Exact One-Periodic and Two-Periodic Wave Solutions to Hirota Bilinear Equations in (2+1) Dimensions

Abstract

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2 + 1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: u t + u xxy − 3uu y − 3u x v = 0 andwhere v x = u y , thereby yielding their one-periodic and two-periodic wave solutions describing one dimensional propagation of waves.

Quasiperiodic waves and asymptotic behavior for the nonisospectral and variable-coefficient KdV equation

“…Thus, we conclude that the one-periodic solution (26) just goes to the one-soliton solution (17) as the amplitude ρ → 0.…”

“…The relation between the periodic wave solution (26) and the one-soliton solution (17) can be established as follows.…”

“…where the p 1 , γ 1 are the same as those in (17). Let us explicitly expand the coefficients of the system (23) as follows…”

“…In 1980s, Nakamura proposed a convenient way to construct a kind of quasiperiodic solutions of nonlinear equation by using the Riemann theta function [14], it gives us a way to obtain periodic wave solutions of NLEEs conveniently. Further Fan, Ma, and one of the authors have extended this method to investigate the breaking soliton equation, ANNV, MNNV, generalized KdV, and the Boussinesq equations respectively [15][16][17][18][19].…”

Hyperelliptic function solutions with finite genus ������ of coupled nonlinear differential equations*

“…Successful methods include inverse scattering transform [1], Lie group [2], Darboux transformation [3], Hirota direct method [4], algebro-geometrical approach [5], et al The algebrogeometrical approach presents quasi-periodic or algebro-geometric solutions to many nonlinear differential equations, which were originally obtained on the Korteweg-de Vries (KdV) equation based inverse spectral theory and algebro-geometric method developed by pioneers such as Novikov, Dubrovin, Mckean, Lax, Its, Matveev, and co-workers [5][6][7][8][9][10] in the late 1970s. Recently, this theory has been extended to a large class of nonlinear integrable equations [11][12][13][14][15][16][17]. By virtue of Riemann theta function, we obtain some quasi-periodic wave solutions of nonlinear equations, discrete equations and supersymmetric equations [43][44][45][46][47][48].…”