We construct, analyze and numerically validate a class of conservative, discontinuous Galerkin schemes for the Generalized Korteweg-de Vries equation. Up to round-off error, these schemes preserve discrete versions of the first two invariants (the integral of the solution, usually identified with the mass, and the L 2 -norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial attributes, such as more faithful reproduction of the amplitude and phase of traveling-wave solutions. The numerical simulations also indicate that the discretization errors grow only linearly as a function of time.

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In this paper, two different techniques will be employed to study the cnoidal wave solutions of the Boussinesq systems. First, the existence of periodic travelling-wave solutions for a large family of systems is established by using a topological method. Although this result guarantees the existence of cnoidal wave solutions in a parameter region in the period and phase speed plane, it does not provide the uniqueness nor the non-existence of such solutions in other parameter regions. The explicit solutions are then found by using the Jacobi elliptic function series. Some of these explicit solutions fall in the parameter region where the cnoidal wave solutions are proved to exist, and others do not; so the method with Jacobi elliptic functions provides additional cnoidal wave solutions. In addition, the explicit solutions can be used in many ways, such as in testing numerical code and in testing the stability of these waves.

For general classes of nonlinear, dispersive wave equations, we discuss the existence of the analogue of the cnoidal-wave solutions of the Korteweg-de Vries equation. The solutions in question have travelling and periodic features. The topological-degree theory of positive operators on Banach spaces is applied.

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This paper discusses the existence and stability of solitary-wave solutions of a general higher-order Benjamin-Bona-Mahony (BBM) equation, which involves pseudo-differential operators for the linear part. One of such equations can be derived from water-wave problems as second-order approximate equations from fully nonlinear governing equations. Under some conditions on the symbols of pseudo-differential operators and the nonlinear terms, it is shown that the general higher-order BBM equation has solitary-wave solutions. Moreover, under slightly more restrictive conditions, the set of solitary-wave solutions is orbitally stable. Here, the equation has a nonlinear part involving the polynomials of solution and its derivatives with different degrees (not homogeneous), which has not been studied before. Numerical stability and instability of solitary-wave solutions for some special fifth-order BBM equations are also given.

The present study is concerned with systemsof Korteweg-de Vries type, coupled through their nonlinear terms. Here, u = u(x, t) and v = v(x, t) are real-valued functions of a real spatial variable x and a real temporal variable t. The nonlinearities P and Q are homogeneous, quadratic polynomials with real coefficients A, B, . . ., viz.the dependent variables u and v. A satisfactory theory of local well-posedness is in place for such systems. Here, attention is drawn to their solitary-wave solutions. Special traveling waves termed proportional solitary waves are introduced and B Jerry L. Bona 123 Appl Math Optim determined. Under the same conditions developed earlier for global well-posedness, stability criteria are obtained for these special, traveling-wave solutions.