Search

Considering the inhomogeneity of propagation media, the nonlinear integrable equations with variable coefficients have attracted much attention in recent years since they can model various physical phenomena in the real world more accurately. In this paper, a nonlinear integrable equation with time-variable coefficients in the Kadomtsev–Petviashvili hierarchy is investigated. By introducing specific variable transformation, the discussed equation is written into one differential equation with respect to the auxiliary function. This process is similar to the bilinearization in the Hirota bilinear method. Based on the perturbation approach, the N-soliton solutions are obtained, and the M-breather solutions are derived by imposing specific constraints on parameters. The lump solution is constructed by using the generalized positive quadratic function method. Besides, the long wave limit method is utilized to explore the T-lump solutions of this integrable equation. In addition, the 3D graphics of the solitons, breathers, lumps and their interaction are drawn to exhibit the physical properties of these solutions. These figures reflect that the time-variable coefficient leads to many novel dynamic behaviors, which are rather different from those for constant coefficient integrable equation. It should be pointed that the solution is the same as the general constant coefficients equations only in the case of the time-variable function as a constant. Therefore, the results obtained in this paper are more advantageous for the analysis of the complex physical applications governed by this model. [ABSTRACT FROM AUTHOR]

We focus on the M-coupled nonlinear Schrödinger system with variable coefficients, describing simultaneous pulse propagation of the M-field components in an inhomogeneous optical fiber. The Riemann–Hilbert problem of this system is investigated based on the (M + 1) × (M + 1) matrix spectral problem. The N-soliton solutions are obtained when the jump matrix is an identity matrix. A variable-coefficient nonlocal nonlinear Schrödinger equation of reverse-time type is proposed with a special reduction of the M-coupled nonlinear Schrödinger system with variable coefficients. The symmetry relations of eigenvectors for one- and two-soliton solutions are given. But it is challenging and difficult to derive such relations for N-soliton solutions when N ≥ 3 . The nonlocal one- and two-solutions exhibit special dynamical properties, such as periodicity and amplitude reduction. The results in this paper might be helpful to study the related physical problem in the field of optical fiber communication. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation which provides overwhelming support for studying the dynamics of high-dimensional nonlinear wave equations. The bilinear form of the equation is obtained based on the Hirota bilinear method, and the N-soliton solutions composed of the higher-order breather, periodic line wave and the mixed forms are constructed. Then, the rational and semi-rational solutions of the equation were acquired by using complex conjugate parameter relations and the long-wave limit method, which mainly consisted of high-order solitons, lumps, breathers and their mixed forms. We analyze the effect of the coefficients of space and time variables on the interaction of solutions to bilinear equations. By classifying these coefficients, we find that these coefficients change the interaction of the solutions by affecting the velocity, position, and trajectory of the waves. In order to describe the dynamics of solutions with different parameters more directly, the time evolution plots and density plots are presented, and the appearance and movement characteristics of the solutions are analyzed. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional generalized shallow water wave equation is systematically investigated in this paper based on the Hirota bilinear method. The N-soliton solution and the higher-order kink-shaped breather solutions of the (3+1)-dimensional generalized shallow water wave equation are first proposed. Then, the line rogue wave solution and various hybrid solutions consisting of the breather, the kink-shaped soliton and the periodic solutions are discussed. Furthermore, the lump solutions of it are derived by using the long wave limit of the N-soliton solution. In addition, the diverse semi-rational solutions composed of lumps, kink solitons, line rogue wave and breathers enrich the research contents of the (3+1)-dimensional generalized shallow water wave equation. The dynamic behaviors of these exact solutions are vividly presented by their respective three-dimensional diagrams and density plots with contours. [ABSTRACT FROM AUTHOR]

