Good Boussinesq Equation

• The good Boussinesq equation is a well-posed bidirectional long-wave model characterized by the sign choice c = -1, ensuring stable dispersive behavior.
• It integrates methods from Hamiltonian dynamics, inverse scattering, and integrability theory with various normalizations and traveling-wave solutions.
• Recent research emphasizes structure-preserving numerical schemes, including spectral and exponential integrators, to effectively capture its dispersive and low-regularity properties.

The good Boussinesq equation denotes the well-posed sign choice within the Boussinesq family of bidirectional long-wave models. In the four-parameter family

$u_{TT}-c\,u_{XX}-a_4\,u_{XXXX}-a_2\,(u^2)_{XX}=0,$

the choice $c=-1$ is called the “good” Boussinesq equation, whereas $c=+1$ gives the ill-posed classical Boussinesq equation (Seilova et al., 2015). In the literature, this designation appears in several shifted and rescaled normalizations, including

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$

as well as shifted forms obtained by removing linear terms through $u=w+\tfrac12$ (Charlier et al., 2020). The equation belongs to the Korteweg–de Vries kind of equations, models long-wave propagation, and occupies a central position at the intersection of dispersive PDE, Hamiltonian dynamics, integrable systems, inverse scattering, and structure-preserving numerics.

1. Formulations, normalizations, and variable conventions

A standard periodic form on the one-dimensional torus $\mathbb T=\mathbb R/(2\pi\mathbb Z)$ is

$$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$

with data in $H^s(\mathbb T)\times H^{s-2}(\mathbb T)$ (Oh et al., 2012). On the line, inverse-scattering treatments often use

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$

or an equivalent first-order system obtained by introducing

$$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$

(Charlier et al., 2020).

A shifted periodic Hamiltonian formulation begins with

$c=-1$0

then introduces

$c=-1$1

so that

$c=-1$2

This form is convenient for Hamiltonian and geometric numerical analysis (Brugnano et al., 2018).

Some expositions also consider a reduced “good” Boussinesq equation obtained after dropping the fourth-order term,

$c=-1$3

as a simplified well-posed model associated with the same sign choice $c=-1$4 (Seilova et al., 2015). A persistent source of terminological ambiguity is therefore not the sign structure itself but the coexistence of several equivalent or reduced normalizations across the analytical and numerical literature.

2. Derivation and physical interpretations

One derivation starts from the two-dimensional irrotational Euler equations for a fluid layer of mean depth $c=-1$5 under gravity, in the long-wave, small-amplitude regime. Introducing the nondimensional variables

$c=-1$6

and expanding the velocity potential and free-surface elevation in powers of $c=-1$7, one obtains, after eliminating the potential and retaining terms up to $c=-1$8, the Boussinesq family

$c=-1$9

(Seilova et al., 2015). In this setting, $c=+1$0 represents the leading-order surface elevation.

The same source connects the equation to the motion of long waves in two dimensions under gravitation and to nonlinear wave propagation in rods or waveguides interacting with an external environment. In such settings one must account for weak nonlinearity $c=+1$1, dispersion due to elasticity or fluid filling, and possible energy leakage at the boundary. For thin rods immersed in a viscous medium, a Hamilton principle followed by asymptotic expansion leads to a third-order corrected Boussinesq equation with two dispersive terms, while the reduced good Boussinesq model captures the leading long-wave balance and predicts solitary pulses and periodic wave trains (Seilova et al., 2015).

In the shifted periodic formulation, the equation admits a Hamiltonian first-order system

$c=+1$2

with Poisson operator

$c=+1$3

and Hamiltonian functional

$c=+1$4

The variational derivatives are

$c=+1$5

and periodic integration by parts yields conservation of $c=+1$6. A second quadratic invariant,

$c=+1$7

is also conserved (Brugnano et al., 2018). These conservation laws are fundamental in both analysis and geometric discretization.

3. Well-posedness, low-regularity theory, and smoothing

For the periodic equation on $c=+1$8,

$c=+1$9

local well-posedness holds in $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$0 for

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$1

under the mean-zero assumption $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$2. The proof uses a normal form approach that explicitly extracts the rougher part of the solution. The nonlinear remainder then lies in

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$3

so the nonlinear part gains up to $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$4 derivatives (Oh et al., 2012).

For the reduced equation

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$5

an energy computation yields the formally conserved quantity

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$6

Because the quadratic form $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$7 is positive definite, one obtains an a priori bound of $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$8. In the linearization about zero, $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad u_{tt}+\tfrac43\,(u^2)_{xx}+\tfrac13\,u_{xxxx}=0,$$9 is the standard wave operator, and a fixed-point or energy-method argument in

$u=w+\tfrac12$0

gives local well-posedness for $u=w+\tfrac12$1, while conservation of $u=w+\tfrac12$2 together with $u=w+\tfrac12$3 gives global existence. By contrast, when $u=w+\tfrac12$4, the energy is indefinite and the problem is ill-posed (Seilova et al., 2015).

