Two families of Liouville integrable lattice equations

The research of gravity solitary waves movement is of great significance to the study of ocean and atmosphere. Baroclinic atmosphere is a complex atmosphere, and it is closer to the real atmosphere. Thus, the study of gravity waves in complex atmosphere motion is becoming increasingly essential. Deriving fractional partial differential equation models to describe various waves in the atmosphere and ocean can open up a new window for us to understand the fluid movement more deeply. Generally, the time fractional equations are obtained to reflect the nonlinear waves and few space-time fractional equations are involved. In this paper, using multiscale analysis and perturbation method, from the basic dynamic multivariable equations under the baroclinic atmosphere, the integer order mKdV equation is derived to describe the gravity solitary waves which occur in the baroclinic atmosphere. Next, employing the semi-inverse and variational method, we get a new model under the Riemann-Liouville derivative definition, i.e., space-time fractional mKdV (STFmKdV) equation. Furthermore, the symmetry analysis and the nonlinear self-adjointness of STFmKdV equation are carried out and the conservation laws are analyzed. Finally, adopting the exp(-Φ(ξ)) method, we obtain five different solutions of STFmKdV equation by considering the different cases of the parameters (η,σ). Particularly, we study the formation and evolution of gravity solitary waves by considering the fractional derivatives of nonlinear terms.

Section: Introductionmentioning

confidence: 99%

Research on Time-Space Fractional Model for Gravity Waves in Baroclinic Atmosphere

Ren

Dong

Meng

et al. 2018

“…In soliton theory, the study of integrability to nonlinear evolution is always a hot topic of interest, which can be regarded as a key step of their exact solvability. Many areas of integrable systems are researched, such as Painlevé analysis [1], Hamiltonian structure [2][3][4][5], Lax pair [6][7][8], Bäcklund transformation (BT) [9][10][11][12], infinite conservation laws [13][14][15][16], and bilinear integrability [17][18][19][20]. Based on the bilinear methods, we have got many kinds of solutions, such as lump solutions [21][22][23][24] and Pfaffian solution [25].…”

Section: Introductionmentioning

confidence: 99%

Hybrid Solutions of (3 + 1)‐Dimensional Jimbo‐Miwa Equation

Zhang

Sun

Dong

2017

The rational solutions, semirational solutions, and their interactions to the (3+1)-dimensional Jimbo-Miwa equation are obtained by the Hirota bilinear method and long wave limit. The hybrid solutions contain rogue wave, lump solution, and the breather solution, in which the breathers which are manifested as growing and decaying periodic line waves show different dynamics in different planes. Rogue waves are localized in time and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods; they arise from a constant background at t≪0 and then disappear in the constant background when time goes on. More importantly, the interactions between some hybrid solutions are demonstrated in detail by the three-dimensional figures, such as hybrid solution between the stripe soliton and breather and hybrid solution between stripe soliton and lump solution.

“…Many methods to solve the equations are proposed, such as traveling wave method [21], Darboux transformation method [22][23][24], Hirota method [25,26], homogeneous balance method [27], Jacobi elliptic function method [28], Symmetry method [29,30], Rational solutions [31][32][33]; meanwhile the features of equations are also discussed [34][35][36]. In this paper, we plan to adopt the trial function method and derive the exact solution of model.…”

Section: Introductionmentioning

confidence: 99%

(2 + 1)‐Dimensional Coupled Model for Envelope Rossby Solitary Waves and Its Solutions as well as Chirp Effect

Chen

Yang

Guo

et al. 2017

Using the method of multiple scales and perturbation method, a set of coupled models describing the envelope Rossby solitary waves in (2 + 1)-dimensional condition are obtained, also can be called coupled NLS (CNLS) equations. Following this, based on trial function method, the solutions of the NLS equation are deduced. Moreover, the modulation instability of coupled envelope Rossby waves is studied. We can find that the stable feature of coupled envelope Rossby waves is decided by the value of . Finally, learning from the concept of chirp in the optical soliton communication field, we study the chirp effect caused by nonlinearity and dispersion in the propagation of Rossby waves.