In this paper, a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation is systematically investigated based on the Hirota bilinear method. The explicit N-soliton solution and the bright and dark multi-soliton solutions of it are first derived. Next, various bright and dark higher-order breather solutions, including the periodic line wave solutions, as well as the hybrid solutions composed of solitons, breathers, and periodic line waves, are proposed by virtue of the complex conjugate constraints on the parameters. Then, applying the long wave limit to the N-soliton solution, the bright and dark lump solutions and line rogue wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation are constructed. The semi-rational solutions composed of breathers, lumps, solitons, and line rogue waves are further discussed. These new exact solutions all appear in pairs of bright and dark, which can be interpreted by the uplifts and collapses of energy. In addition, the dynamic behaviors of these exact nonlinear wave solutions are vividly demonstrated by their corresponding three-dimensional diagrams, sectional drawings, and density plots with contours. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study soliton solutions for nonlocal Kundu-nonlinear Schrödinger (Kundu-NLS) equation via the Darboux transformation. The nonlocal Kundu-NLS equation can be obtained through a symmetry reduction r (x , t) = q ∗ (- x , t) . The form of N-soliton solutions for the nonlocal Kundu-NLS equation can be investigated via the one-fold and n-fold Darboux transformation. Particularly, from the Darboux transformation of the nonlocal Kundu-NLS equation, we obtain some exact solutions for the nonlocal Kundu-NLS equation with different spectral parameters and corresponding graphs are given. [ABSTRACT FROM AUTHOR]

In this paper, we use the Riemann–Hilbert (RH) approach to examine the integrable three-coupled Lakshmanan–Porsezian–Daniel (LPD) model, which describes the dynamics of alpha helical protein with the interspine coupling at the fourth-order dispersion term. Through the spectral analysis of Lax pair, we construct the higher-order matrix RH problem for the three-coupled LPD model, when the jump matrix of this particular RH problem is a 4 × 4 unit matrix, the exact N-soliton solutions of the three-coupled LPD model can be exhibited. As special examples, we also investigate the nonlinear dynamical behaviors of the single-soliton, two-soliton, three-soliton and breather soliton solutions. Finally, an integrable generalized N-component LPD model with its linear spectral problem is discussed. [ABSTRACT FROM AUTHOR]

The Boussinesq equation can describe wave motions in media with damping mechanism, e.g., the propagation of long waves in shallow water and the oscillations of nonlinear elastic strings. To study the propagation of gravity waves on the surface of water, a second spatial variable (say, y) is weakly dependent, and an alternative form of generalized two-dimensional Boussinesq equation is investigated in this paper. Four families of lump solutions are derived by searching for positive quadratic function solutions to the associated bilinear equation. To guarantee the analyticity and rational localization of the lumps, some conditions are posed on both the lump parameters and the coefficients of the generalized two-dimensional Boussinesq equation. Localized structures and energy distribution of the lumps are analyzed as well. [ABSTRACT FROM AUTHOR]

This paper is concerned with the variable-coefficient nonlinear evolution equations, which include the cylindrical KdV types of equations. By the combination of Painlevé analysis and Lie group classification method, the integrable conditions, Bäcklund transformations and Lax pairs of the equations are obtained, all of the geometric vector fields of the equations are presented. Then, the relationship among Lie group classification, Painlevé analysis and CK transformation method is considered. Furthermore, the exact solutions generated from the Bäcklund transformations and symmetry reductions of the vc-KdV types of equations are investigated. [ABSTRACT FROM AUTHOR]

We comment on a recent paper by Wazwaz. We show that two fourth-order integrable nonlinear equations presented by him are related to the potential Ito equation by simple changes of variables. [ABSTRACT FROM AUTHOR]

This paper is concerned with the variable-coefficient Gardner (vc-Gardner) types of equations, which arise in fluid dynamics, nonlinear lattice and plasma physics. As its special cases, the generalized cylindrical KdV types of equations are considered simultaneously. By using the combination of Painlevé analysis and Lie group classification method, the integrable conditions, Bäcklund transformations and complete group classifications of the vc-Gardner types of equations are obtained. Then, the exact solutions generated from the Painlevé analysis and symmetry reductions are investigated. [ABSTRACT FROM AUTHOR]

