Dimensional Sawada Kotera Equation Research Articles

Soliton molecules, asymmetric solitons and some new types of hybrid solutions in (2+1)-dimensional Sawada–Kotera model

Soliton molecules can be formed in both theoretical and experimental situations. In this paper, a new velocity resonance is introduced, which can form soliton molecules for the (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada–Kotera equation. By selecting some suitable parameters for soliton molecules, the asymmetric solitons of the (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada–Kotera equation can be obtained. And, the interactions among multiple soliton molecules are elastic. Furthermore, some new types of hybrid solutions consisting of soliton molecules, lump wave and breather wave can be derived by utilizing velocity resonance, module resonance of wave numbers and long wave limits method. This method of solving the soliton molecules, asymmetric solitons and some new hybrid solutions can also be applied to other nonlinear evolution equations.

Nonlinear dynamics behavior of the (2+1)-dimensional Sawada–Kotera equation

In this paper, we mainly analyze the nonlinear dynamics behavior of the (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada–Kotera (S–K) equation, which can be usually used to describe shallow water phenomena from natural science. First, the multiple resonant wave and complexiton solutions are constructed with the help of the linear superposition principle, under different domain fields, such as real and complex domain fields, respectively. Next, we apply a new ansatz method to obtain a class of rogue wave solutions (one-rogue wave and two-rogue wave solutions). Finally, the 3-dimensional and 2-dimensional density graphs are plotted for the yielded results in the above texts to better illustrate the dynamics processes to them.

General $${\varvec{M}}$$-lump, high-order breather and localized interaction solutions to the $$\varvec{2+1}$$-dimensional Sawada–Kotera equation

The $$2+1$$ -dimensional Sawada–Kotera equation is an important physical model. Here, by taking a long limit and restricting a conjugation condition to the related solitons, the general M-lump, high-order breather and localized interaction hybrid solutions are constructed, correspondingly. In order to study the dynamical behaviors, numerical simulations are implemented, which show that the parameters selected have great impacts on the types, dynamical behaviors and propagation properties of the solutions. The method proposed can be effectively applied to construct M-lumps, high-order breathers and interaction solutions of many nonlinear equations. The results obtained can be used to study the propagation phenomena of other nonlinear localized waves.

Resonant multi-soliton solutions to the (2 + 1)-dimensional Sawada–Kotera equations via the simplified form of the linear superposition principle

In this work, a simplified form of the linear superposition principle is proposed to facilitate the computational work and make the resonant multi-soliton solutions easily generated. The (2 + 1)-dimensional Sawada–Kotera (SK) equation, one of fifth-order KdV-like equations describing the nonlinear wave phenomena in shallow water, ion-acoustic waves in plasmas, etc., is investigated. Moreover, in order to demonstrate the power of the proposed method, a new version of the SK equation is further considered and examined. The general forms of resonant multi-soliton solutions are formally established. Furthermore, by taking about reverse engineering of the generated solutions, various versions of the (2 + 1)-dimensional SK equation can be derived that may make great contributions to real physical phenomena and enrich the related nonlinear sciences. Finally, the propagations of two- and three-soliton waves are presented.

Interaction behaviors for the ( $$\varvec{2+1}$$ 2 + 1 )-dimensional Sawada–Kotera equation

In this work, in this paper, we mainly study two kinds of interaction solutions of the ( $$2+1$$ )-dimensional Sawada–Kotera equation, one of which is the interaction solution between rational function and trigonometric function, including $$\cos $$ function and $$\sin $$ function, and the other one is the interaction solution between rational function and double exponential function. The dynamic properties and characteristics of these derived solutions are presented by the three-dimensional plots and corresponding contour lines.

