Dimensional KdV-type Equation Research Articles

Symmetry group analysis of several coupled fractional partial differential equations

Under investigations were several coupled time–space fractional integrable equations which are called the (1+1)-dimensional KdV-type equations, the (2+1)-dimensional Burgers equations and the (3+1)-dimensional unsteady Euler equations of gas dynamics. In this article, these above considered time–space fractional models through the symmetry analysis approach were explored. As the most basic results, symmetries of these researched equations, were obtained. With the help of the multiple-parameters Erdélyi-Kober fractional operators, they can be reduced into lower dimensional systems. We noticed that this reduction process was somewhat different from the integer case. Besides, we also obtained the explicit power series solution and analyze the convergence to it. By considering the new fractional conservation theorem, conserved quantity of time and space of these time–space fractional systems, were also obtained.

GROUP ANALYSIS OF THE TIME FRACTIONAL (3 + 1)-DIMENSIONAL KDV-TYPE EQUATION

Under investigations into this paper is a higher-dimensional model, namely the time fractional [Formula: see text]-dimensional Korteweg–de Vries (KdV)-type equation, which can be usually used to express shallow water wave phenomena. At the beginning, the symmetry of the time fractional [Formula: see text]-dimensional KdV-type equation via the group analysis scheme is obtained. The definition of the fractional derivative in the sense of the Riemann–Liouville is considered. Then, the one-parameter Lie group and invariant solutions of this considered equation are constructed. Subsequently, we applied a direct method to construct the optimal system of one-dimensional of this considered equation. Next, this considered higher-dimensional model can be reduced into the lower-dimensional fractional differential equations (FDEs) with the help of the three-parameter and two-parameter Erdélyi–Kober fractional differential operators (FDOs). Lastly, conservation laws of this discussed equation by using a new conservation theorem are also found. A series of results of the above obtained can provide strong support for us to reveal the mysterious veil of this viewed equation.

Lie symmetries, optimal system, group-invariant solutions and dynamical behaviors of solitary wave solutions for a (3+1)-dimensional KdV-type equation

Lie symmetries, optimal system, group-invariant solutions and dynamical behaviors of solitary wave solutions for a (3+1)-dimensional KdV-type equation

Homoclinic breather wave, rouge wave and interaction solutions for a (3+1)-dimensional KdV-type equation

Based on symbolic computations, homoclinic breather wave, rouge wave and interaction solutions to the (3+1)-dimensional KdV-type equation are constructed. Two homoclinic breather solutions and a rouge wave solution are obtained by using the extended homoclinc test. Subsequently, we study interaction solutions between a lump solution and 1-kink soliton or 2-kink soliton of the KdV-type equation. In addition, periodic interaction solutions of the KdV-type equation are also studied. Moreover, dynamics behavior of these obtained solutions are exhibited using graphical analysis.

Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations

In this paper, we implemented an improved tanh function Method for multiple soliton solutions of new coupled Konno-Oono equation and extended (3+1)-dimensional KdV-type equation.

Well-Posedness Results and Dissipative Limit of High Dimensional KdV-Type Equations

Considered in this work is an n-dimensional dissipative version of the Korteweg–de Vries (KdV) equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study well-posedness behavior of this equation when the dissipative effects are reduced.

Integrability test of a higher-order (2+1)-dimensional KdV-type equation

The so-called KdV6 equation has recently generated much interest. Here we show that a recently considered (2+1)-dimensional extension of this equation is in fact nonintegrable.

Exact three-wave solution for higher dimensional KdV-type equation

A new periodic type of three-wave solutions including periodic two-solitary solution, doubly periodic solitary solution and breather type of two-solitary solution for the (1+2)-dimensional and (1+3)-dimensional KdV-type equations are obtained using Hirota’s bilinear form and generalized three-wave type of ansatz approach.

The Structures and the Interactions of Soliton in two (2+1)-dimensional KdV-Type Equations

Exact solutions in two (2+1)-dimensional KdV-type (Sawada-Kodera and Boussinesq) equations are presented by using the bilinear method. The N-breather solution, the solution to describe the interaction between a line soliton and a y-periodic soliton, and the solution to express the interaction between two y-periodic solitons are included in our results. Detailed behavior of interactions between a line soliton and a y-periodic soliton for the SK equation and between two y-periodic solitons for the BS equation are illustrated both analytically and graphically. For these two equations, we only discuss the repulsive interaction keeping the shapes of the soliton unchanged.