This research explores a (2 + 1)-D generalized Camassa–Holm–Kadomtsev–Petviashvili model. We use a probable transformation to build bilinear formulation to the model by Hirota bilinear technique. We derive a single lump waves, multi-soliton solutions to the model from this bilinear form. We present various dynamical properties of the model such as one, two, three and four solitons. The double periodic breather waves, periodic line rogue wave, interaction between bell soliton and double periodic rogue waves, rogue and bell soliton, rogue and two bell solitons, two rogues, rogue and periodic wave, double periodic waves, two pair of rogue waves as well as interaction of double periodic rogue waves in a line are established. Among the results, most of the properties are unexplored in the prior research. Furthermore, with the assistance of Maple software, we put out the trajectory of the obtained solutions for visualizing the achieved dynamical properties. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Since the outbreak of coronavirus disease in 2019 (COVID-19), the disease has rapidly spread to the world, and the cumulative number of cases is now more than 2.3 million. We aim to study the spread mechanism of rumors on social network platform during the spread of COVID-19 and consider education as a control measure of the spread of rumors. Firstly, a novel epidemic-like model is established to characterize the spread of rumor, which depends on the nonautonomous partial differential equation. Furthermore, the registration time of network users is abstracted as 'age,' and the spreading principle of rumors is described from two dimensions of age and time. Specifically, the susceptible users are divided into higher-educators class and lower-educators class, in which the higher-educators class will be immune to rumors with a higher probability and the lower-educators class is more likely to accept and spread the rumors. Secondly, the existence and uniqueness of the solution is discussed and the stability of steady-state solution of the model is obtained. Additionally, an interesting conclusion is that the education level of the crowd is an essential factor affecting the final scale of the spread of rumors. Finally, some control strategies are presented to effectively restrain the rumor propagation, and numerical simulations are carried out to verify the main theoretical results. [ABSTRACT FROM AUTHOR]

For a Cm+1 differential system on Rn, we study the limit cycles that can bifurcate from a zero-Hopf singularity, i.e., from a singularity with eigenvalues ±bi and n-2 zeros for n≥3. If the singularity is at the origin and the Taylor expansion of the differential system (without taking into account the linear terms) starts with terms of order m, then ℓ limit cycles can bifurcate from the origin with ℓ∈{0,1,…,2n-3} for m=2 [see Llibre and Zhang (Pac J Math 240:321-341, 2009)], with ℓ∈{0,1,…,3n-2} for m=3, with ℓ≤6n-2 for m=4, and with ℓ≤4·5n-2 for m=5. Moreover, ℓ∈{0,1,2} for m=4 and n=3, and ℓ∈{0,1,2,3,4,5} for m=5 and n=3. In particular, the maximum number of limit cycles bifurcating from the zero-Hopf singularity grows up exponentially with n for m=2,3. [ABSTRACT FROM AUTHOR]

Using a uniform algebraic method, new exact solitary wave solutions and periodic wave solutions for 2D Ginzburg-Landau equation are obtained. Moreover, three-dimensional and two-dimensional graphics of some solutions have been plotted. [ABSTRACT FROM AUTHOR]

Associated with the prime number $$p=3$$ , a combined model of generalized bilinear Kadomtsev-Petviashvili and Boussinesq equation (gbKPB for short) in terms of the function f is proposed, which involves four arbitrary coefficients. To guarantee the existence of lump solutions, a constraint among these four coefficients is presented firstly, and then, the lump solutions are constructed and classified via searching for positive quadratic function solutions to the gbKPB equation. Different conditions posed on lump parameters are investigated to keep the analyticity and rational localization of the resulting solutions. Finally, 3-dimensional plots, density plots and 2-dimensional curves with particular choices of the involved parameters are given to show the profile characteristics of the presented lump solutions for the potential function $$u=2(\mathrm{{ln}}f)_x$$ . [ABSTRACT FROM AUTHOR]