New soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation

N-soliton solutions and asymptotic analysis for a Kadomtsev–Petviashvili–Schrödinger system for water waves

“…1(2a), (2b)]. Figures 3 and 4 illustrate the successive elastic collisions among three bright solitons and three dark solitons [via Expressions (23)] with different amplitudes and velocities. The solitons collide with one another without any change in the physical quantities except for some small phase shifts during the process of each collision.…”

“…Solutions for Eq. (3) can describe the long waves in shallow water and explain the phenomenon that the waves maintain their original properties after the collision [22,23]. Periodic wave solutions and single-soliton solutions for Eq.…”

“…Traveling wave solutions for Eq. (3) have been displayed [23]. Soliton solutions for the nonlinear evolution equations (NLEEs) can describe the nonlinear wave phenomenon in many fields [17][18][19][20][21][22][24][25][26][27][28][29][30].…”

Variational approach, soliton solutions and singular solitons for new coupled ZK system

“…So we should search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. In recent years, powerful and efficient methods explored to find analytic solutions of nonlinear equations have drawn a lot of interest by a variety of scientists, such as Adomian decomposition method [2], the homotopy perturbation method [3,4], some new asymptotic methods searching for solitary solutions of nonlinear differential equations, nonlinear differential-difference equations and nonlinear fractional differential equations using the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform and ancient Chinese mathematics [4], the variational iteration method [5,6] which is used to introduce the definition of fractional derivatives [7,4], the He's variational approach [8], the extended homoclinic test approach [9,10], homogeneous balance method [11][12][13][14], Jacobi elliptic function method [15][16][17][18], Băclund transformation [19,20], G ′ /G expansion method for nonlinear partial differential equation [21,22], and fractional differential-difference equations of rational type [23][24][25] It is important to point out that a new constrained variational principle for heat conduction is obtained recently by the semi-inverse method combined with separation of variables [26], which is exactly the same with He-Lee's variational principle [27]. A short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [28].…”

Exact solution of sixth-order thin-film equation by GʹG-expansion method

“…The powerful and efficient methods to find analytic solutions of nonlinear partial differential equations NLPDEs by using various method has become the main aim for many authors. Many powerful methods have been created and developed to obtain analytic solutions of NLPDEs, such as the the tanh-method [1], sinecosine method [2], tanh-coth method [3], exp-function method [4], homogeneous-balance method [5], Jacobielliptic function method [6], first-integral method [7] and so on. One of the most effective direct method to build traveling wave solution of NLPDEs is the G Gexpansion method, which was first proposed by Wang et al [8].…”