Bäcklund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota-Satsuma-Ito-like equation

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.

Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types

Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws

Dispersive of propagation wave structures to the dullin-Gottwald-Holm dynamical equation in a shallow water waves

Localized characteristics of lump and interaction solutions to two extended Jimbo-Miwa equations

In this paper, we study abundant exact solutions including the lump and interaction solutions to the (2 + 1)-dimensional Yu–Toda–Sasa–Fukuyama equation. With symbolic computation, lump solutions and the interaction solutions are generated directly based on the Hirota bilinear formulation. Analyticity and well-definedness is guaranteed through some conditions posed on the parameters. With special choices of the involved parameters, the interaction phenomena are simulated and discussed. We find the lump moves from one hump to the other hump of the two-soliton, while the lump separates from the hump of the one-soliton.

Abundant exact solutions and interaction phenomena of the (2 + 1)-dimensional YTSF equation

This paper is concerned with the integrability of a (2+1)-dimensional nonlinear evolution equation. The Painlevé analysis proves that this equation possesses the Painlevé property. Other integrable properties, including the bilinear Bäcklund transformation, Bell-polynomial-typed Bäcklund transformation, Lax pairs and infinite conservation laws, are derived directly by virtue of the Hirota bilinear method and Bell polynomials. The general form of the resonant soliton solutions are constructed based on the linear superposition principle. The resonant two-soliton solutions consist of three waves, each of which is one-soliton profile. For the resonant three-soliton solutions, the resonance of waves may cause some waves to disappear or appear. We hope that the various resonant phenomena discussed here will be helpful to understand the propagation of nonlinear waves. • Integrability is investigated for a (2+1)-dimensional nonlinear evolution equation, which is a generalized one of the Boussinesq equation. • The general form of the resonant soliton solutions are constructed based on the linear superposition principle. • Various resonant phenomena discussed here will be helpful to understand the propagation of nonlinear waves. 

Observation of resonant solitons and associated integrable properties for nonlinear waves

In this paper, we propose a combined form of the bilinear Kadomtsev–Petviashvili equation and the bilinear extended (2+1)-dimensional shallow water wave equation, which is linked with a novel (2+1)-dimensional nonlinear model. This model might be applied to describe the evolution of nonlinear waves in the ocean. Under the effect of a novel combination of nonlinearity and dispersion terms, two cases of lump solutions to the (2+1)-dimensional nonlinear model are derived by searching for the quadratic function solutions to the bilinear form. Moreover, the one-lump-multi-stripe solutions are constructed by the test function combining quadratic functions and multiple exponential functions. The one-lump-multi-soliton solutions are derived by the test function combining quadratic functions and multiple hyperbolic cosine functions. Dynamic behaviors of the lump solutions and mixed solutions are analyzed via numerical simulation. The result is of importance to provide efficient expressions to model nonlinear waves and explain some interaction mechanism of nonlinear waves in physics.

Si-Jia Chen

Papers

Bäcklund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota-Satsuma-Ito-like equation

Abundant exact solutions and interaction phenomena of the (2 + 1)-dimensional YTSF equation

Observation of resonant solitons and associated integrable properties for nonlinear waves

Dynamic behaviors of the lump solutions and mixed solutions to a (2+1)-dimensional nonlinear model

Elastic collision between one lump wave and multiple stripe waves of nonlinear evolution equations