Hirota bilinear equations with linear subspaces of solutions

Some Exact and Explicit Solutions for Nonlinear Schrödinger Equations

“…In this case, from (16) and (17), we conclude that deg B(z) = 2 and deg A 0 (z) = 6. Assuming (17), we get b 2 = 3a 3 , b 1 = 2a 2 , and b 0 = a 1 .…”

“…Through the wave transformation u(x, t) = U (ξ), ξ = kx − wt + ξ 0 , where k, w, and (16) ξ 0 are arbitrary constants, Eq. (1) can be converted into an ordinary dierential equation (ODE) of the form…”

“…More lately, the superposition principle was successfully used to nd exponential traveling wave solutions to the Hirota bilinear equations [16]. But, it is usually hard and time consuming to tackle various kinds of nonlinear problems via the well-known traditional approaches because they are usually restricted and cannot be im- * e-mail: ismailaslan@iyte.edu.tr plemented to numerous realistic mathematical/physical scenarios.…”

Symbolic computation of exact solutions for fractional differential-difference equation models

“…Recently, the discrete Jacobi subequation method was presented in [34] for solving nonlinear DDEs via Jacobi elliptic functions sn, cn and dn. More lately, the superposition principle was successfully used to construct exponential traveling wave solutions to bilinear equations [23]. Another interesting technique is the Frobenius integrable decompositions method [22] for nonlinear PDEs.…”

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

“…Now a more general question is: When is a multivariate polynomial of even order positive (or non-negative)? A more specific question is: When does a multivariate polynomial of even order have a unique zero [66]? Equivalently, how can one judge if…”