Generalized KdV Equation Research Articles

The structure of algebraic solitons and compactons in the generalized Korteweg–de Vries equation

We analyze the main properties of soliton solutions to the generalized KdV equation ut+[F(u)]x+uxxx=0, where the leading term F(u)∼quα, α>0, q∈R. The far field of such solitons may have three options. For q>0 and α>1 the analysis re-confirmed that all traveling solitons have “light” exponentially decaying tails and propagate to the right. If q<0 and α<1, the traveling solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated Fu and α>1 (e.g., the Gardner equation) standing algebraic solitons with “heavy” power-law tails may appear. If the leading term of Fu is negative, the set of solutions may include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic solitons and compactons with any of the three types of tails. The solutions usually have a single-hump structure but if Fu represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.

Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types

Interaction solutions between lump and soliton are analytical exact solutions to nonlinear partial differential equations. The explicit expressions of the interaction solutions are of value for analysis of the interacting mechanism. We analyze the one-lump-multi-stripe and one-lump-multi-soliton solutions to nonlinear partial differential equations via Hirota bilinear forms. The one-lump-multi-stripe solutions are generated from the combined solution of quadratic functions and N exponential functions, while the one-lump-multi-soliton solutions from the combined solution of quadratic functions and N hyperbolic cosine functions. Within the context of the derivation of the lump solution and soliton solution, necessary and sufficient conditions are presented for the two types of interaction solutions, respectively, based on the combined solutions to the associated bilinear equations. Applications are made for a (2+1)-dimensional generalized KdV equation, the (2+1)-dimensional NNV system and the (2+1)-dimensional Ito equation.

Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

Numerical analytic method for solving a large class of generalized non-homogeneous variable coefficients KdV problems based on lie symmetries

In this study, we use Lie symmetry method to reduce a generalized KdV equation with initial and boundary conditions with non-homogeneous variable coefficients to initial value problem. In particular, we concentrate on the cases that the reduced IVP cannot be solved analytically. We also compare the approximated solutions of IVP using numerical methods with IBVP, and note that they are more efficient than doing the same procedure for IBVP. In fact, it is shown that reducing IBVP to IVP and solving the reduced problem numerically will lead to more accurate solutions.

Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations

In this study we investigate for the first time the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards we focus on higher order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. The direct cascade structure remains invariant regardless of the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. The results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics.

Full Family of Flattening Solitary Waves for the Critical Generalized KdV Equation

For the critical generalized KdV equation partial _t u + partial _x (partial _x^2 u + u^5)=0 on {mathbb {R}}, we construct a full family of flattening solitary wave solutions. Let Q be the unique even positive solution of Q''+Q^5=Q. For any nu in (0,frac{1}{3}), there exist global (for tge 0) solutions of the equation with the asymptotic behavior u(t,x)=t-ν2Qt-ν(x-x(t))+w(t,x)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} u(t,x)= t^{-\\frac{\\nu }{2}} Q\\left( t^{-\\nu } (x-x(t))\\right) +w(t,x) \\end{aligned}$$\\end{document}where, for some c>0, x(t)∼ct1-2νand‖w(t)‖H1(x>12x(t))→0ast→+∞.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} x(t)\\sim c t^{1-2\\nu } \\quad \\text{ and }\\quad \\Vert w(t)\\Vert _{H^1(x>\\frac{1}{2} x(t))} \\rightarrow 0\\quad \\text{ as } t\\rightarrow +\\infty . \\end{aligned}$$\\end{document}Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.

VARIATIONAL PRINCIPLE FOR A GENERALIZED KdV EQUATION IN A FRACTAL SPACE

A generalized KdV equation with fractal derivatives is suggested, and a special function is introduced to establish a fractal variational principle. A detailed derivation is elucidated step by step by the semi-inverse method, and some special cases are discussed.

SYMMETRY ANALYSIS FOR A SEVENTH-ORDER GENERALIZED KdV EQUATION AND ITS FRACTIONAL VERSION IN FLUID MECHANICS

KdV types of equations play an important role in many fields. In this paper, we study a seventh-order generalized KdV equation and its fractional version in fluid mechanics using symmetry. From symmetry, the corresponding vectors, symmetry reduction and conservation laws are derived. Potential equation is also analyzed with regard to the symmetry method. Based on the symmetry, similarity reductions and conservation laws are also presented. Subsequently, the fractional version of the seventh-order KdV equation is discussed. Finally, differential invariants are constructed for the special case.