Dromion Interactions of (2+1)-Dimensional KdV-type Equations

Starting from a two line soliton solution of an integrable (2+1)-dimensional equation in bilinear form, one can find a dromion solution, that is localized in all directions, for the physical field and/or the suitable potential (the physical field's derivatives). The interaction properties between two dromions are studied both analytically and numerically for (2+1)-dimensional KdV-type equations. Our analysis shows that the shape of dromions after interaction may be changed or unchanged for different form of solutions.

Soliton interactions of integrable models

The solution of integrable ( n+1)-dimensional KdV system in bilinear form yields a dromion solution that is localized in all directions. The interactions between two dromions are studied both in analytical and in numerical for three ( n+1)-dimensional KdV-type equations ( n=1, 2, 3). The same interactive properties between two dromions (solitons) are revealed for these models. The interactions between two dromions (solitons) may be elastic or inelastic for different form of solutions.

Reconstruction of multi-soliton interactions using crest data for (2+1)-dimensional KdV type equations

To predict the wave parameters from the interaction patterns of waves (the inverse problem), the corresponding direct problem needs to be solved. In this paper, interaction patterns of multi-soliton solutions of KdV type equations are constructed. The connection between the multi-soliton interaction patterns and the decomposition of a multi-soliton solution into a linear superposition of solitons and interaction solitons is demonstrated. All new concepts are illustrated for three-soliton solutions of the Kadomtshe–Petviashvili equation.

Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method

To get the similarity solutions of a nonlinear physical equation, one may use the classical Lie group approach, nonclassical Lie group approach and the Clarkson and Kruskal (CK) direct method. In this paper the direct method is modified to get the similarity and conditional similarity reductions of a (2+1) dimensional KdV-type equation. Ten types of usual similarity reductions [including the (1+1)-dimensional shallow water wave equation and the variable KdV equation] and six types of conditional similarity reductions of the (2+1)-dimensional KdV equation are obtained. Some special solutions of the conditional similarity reduction equations are found to show the nontriviality of the conditional similarity reduction approach. The conditional similarity solutions cannot be obtained by using the nonclassical Lie group approach in its present form. How to modify the nonclassical Lie group approach to obtain the conditional similarity solutions is still open.

Dromion interactions of (2+1)-dimensional nonlinear evolution equations

Starting from two line solitons, the solution of integrable (2+1)-dimensional mKdV system and KdV system in bilinear form yields a dromion solution or a "Solitoff" solution. Such a dromion solution is localized in all directions and the Solitoff solution decays exponentially in all directions except a preferred one for the physical field or a suitable potential. The interactions between two dromions and between the dromion and Solitoff are studied by the method of figure analysis for a (2+1)-dimensional modified KdV equation and a (2+1)-dimensional KdV type equation. Our analysis shows that the interactions between two dromions may be elastic or inelastic for different forms of solutions.

On the Dromion Solutions of a (2+l)-Dimensional KdV-Type Equation11The project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province of China

A general curve soliton which is finite on a curved line and localized apart from the curve for a (2+1)-dimensional KdV-type equation is found. For the KdV-type equation, we find that the dromion solutions can be obtained not only by two perpendicular line solitons, two nonperpendicular (with one is parallel to x-axis) line solitons, but also by one line soliton and one curve soliton. Various types of multi-dromion solutions which are constituted by n straight line solitons parallel to the x axis and one curve soliton can be cast in a simple formula with two arbitrary functions. The KdV-type equation is not integrable because it cannot pass through the three nonparallel line soliton test.

Dromion-like structures in a (3+1)-dimensional KdV-type equation

Dromion structures have been found in many (2 + 1)-dimensional models. The similar structure in (3 + 1) dimensions is studied in this paper for a KdV-type equation, . Starting from the bilinear form of the model, we found that there are five types of multi-dromion solutions for its potentials, say, . The first type of multi-dromion solution is driven by multi-camber solitons with one of them being non-parallel to the x-axis. The second, third and fourth types of multi-dromion solutions are driven by two camber solitons and one, two and three sets of parallel plane solitons, respectively. The fifth type of multi-dromion solution is driven by one set of parallel plane solitons, one set of camber solitons which are parallel to the x-axis and one camber soliton which is non-parallel to the x-axis. Phase shifts may be involved in the interactions among the multi-dromions for the first and fifth types of solutions. A single dromion solution of the model may possess quite a free shape. For instance, the point-like dromions, ring-type dromions, extended and sharp dromions and oscillatory dromions can be obtained by selecting some arbitrary functions appropriately.