We consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gröbner bases method and others. New potentials and corresponding solutions of the Sawada–Kotera, Kaup– Kupershmidt, Boussinesq eq...

In this paper, the projective Riccati equation method is applied to find exact solutions for fractional partial differential equations in the sense of modified RiemannLiouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the vali...

In this paper, a new analytic iterative technique, called the residual power series method (RPSM), is applied to time fractional Whitham–Broer–Kaup equations. The explicit approximate traveling solutions are obtained by using this method. The efficiency and accuracy of the present method is demonstrated by two aspects. One is analyzing the approximate solutions graphically. The other is compari...

In this letter, He’s Variational iteration method (VIM) is implemented for solving the nonlinear Whitham-Broer-Kaup partial differential equations. This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which tends to the exact solution of the problem. The results are compared with those obtained by t...

We take into account here the stochastic-fractional Broer–Kaup equations (SFBKEs) perturbed by multiplicative Wiener process. To get rational, hyperbolic, and elliptic stochastic solutions for SBKEs, we utilize Jacobi function method. The derived are significantly more useful effective in comprehending various important challenging physical phenomena due to of SFBKEs describing propagation shal...

We construct integrable hierarchies of flows for curves in centroaffine R through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and prove that the restricted hierarchy of flows on curves that project to conics in RP induces...

In this present work we applied the direct algebraic method to Broer-Kaup-Kupershmidt system. Then new types of complex solutions are obtained to the Broer-Kaup-Kupershmidt system.

We develop a Cauchy matrix reduction technique that enables us to obtain solutions for the reduced local and nonlocal complex equations from of original nonreduced systems. Specifically, by imposing reductions on some Ablowitz–Kaup–Newell–Segur-type equations, we study involving modified Korteweg–de Vries equation, sine-Gordon potential nonlinear Schrödinger equation. matrix-type soliton Jordan...