Diversity of Interaction Solutions to the (2 + 1)-Dimensional Sawada-Kotera Equation

In this paper, based on Hirota bilinear form, we aim to show the diversity of interaction solutions to the (2 + 1)-dimensional Sawada-Kotera (SK) equation. By introducing an arbitrary differentiable function in assumption form, we can obtain abundant interaction solutions which can provide the possibility for exploring the interactions between lump waves and other kinds of waves. By choosing some particular functions and values of the involved parameters, we give four illustrative examples of the resulting solutions, and explore some novel interaction behaviors in (2 + 1)-dimensional SK equation.

Solitons and breather waves for a (2+1)-dimensional Sawada–Kotera equation

Under investigation in this letter is a (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada-Kotera equation. With the aid of the bilinear forms derived from the Bell polynomials, the Nth-order soliton solutions are obtained via the Pffafian method, and breather solutions are derived with the ansätz method. Analytic solutions obtained via the Pffafian method are the bell-type solitons. Two different kinds of the homoclinic breathers are seen, one of which is real and the other of which is complex, with two breathers interacting with each other. Homoclinic breather wave can evolve periodically along a straight line with a certain angle with the x axis and y axis, and its velocity, amplitude and width remain unchanged during the propagation. Homoclinic breather wave is not only space-periodic but also time-periodic. Interaction between the two breathers is elastic, which is similar to that of the solitons.

Pfaffian solutions for the (3+1)-dimensional nonlinear evolution equation in a fluid/plasma/crystal and the (2+1)-dimensional Sawada–Kotera equation in a liquid

For the (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equation in a plasma, crystal or fluid, the Pfaffian solutions are obtained via the Bell polynomials, symbolic computation and Hirota method. Amplitudes and velocities of the solitons are discussed, respectively, as well as the conditions on whether the collisions are overtaking or head-on in the fluid/plasma/crystal. If the product of four wave numbers is greater than 0, collisions are overtaking, or else, head-on. For the (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada–Kotera equation in a liquid, we discuss that amplitudes and velocities of the solitons, as well as the conditions of solitonic collisions. Hereby, there only exist the overtaking collisions between the solitons in such a liquid because the sign of [Formula: see text] is the same as [Formula: see text], where [Formula: see text] and [Formula: see text] are the wave numbers in the liquid. Figures showing the overtaking and head-on collisions for the two and three solitons in the fluid/plasma/crystal are also presented.

Solitons for a (2+1)-dimensional Sawada–Kotera equation via the Wronskian technique

Under investigation in this letter is a (2+1)-dimensional Sawada–Kotera equation. Solitons are obtained by virtue of the Wronskian technique. Via the fourth- and sixth-order Plücker relations for the Wronskian, we give a proof for the N -soliton solutions. Interactions between/among the two/three solitons are investigated, and it seems that those interactions are elastic.

Lump Solutions and Interaction Phenomenon for (2+1)-Dimensional Sawada–Kotera Equation**Supported by the Global Change Research Program of China under Grant No. 2015CB953904, National Natural Science Foundation of China under Grant Nos. 11675054 and 11435005, Outstanding Doctoral Dissertation Cultivation Plan of Action under Grant No. YB2016039, and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No. ZF1213

In this paper, a class of lump solutions to the (2+1)-dimensional Sawada–Kotera equation is studied by searching for positive quadratic function solutions to the associated bilinear equation. To guarantee rational localization and analyticity of the lumps, some sufficient and necessary conditions are presented on the parameters involved in the solutions. Then, a completely non-elastic interaction between a lump and a stripe of the (2+1)-dimensional Sawada–Kotera equation is obtained, which shows a lump solution is drowned or swallowed by a stripe soliton. Finally, 2-dimensional curves, 3-dimensional plots and density plots with particular choices of the involved parameters are presented to show the dynamic characteristics of the obtained lump and interaction solutions.

Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

Based on the symbolic computation, a class of lump solutions to the (2+1)-dimensional Sawada-Kotera (2DSK) equation is obtained through making use of its Hirota bilinear form and one positive quadratic function. These solutions contain six parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are free. Then by adding an exponential function into the original positive quadratic function, the interaction solutions between lump solutions and one stripe soliton are derived. Furthermore, by extending this method to a general combination of positive quadratic function and hyperbolic function, the interaction solutions between lump solutions and a pair of resonance stripe solitons are provided. Some figures are given to demonstrate the dynamical properties of the lump solutions, interaction solutions between lump solutions, and stripe solitons by choosing some special parameters.

Lump solutions to the ( $$\mathbf 2+1 $$ 2 + 1 )-dimensional Sawada–Kotera equation

In this paper, via generalized bilinear forms, we consider the (\(2+1\))-dimensional bilinear p-Sawada–Kotera (SK) equation. We derive analytical rational solutions in terms of positive quadratic functions. Through applying the dependent transformation, we present a class of lump solutions of the (\(2+1\))-dimensional SK equation. Those rationally decaying solutions in all space directions exhibit two kinds of characters, i.e., bright lump wave (one peak and two valleys) and bright–dark lump wave (one peak and one valley). In addition, we also obtain three families of bright–dark lump wave solutions to the nonlinear p-SK equation for \(p=3\).

OPTIMAL SYSTEM AND SYMMETRY REDUCTION OF THE (1+1) DIMENSIONAL SAWADA-KOTERA EQUATION

We study the nonlinear fifth order (1 + 1) dimensional Sawada-Kotera equation using Lie symmetry group. For this equation Lie point symmetry operators and optimal sys- tem are obtained. We determine the corresponding invariant solutions and reduced equations using obtained infinitesimal generators.

Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy

In this paper, we construct the bilinear identities for the wave functions of an extended B-type Kadomtsev-Petviashvili (BKP) hierarchy, which contains two types of (2+1)-dimensional Sawada-Kotera equation with a self-consistent source (2d-SKwS-I and 2d-SKwS-II). By introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the tau-function and the bilinear identities for this extended BKP hierarchy. The bilinear identities can generate all the Hirota's bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, the Hirota's bilinear equations for the two types of 2d-SKwS (both 2d-SKwS-I and 2d-SKwS-II) will be given explicitly.

Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation

Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of(2+1)-dimensional nonlinear partial differential equation. The resulting theory is applied to the(2+1)-dimensional Sawada-Kotera equation, thereby yielding its double periodic wave solutions. The relations between the periodic wave solutions and soliton solutions are rigorously established by a limiting procedure.

The Multisoliton Solutions for the (2+1)-Dimensional Sawada-Kotera Equation

Applying bilinear form and extended three-wavetype of ansätz approach on the (2+1)-dimensional Sawada-Kotera equation, we obtain new multisoliton solutions, including the double periodic-type three-wave solutions, the breather two-soliton solutions, the double breather soliton solutions, and the three-solitary solutions. These results show that the high-dimensional nonlinear evolution equation has rich dynamical behavior.

New exact solutions for the (2 + 1)-dimensional Sawada–Kotera equation

A new test function is proposed to construct new exact solitary solution for nonlinear evolution equation. The (2+1)-dimensional Sawada–Kotera equation is employed as an example to illustrate the effectiveness of the suggested method and some new wave solutions with three different velocities and frequencies are obtained. Obviously, the method can be applied to solve other type of nonlinear evolution equations as well.

Binary Bell Polynomials, Bilinear Approach to Exact Periodic Wave Solutions of (2 + 1)-Dimensional Nonlinear Evolution Equations

In the present letter, we get the appropriate bilinear forms of (2 + 1)-dimensional KdV equation, extended (2 + 1)-dimensional shallow water wave equation and (2 + 1)-dimensional Sawada—Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

Symmetry Analysis and Exact Solutions of (2+1)-Dimensional Sawada–Kotera Equation

Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the (2+1)-dimensional Sawada–Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou's direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada–Kotera and Konopelchenko–Dubrovsky equations, respectively.