On the half-line, the initial-boundary-value problem

$u=w+\tfrac12$5

with data

$u=w+\tfrac12$6

is locally well-posed in $u=w+\tfrac12$7 for

$u=w+\tfrac12$8

The boundary data belong to

$u=w+\tfrac12$9

with compatibility conditions when $\mathbb T=\mathbb R/(2\pi\mathbb Z)$0 and $\mathbb T=\mathbb R/(2\pi\mathbb Z)$1. The proof uses Bourgain-type restricted-norm spaces $\mathbb T=\mathbb R/(2\pi\mathbb Z)$2, linear estimates for the free and boundary-forcing operators, and a bilinear estimate for

$\mathbb T=\mathbb R/(2\pi\mathbb Z)$3

Moreover, the nonlinear part of the solution is smoother than the data: $\mathbb T=\mathbb R/(2\pi\mathbb Z)$4 and in particular gains half a derivative in $\mathbb T=\mathbb R/(2\pi\mathbb Z)$5 in some cases. Within the restricted norm method, the threshold $\mathbb T=\mathbb R/(2\pi\mathbb Z)$6 and the half-derivative smoothing are sharp (Compaan et al., 2016).

4. Traveling waves, bilinearization, and integrable structures

A traveling-wave reduction for the reduced good Boussinesq equation takes

$\mathbb T=\mathbb R/(2\pi\mathbb Z)$7

Then

$\mathbb T=\mathbb R/(2\pi\mathbb Z)$8

After integration, the analysis in the cited paper yields three principal families of exact solutions in Jacobi elliptic functions: a periodic $\mathbb T=\mathbb R/(2\pi\mathbb Z)$9-wave, a periodic $$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$0-wave, and a $$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$1-family whose $$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$2 limit is the solitary $$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$3 wave. The latter can be written as

$$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$4

in a particular normalization, or more generally

$$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$5

(Seilova et al., 2015).

For the normalized fourth-order form

$$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$6

the binary Bell polynomial method yields a bilinear representation. Introducing

$$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$7

one obtains

$$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$8

The same framework produces the $$u_{tt}+u_{xxxx}-u_{xx}+\partial_{xx}(u^2)=0, \qquad u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),$$9-soliton $H^s(\mathbb T)\times H^{s-2}(\mathbb T)$0-function, a bilinear Bäcklund transformation, a Lax pair, an infinite hierarchy of conservation laws, and Wronskian determinant solutions. The authors state that these constructions fully demonstrate the complete integrability of the good Boussinesq equation in this normalization (Dai et al., 2023).

A distinct integrable-system development establishes a Miura-type transformation from the second-order equation

$H^s(\mathbb T)\times H^{s-2}(\mathbb T)$1

to the fourth-order good Boussinesq equation

$H^s(\mathbb T)\times H^{s-2}(\mathbb T)$2

The correspondence is derived by comparing a regular and a singular $H^s(\mathbb T)\times H^{s-2}(\mathbb T)$3 Riemann–Hilbert problem, so the Miura map exists simultaneously at the PDE, Lax-pair, and RH levels (Charlier et al., 2023).

5. Lax pairs, Riemann–Hilbert formulations, and inverse spectral theory

For the rescaled good Boussinesq equation

$H^s(\mathbb T)\times H^{s-2}(\mathbb T)$4

a line-based inverse scattering transform is formulated through the Lax pair

$H^s(\mathbb T)\times H^{s-2}(\mathbb T)$5

or equivalently through a $H^s(\mathbb T)\times H^{s-2}(\mathbb T)$6 matrix system with spectral parameter $H^s(\mathbb T)\times H^{s-2}(\mathbb T)$7, cubic-root symmetry $H^s(\mathbb T)\times H^{s-2}(\mathbb T)$8, and contour

$H^s(\mathbb T)\times H^{s-2}(\mathbb T)$9

Under no-soliton and nondegeneracy assumptions, the RH problem is determined by two reflection coefficients,

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$0

and the solution is reconstructed by

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$1

The formulation is specifically designed to support Deift–Zhou steepest descent analysis (Charlier et al., 2020).

On the half-line, assuming existence of a sufficiently smooth rapidly decaying solution, the good Boussinesq equation can likewise be recovered from a $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$2 Riemann–Hilbert problem depending only on initial and boundary values. In that setting the jump contour consists of twelve half-lines, the unknown matrix is sectionally analytic in twelve sectors, and the reconstruction formula again uses the large-$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$3 behavior of the $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$4-entry. The analysis is carried out under “no-soliton” and “generic at $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$5” hypotheses (Charlier et al., 12 Mar 2026).