Deformation characteristics of three-wave solutions and lump N-solitons to the (2 + 1)-dimensional generalized KdV equation

In this paper, the three-wave solution of ($$2+1$$)-dimensional generalized Korteweg-de Vries equation is obtained by using Hirotas bilinear method and three-wave method. We study the deformation characteristics of three-wave solution by taking different parameter values. A new lump solution is obtained when we study the degenerate behavior of three-wave solutions. Besides, we give an existence theorem of the interaction between lump solution and different forms of N-solitons ($$N\rightarrow \infty $$) and give a detailed calculation and proof process. Some new interaction solutions, such as lump N-solitons type, lump N-logarithmic type, lump cos–sin–exponential type, higher-order lump-type solutions, are used as examples to illustrate the correctness and effectiveness of the description theorem. We also give the evolutionary structure plots of the superposition behavior between lump solutions and solitons, and study the interaction behavior between lump-type solutions and solitons.

Periodic solutions of the generalized KdV equation

This paper concerns the existence and linear stability of periodic solutions of the generalized KdV equation. The proof is based on an abstract KAM theorem for infinite dimensional Hamiltonian systems.

A nonlocal variable coefficient KdV equation: Bäcklund transformation and nonlinear waves

A nonlocal two-layer fluid model is constructed through a simple symmetry reduction from the local one. Then, a general variable coefficients nonlocal KdV (VCKdV) equation with shifted space parity and delayed time reversal is derived from it by using multi-scale expansion method, with and without the so-called $$y-$$average method, respectively. A non-auto-Backlund transformation between the VCKdV equation and a constant coefficients KdV (CCKdV) equation is established. By using this transformation, various exact solutions of the VCKdV equation can be obtained from the seed solutions of the CCKdV equation. As some concrete examples, one solitary wave solution and two kinds of periodic wave solutions are given. Due to the inclusion of arbitrary functions in these solutions, they possess abundant dynamical behaviors with some of them analyzed graphically.

Generalized KdV equation involving Riesz time-fractional derivatives: constructing and solution utilizing variational methods

In this work, the time-fractional generalized Korteweg de Vries (TFGKdV) is derived by utilizing the method of a semi-inverse and the variational principles. Based on the initial condition relying on the dispersion and nonlinear coefficients, we can apply the He’s variational iteration to construct an approximated solution for the TFGKdV equation. Finally, we study the impact of the fractional derivatives on the propagation and the structure of the solitary waves obtained from the solution of TFGKdV equation.

Collocation Method for Solving the Generalized KdV Equation

In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including; single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, L2 and L∞ and invariants I1, I2 and I3 have been calculated. Our numerical results are compared with some of those available in the literature.

Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system

Abstract This research paper investigates the analytical and numerical solutions of the generalized formula of Hirota-Satsuma coupled KdV system which is also known as the generalized KdV equation that is derived by R. Hirota and J. Satsuma. The modified Khater method and B-spline scheme are used to earn abundant of computational and approximate solutions on this model. This equation characterizes an interaction of two long undulations with diverse dispersion kinsmen. The comparison between our obtained computational and numerical solutions to clarify the convergence of solutions is explained and discussed. For more explanation of the model’s physical properties, some of the obtained solutions are sketched in different types, and the comparison between the computational and numerical solutions are explained by showing the values of absolute error between them. The performance of both used method is effective, powerful, and shows its ability to apply to many nonlinear evolution equations.

A method for generating isospectral and nonisospectral hierarchies of equations as well as symmetries

We have known that the Tu scheme is a powerful tool for generating isospectral integrable hierarchies of evolution equations. However, it is not clear that how to apply it to generate nonisospectral integrable hierarchies along with the time-evolution of the spectral parameter as the polynomial form in the spectral parameter λ. In the paper, we propose a method for generating isospectral and nonisospectral integrable hierarchies by improved the Tu scheme. As one of applications, a spectral–nonisospectral problem is introduced for which a kind of isospectral plus nonisospectral integrable hierarchy is presented, including the generalized KdV equations and the cylinder-KdV equation with variable coefficients. One of the generalized KdV equations generalizes the variable-coefficient integrable equation given by Li Yishen. Therefore, we choose it to discuss and obtain its various symmetries with the help of Lie symmetry analysis which looks complicated, but another two sets of infinite symmetries are produced by the infinite conserved densities of the integrable equation. As the second application, an isospectral–nonisospectral AKNS-Kaup-Newell soliton hierarchy (briefly, AKNS-K-N-SH) is derived from an appropriate spectral problem whose K symmetries and a set of new τ symmetries are generated by applying Lie group analysis. As a reduction of the AKNS-K-N-SH, a generalized sine-Gordon equation is singled out.

