Dimensional KdV Research Articles

Rogue wave, lump, kink, periodic and breather-like solutions of the (2+1)-dimensional KdV equation

In this paper, the ([Formula: see text])-dimensional KdV equation is investigated by using the bilinear neural network method (BNNM). We construct six neural network models, extending beyond single hidden layer models to create deeper and broader network structures (e.g. [3-3-1], [3-4-1], [3-1-3-1], [3-4-1-1], [3-2-2-1] and [3-2-3-1-1] models). Introducing specific activation functions into the neural network model enables the generation of various test functions, resulting in novel solutions for equations that include rogue wave solutions, lump-kink solutions, periodic soliton solution, breather-like solutions and lump solutions. The physical properties of these novel solutions are vividly depicted through three-dimensional plots, density plots, and curve plots. The findings contribute to a better understanding of the propagation behavior of shallow water waves.

Various nonlinear characteristics of breather/rogue waves and controllable interaction phenomena for a new KdV equation with variable coeffcients

In this paper, we investigate and analyze various nonlinear phenomena of a new (2+1)-dimensional KdV equation with variable coefficients, and successfully obtain breather/rogue wave solutions and interaction solutions of the KdV equation by using the bilinear neural network method and symmetry transformation. Subsequently, we analyze the dynamical characteristics and evolution process of these obtained solutions through the 3-D animations, and find a series of interesting nonlinear phenomena concerning breather/rogue waves, such as fission, regeneration, annihilation, collision, and controllable interaction phenomena on nonzero backgrounds. This paper provides a more intuitive understanding for the nonlinear phenomena of these obtained solutions, and these nonlinear phenomena have potential application value in fluid dynamics, elastic mechanics and other fields of nonlinear science.

Unveiling single soliton solutions for the (3+1)-dimensional negative order KdV–CBS equation in a long wave propagation

Unveiling single soliton solutions for the (3+1)-dimensional negative order KdV–CBS equation in a long wave propagation

New periodic-wave, periodic-cross-kink wave, three wave and other analytical wave solitons of new (2+1)-dimensional KdV equation

New periodic-wave, periodic-cross-kink wave, three wave and other analytical wave solitons of new (2+1)-dimensional KdV equation

Exact solutions to the $$(3 + 1)$$-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients

Exact solutions to the $$(3 + 1)$$-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients

Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials

Abstract In this article, the (2+1)-dimensional KdV equation by Hirota’s bilinear scheme is studied. Besides, the binary bell polynomials and then the bilinear form is created. In addition, an interaction lump with k k -soliton solutions of the addressed system with known coefficients is presented. With the assistance of the stated methodology, a cloaked form of an analytical solution is discovered in expressions of lump-soliton rational functions with a few lovable parameters. Solutions to this study’s problems are identified specifically as belonging to the lump-one, two, three, and four soliton solutions. By defining the specific advantages of the epitomized parameters by the depiction of figures and by interpreting the physical occurrences are established acceptable soliton arrangements and dealt with the physical importance of the obtained arrangements. Finally, under certain conditions, the physical behavior of solutions is analyzed by using the mentioned method. Moreover, the graphs with high resolutions including three-dimensional plots, density plots, and two-dimensional plots to determine a deep understanding of plotted solutions that will arise in the applied mathematics and nonlinear physics are employed.

Dynamics of Lump, Breather, Two-Waves and Other Interaction Solutions of (2+1)-Dimensional KdV Equation

Dynamics of Lump, Breather, Two-Waves and Other Interaction Solutions of (2+1)-Dimensional KdV Equation

On the lump interaction phenomena to the conformable fractional (2+1)-dimensional KdV equation

This article pays attention to the interaction of waves for the (2+1)-dimensional KdV equation arising in the diversity of fields with the properties of conformable fractional derivatives. The KdV equation is notably significant as a prototypical example of an exactly solvable nonlinear system (that is, an infinite-dimensional system that is completely integrable). In a density-stratified ocean, the KdV equation characterizes shallow water waves that interact weakly and nonlinearly with long internal waves. The Hirota bilinear method (HBM) is successfully adopted with different test approaches. Different kinds of solutions like lump-periodic, breather-type, and two-wave solutions, have been obtained. The method used adequately describes NLPDEs since it both provides solutions that were previously confirmed and generates fresh exact solutions by combining the results of several operations. We also plot the graphs using the corresponding parameter values to demonstrate the graphical representation of selected solutions. These findings demonstrate the beneficial effects of the approach in improving the system’s nonlinear dynamical behavior. The outcomes demonstrate the efficiency, swiftness, ease, and adaptability of the used algorithm, even when applied to intricate systems.

Different solitary wave solutions and bilinear form for modified mixed-KDV equation

This study we handled (1+1)-dimensional modified-mixed KDV (mmKdV) equation for unique soliton solutions. For this purpose Firstly, auto-Bäcklund and Cole-Hopf transformations are created through the homogeneous balance method. Various soliton-like solutions are given by the trigonometric, hyperbolic, and exponential function waves and that leads to the extraction of novel periodic, dark, and bright soliton solutions. Moreover, the bilinear form of the model is found by utilizing the logarithmic transformation. Density, 3-D, and 2-D graphics are displayed to illustrate the dynamics of the acquired solutions. It is determined that the approach employed to obtain analytic solutions to nonlinear partial differential equation is efficient and powerful.

