The dynamic behaviors between multi-soliton of the generalized $$\pmb {(3+1)}$$ ( 3 + 1 ) -dimensional variable coeffi
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ORIGINAL PAPER
The dynamic behaviors between multi-soliton of the generalized (3 + 1) 1)-dimensional variable coefficients Kadomtsev–Petviashvili equation Yanxia Wang · Ben Gao
Received: 19 June 2020 / Accepted: 14 August 2020 © Springer Nature B.V. 2020
Abstract In this work, the generalized (3 + 1)dimensional variable coefficients Kadomtsev–Petviashvili equation, widely used in fluids or plasmas, is analyzed via the unified method and its general form. The multi-soliton rational solutions are obtained including single- and double-soliton rational solutions. Singlesoliton shaping and the interactions of double-soliton are graphically discussed in different choices of coefficients. Single-soliton wave keeps its shape, velocity and amplitude unchanged and propagates periodically in a certain direction. The double-soliton waves do not change in shapes, velocities and amplitudes before and after the collisions. We conclude that collisions between the double-soliton waves are elastic and they are not affected by the coefficients of the equation. Keywords The generalized (3 + 1)-dimensional Kadomtsev–Petviashvili equation · Variable coefficients · The generalized unified method · Dynamic behavior
1 Introduction With the development of science and technology, nonlinear evolution equations (NLEEs) have been widely Y. Wang · B. Gao (B) College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, People’s Republic of China e-mail: [email protected]
used in models to describe complex physical phenomena, such as fluid mechanics, optical fiber communications, plasma physics and chemical physics [1–5]. The study of the exact solutions of NLEEs is helpful for us to understand the regularity of matter motion, at the same time it also provides the imperative foundation for the approximate calculation, theorem analysis and other practical problems. Several analytic techniques have been proposed to study exact solutions, such as the inverse scattering method [6–8], the Hirota’s bilinear algorithm and its simplified form [9–11], the generalized unified method (GUM) [12–23] and Darboux transformation [24–27]. The essence of the inverse scattering method [6] is to transform the nonlinear partial differential equations into several linear problems, and its basis is the function transformations. In 1971, for obtaining soliton solutions of nonlinear partial differential equations, Hirota proposed a simple and direct method called Hirota’s bilinear algorithm [9] by using the bilinear transformation. To simplify the calculation, a simplified form of the Hirota’s bilinear method [9] was introduced which did not depend on the construction of bilinear forms. For the sake of searching for multi-soliton solutions of NLEEs, a simple algorithm was proposed called the generalized unified method (GUM) [12–21]. The GUM can not only give a unified method to construct multi-soliton solutions, but also urge the classification of these solutions based on the given parameters.
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Y.X. Wang, B. Gao
It is vital for us to study the NLE
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