Multi-symplectic integrator of the generalized KdV-type equation based on the variational principle

The variational principle is used to construct a multi-symplectic structure of the generalized KdV-type equation. Accordingly, the local energy conservation law, the local momentum conservation law, and the Cartan form of the generalized KdV-type equation are given. An explicit multi-symplectic scheme for the generalized KdV equation based on the Fourier pseudo-spectral method and the symplectic Euler scheme is constructed. Through a numerical examination, the explicit multi-symplectic Fourier pseudo-spectral scheme for the generalized KdV equation not only preserve the discrete global energy conservation law and the global momentum conservation law with high accuracy, but show long-time numerical stability as well.

Lie Symmetry Analysis of the Time Fractional Generalized KdV Equations with Variable Coefficients

The group classification of a class of time fractional generalized KdV equations with variable coefficient is presented. The Lie symmetry analysis method is extended to the certain subclasses of time fractional generalized KdV equations with initial and boundary values. Under the corresponding similarity transformation with similarity invariants, KdV equations with initial and boundary values have been transformed into fractional ordinary differential equations with initial value. Then we use the power series method to obtain the exact solution of the reduced equation with the Erdélyi-Kober fractional differential operator.

On a class of solutions to the generalized KdV type equation

We consider the IVP associated to the generalized KdV equation with low degree of nonlinearity [Formula: see text] By using an argument similar to that introduced by Cazenave and Naumkin [Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Commun. Contemp. Math. 19(2) (2017) 1650038, MR3611666], we establish the local well-posedness for a class of data in an appropriate weighted Sobolev space. Also, we show that the solutions obtained satisfy the propagation of regularity principle proven in [P. Isaza, F. Linares and G. Ponce, On the propagation of regularity and decay of solutions to the [Formula: see text]-generalized Korteweg–de Vries equation, Comm. Partial Differential Equations 40(7) (2015) 1336–1364, MR3341207] in solutions of the [Formula: see text]-generalized KdV equation.

Lie group analysis for initial and boundary value problem of time-fractional nonlinear generalized KdV partial differential equation

The Lie group analysis or in other word the symmetry analysis method is extended to deal with a time-fractional order derivative nonlinear generalized KdV equation. Our research in this work aims to use transformation methods and their analysis to search for exact solutions to the nonlinear generalized KdV differential equation. It is shown that this equation can be reduced to an equation with Erdelyi-Kober fractional derivative. In this study, we research the initial and boundary conditions, considering them invariant, and so we get two ordinary initial value problems, i.e. two Cauchy problems. Conservation laws for the given equation are also investigated. In this work, we introduce symmetry analysis and find conservation laws for the nonlinear generalized time-fractional KdV equation by the Lie groups method using fractional derivatives in the Riemann-Liouville sense.

State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation

In this paper, we are mainly concerned with the (2+1)-dimensional generalized Korteweg–de Vries equation in fluid dynamics. Based on the translation transformation and Hirota bilinear method, we study the excitations of nonlinear lump-type waves on a constant background. A remarkable feature of these lump-type waves is that under some parameter conditions, these lump-type wave solutions can be converted into some amusing nonlinear wave structures, including the W-shaped solitary wave, double-peak solitary wave, parallel solitary wave, multi-peak solitary wave and periodic wave solutions. These results do not have an analog in the standard Kadomtsev–Petviashvili equation. The transition condition between the lump-type wave and other nonlinear wave solutions is presented. The dynamical behaviors of these nonlinear wave solutions are investigated analytically and illustrated graphically. Furthermore, the existence conditions for these nonlinear wave solutions are exhibited explicitly. Our results further enrich the nonlinear wave theories for the (2+1)-dimensional generalized Korteweg–de Vries equation.