A reduction technique to solve the (2+1)-dimensional KdV equations with time local fractional derivatives

A reduction technique to solve the (2+1)-dimensional KdV equations with time local fractional derivatives

(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters, the amplitude parameter α, the long-wavelength parameter β, the transverse wavelength parameter γ, and the bottom variation parameter δ. Such an approach, known in (1+1)-dimensional theory, is extended for the first time for (2+1)-dimensional shallow water waves. Using this procedure, we derived the only possible (2+1)-dimensional extensions of the Korteweg–de Vries equation, the fifth-order KdV equation and the Gardner equation in three cases of the relationship between these parameters. All these equations are non-local. When the bottom is flat, the (2+1)-dimensional KdV equation can be transformed to the Kadomtsev–Petviashvili equation in a fixed reference frame and next to the classical KP equation in a moving frame. Thus, the Kadomtsev–Petviashvili equation gained a derivation from the fundamental laws of hydrodynamics. We have found soliton, cnoidal, and superposition solutions (essentially one-dimensional) to the (2+1)-dimensional Korteweg–de Vries equation and the Kadomtsev–Petviashvili equation.

N-soliton, breather, M-lump and interaction dynamics for a (2 + 1)-dimensional KdV equation with variable coefficients

The main purpose of this paper is to learn N-soliton, M-lump, breather solutions and interaction solutions for a KdV equation with variable coeffificients. Using the Hirota bilinear method, one-soliton, two-soliton, three-soliton and N-soliton are obtained. By taking the conjugate parameters for two-soliton amd four-soliton, one-breather and two-breather waves are obtained, respectively. The long wave limit technique is applied to two-soliton, four-soliton and six-soliton, one-lump, two-lump and three-lump solutions are obtained, respectively. Furthermore, the interaction solutions are constructed, including one-breather and one-soliton, one-breather and two-soliton, one-lump and one-soliton, one-lump and two-soliton. In order to discussing the dynamical properties of the above solutions, some 3D-plots, 2D-plots, contour plots and density plots of these solutions are given.

A survey of (2+1)-dimensional KdV–mKdV equation using nonlocal Caputo fractal–fractional operator

We analyze the nonlinear (2+1)-dimensional KdV–mKdV equation with Caputo fractal–fractional operator. Some theoretical features are demonstrated via fixed point results. The solution of the considered KdV–mKdV is studied by the composition of the J-transformation and decomposition method. For the validity and effectiveness of the considered method, two examples with suitable initial conditions are solved, where best agreements observed. The validity of the suggested approach is verified by convergence analysis and Picard stability. From the simulations of the obtained results, it is noted that fractional order and fractal dimension significantly affects the amplitude and shape of wave solutions.

Traveling wave behavior of new (2+1)-dimensional combined KdV–mKdV equation

The KdV and mKdV equations have many significant applications in the field of plasma physics and ocean dynamics. In this work, the traveling wave behavior of the newly proposed (2+1)-dimensional combined KdV–mKdV equation has been examined using two direct analytical approaches. The two variable G′G,1G-expansion method and generalized projective Riccati equation method are used to determine a variety of solitons and solitary waves for the governing model. The graphical observations are made for suitable choice of parametric values. The obtained results will provide useful insight in further explorations of the wave phenomena depicted by the considered equation.

Lump Solutions, Interaction Solutions and Breather Solutions of Generalized (3+1)-Dimensional KdV Equations

Lump Solutions, Interaction Solutions and Breather Solutions of Generalized (3+1)-Dimensional KdV Equations

Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation

<abstract><p>The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $ N $-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation.</p></abstract>

Overtaking collisions of m shock waves and interactions of n(n→∞)-lump, m(m→∞)-solitons, τ(τ→∞)-periodic waves solutions to a generalized (2+1)-dimensional new KdV model

We consider a new generalized (2+1)-dimensional KdV model to investigate m ( m → ∞ ) shock and n ( n → ∞ ) breather wave solutions via two integral schemes. For the treatment of the model in an auxiliary equation approach, we first convert a nonlinear Burger equation to an ordinary differential equation (ODE) through a certain transformation. This ODE is used as an auxiliary equation of the method to obtain m ( m → ∞ ) shock wave solutions of the model. For different values of the parameters, we present head on and overtaking collisions with scattering ways of particle of the m ( m → ∞ ) shock wave solutions. We construct n soliton solutions of the model by using Hirota-bilinear approach. We obtain one lump type breather waves, interactions of one breather wave with a kink wave, interactions of two lump type breather waves by choosing complex conjugate values of free parameters in the n -soliton solutions of the model. Finally, we introduce two lemmas, a theorem and few corollaries on the hybrid interaction ( n → ∞ lumps, m → ∞ solitons and τ → ∞ periodic waves) solutions of the model. The theories and results are illustrated with adequate examples and suitable graphs. • We propose a generalized (2+1)- dimensional new KdV model for investigation waves solutions. • Derive m (m → ∞ ) shock wave solutions of the model. • Present head on and overtaking collisions with scattering ways of particle of the m (m → ∞ ) shock wave solutions. • We introduce two lemmas, a theorem and few corollaries on the hybrid interaction (n → ∞ lumps, m → ∞ solitons and τ → ∞ periodic waves) solutions of the model. • The theories and results are illustrated with adequate examples and suitable graphs.

Integrability and Exact Solutions of the (2+1)-dimensional KdV Equation with Bell Polynomials Approach

Integrability and Exact Solutions of the (2+1)-dimensional KdV Equation with Bell Polynomials Approach

Interactions among two-dimensional nonlinear localized waves and periodic wave solution for a novel integrable $$(2+1)$$-dimensional KdV equation

Interactions among two-dimensional nonlinear localized waves and periodic wave solution for a novel integrable $$(2+1)$$-dimensional KdV equation

Integrability and lump solutions to an extended (2+1)-dimensional KdV equation

Integrability and lump solutions to an extended (2+1)-dimensional KdV equation