General high-order rational solutions and their dynamics in the (3+1)-dimensional Jimbo–Miwa equation

General high-order rational solutions are derived for the (3+1)-dimensional Jimbo–Miwa equation based on the Hirota bilinear form. The solutions are presented in terms of Gram determinants; the elements of determinants are connected to Schur polynomials and have simple algebraic expressions. Their dynamic behaviors are researched using three-dimensional imagery and contour plots. It is revealed that different kinds of solutions appear in (x, y) plane and (y, z) plane. When one of these internal parameters in the rational solutions is sufficiently large, in (x, y) plane Lump solutions appear with obvious geometric structures, which are deconstructed by a first-order Lump such as triangle, pentagon, and nonagon, among others; in (y, z) plane rational line soliton solutions with maximum background amplitude changing over time appear. These findings might help us comprehend the nonlinear wave propagation processes in the many nonlinear physical systems.

     

Exact solutions and Painlevé analysis of a new (2+1)-dimensional generalized KdV equation

The new (2+1)-dimensional generalized KdV equation which exists the bilinear form is mainly discussed. We prove that the equation does not admit the Painlevé property even by taking the arbitrary constant a=0. However, this result is different from Radha and Lakshmanan??s work. In addition, based on Hirota bilinear method, periodic wave solutions in terms of Riemann theta function and rational solutions are derived, respectively. The asymptotic properties of the periodic wave solutions are analyzed in detail.     

New interaction of high-order breather solutions,lump solutions and mixed solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like equation

Nonlinear Dynamics - Under investigation in this letter is an (3+1)-dimensional Hirota–Satsuma–Ito-like equation, which provide strong support for studying the dynamic behavior of...     

Abundant fission and fusion solutions in the ( $$2+1$$ 2 + 1 )-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Nonlinear Dynamics - Fission and fusion are important phenomena, which have been observed experimentally in many physical areas. In this paper, we study the above two phenomena in the ( $$2+1$$...     

Bell-polynomial approach and soliton solutions for the Zhiber–Shabat equation and (2+1)-dimensional Gardner equation with symbolic computation

Under investigation in this paper are the Zhiber?CShabat and (2+1)-dimensional Gardner equations in quantum fields, fluids and plasmas. Via the Hirota method and symbolic computation, the Bell-polynomial approach is performed to directly bilinearize those equations. For the Zhiber?CShabat equation, based on the bilinear form with an auxiliary variable, the bell-shaped soliton, upside-down bell-shaped soliton and breather-like solutions are obtained. Figures are plotted to illustrate the elastic interactions between two upside-down bell-shaped solitons and the interaction between the breather-like. As to the (2+1)-dimensional Gardner equation, bilinear form, B?cklund transformation, one- and two-shock wave solutions are derived. Amplitude-compression and amplification interactions are investigated analytically and graphically.     

Resonance and deflection of multi-soliton to the (2+1)-dimensional Kadomtsev–Petviashvili equation

In this work, a modified three-soliton method with a perturbation parameter is proposed, and it is applied to the (2+1)-dimensional Kadomtsev–Petviashvili equation (KP), and new breather multi-soliton solutions are obtained. The dependence of new mechanical structures on the perturbed parameter for multi-soliton including resonance and deflection for KP equation are investigated and exhibited.     

A (2+1)-dimensional combined KdV–mKdV equation: integrability,stability analysis and soliton solutions

Nonlinear Dynamics - In this study, the (2+1)-dimensional combined Korteweg–de Vries and modified Korteweg–de Vries equation has been considered for the first time. Firstly, we check...     

Interaction solutions between lump and stripe soliton to the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

Nonlinear Dynamics - In this paper, we apply the ansatz method to the multi-linear form of the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation for constructing interaction...     

Nonlinear localized waves resonance and interaction solutions of the (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation

Nonlinear Dynamics - This paper deals with localized waves in the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation in the incompressible fluid. Based on Hirota’s...     

Solutions and conservation laws of a (3+1)-dimensional Zakharov–Kuznetsov equation

In this paper, we study a nonlinear evolution partial differential equation, namely the (3+1)-dimensional Zakharov–Kuznetsov equation. Kudryashov method together with Jacobi elliptic function method is used to obtain the exact solutions of the (3+1)-dimensional Zakharov–Kuznetsov equation. Furthermore, the conservation laws of the (3+1)-dimensional Zakharov–Kuznetsov equation are obtained by using the multiplier method.     

New mixed solutions generated by velocity resonance in the $$(2+1)$$ ( 2 + 1 ) -dimensional Sawada–Kotera equation

Nonlinear Dynamics - Based on the N-soliton solutions of the $$(2+1)$$ -dimensional Sawada–Kotera equation, the collisions among lump waves, line waves, and breather waves are studied in this...     

A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

     

Abundant lump and lump–kink solutions for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

Nonlinear Dynamics - By utilizing the Hirota’s bilinear form and symbolic computation, abundant lump solutions and lump–kink solutions of the new (3 + 1)-dimensional...