave amplitude.

A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations

Abstract This article presents a new method of solving partial differential equations. The method is an improvement of the previously reported compact finite difference quasilinearization method (CFDQLM) which is a combination of compact finite difference schemes and quasilinearization techniques. Previous applications of compact finite difference (FD) schemes when solving parabolic partial differential equations has been solely on discretizing the spatial variables and another numerical technique used to discretize temporal variables. In this work we attempt, for the first time, to use the compact FD schemes in both space and time. This ensures that the rich benefits of the compact FD schemes are carried over to the time variable as well, which improves the overall accuracy of the method. The proposed method is tested on four nonlinear evolution equations. The method produced highly accurate results which are portrayed in tables and graphs.

A Multi-Domain Bivariate Pseudospectral Method for Evolution Equations

In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre–Gauss–Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV–Burgers equation, modified KdV equation, Fisher’s equation, Burgers–Fisher equation, Burgers–Huxley equation and the Fitzhugh–Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.