In periodic inverse-spectral theory, the good Boussinesq equation is written in Lax form with the third-order operator

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$6

on $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$7 under the three-point Dirichlet conditions

$$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$8

Its discrete Dirichlet spectrum constitutes the auxiliary spectrum of the equation, in analogy with the Hill operator for periodic KdV. For $$u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0, \qquad x\in\mathbb R,\ t>0,$$9 near zero, each neighborhood $$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$0 contains exactly one simple real eigenvalue $$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$1, and the corresponding norming constants $$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$2 admit uniform high-energy asymptotics. The spectral data determine $$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$3 uniquely near the zero potential through a nonlinear Riemann–Hilbert or nonlinear Gelʹfand–Levitan–Marchenko scheme adapted to third-order operators. This work is described as the first in a series devoted to solving the inverse problem for the Boussinesq equation (Badanin et al., 2024).

6. Long-time asymptotics and dispersive regimes

For Schwartz-class initial data on the line, under no-soliton and generic-pole-at-zero assumptions, the inverse-scattering solution can be analyzed by Deift–Zhou steepest descent. In the region

$$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$4

the solution exhibits modulated oscillatory asymptotics of size $$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$5. Writing

$$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$6

one obtains an explicit leading term with amplitude proportional to

$$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$7

and phase containing $$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$8, $$v(x,t)=\int_{-\infty}^x u_t(x',t)\,dx', \qquad u_t=v_x$$9, $c=-1$00, and an integral correction, with error

$c=-1$01

uniformly for $c=-1$02 in compact subsets of $c=-1$03 (Charlier et al., 2020).

A distinct asymptotic regime arises in the Painlevé region

$c=-1$04

There the associated $c=-1$05 RH problem deforms to a Painlevé IV model problem, and the modified Boussinesq fields $c=-1$06 and $c=-1$07 are described by the Clarkson–McLeod solution $c=-1$08 of Painlevé IV with parameters

$c=-1$09

Via the Miura transformation, the good Boussinesq field $c=-1$10 has an explicit leading term of order $c=-1$11, with uniform remainder

$c=-1$12

for $c=-1$13. The same work states that the theoretical asymptotic solutions were validated against direct numerical simulations (Wang et al., 17 Nov 2025).

Taken together, these results separate at least two analytically distinct dispersive regimes: an oscillatory region governed by stationary-phase analysis around nonzero critical points, and a near-origin Painlevé region governed by the small-$c=-1$14 structure of the $c=-1$15 RH problem.

7. Numerical analysis and structure-preserving discretization

A fully discrete Fourier pseudospectral method with second-order temporal accuracy has been analyzed for the periodic good Boussinesq equation

$c=-1$16

Using a specially designed second-order time-stepping with an auxiliary variable $c=-1$17, the analysis proves unconditional nonlinear stability, with no restriction of the form $c=-1$18, and establishes the convergence estimate

$c=-1$19

under the stated regularity assumptions (Cheng et al., 2014).

A geometric alternative combines Fourier spectral semi-discretization in space with Hamiltonian Boundary Value Methods in time. In the shifted Hamiltonian formulation, HBVM$c=-1$20 has order $c=-1$21, is symmetric for all $c=-1$22, and exactly conserves polynomial Hamiltonians of degree $c=-1$23. For the cubic discrete Hamiltonian $c=-1$24, exact energy conservation requires

$c=-1$25

The SHBVM variant, with large $c=-1$26, acts as a spectral integrator in time and, in the reported tests, preserved both the Hamiltonian and momentum to round-off (Brugnano et al., 2018).

Exponential-type integrators have also been developed for the periodic equation. One first-order scheme proves convergence in $c=-1$27,

$c=-1$28

and in particular first-order convergence when the exact solution remains in $c=-1$29. A second scheme proves

$c=-1$30

when the exact solution belongs to $c=-1$31, yielding second order in $c=-1$32 for $c=-1$33, and

$c=-1$34

for $c=-1$35-solutions. The same paper emphasizes that the regularity requirements are lower than those of classical exponential integrators and states an $c=-1$36 cost per step (Ostermann et al., 2019).

A Deuflhard-type exponential integrator Fourier pseudospectral method rewrites the equation as

$c=-1$37

and combines a trapezoidal-type exponential integrator in time with FFT-based mode updates in space. For sufficiently smooth solutions, the method satisfies

$c=-1$38

so it is quadratically convergent in time and spectrally accurate in space, without any CFL-type condition (Su et al., 2019).

More recently, a filtered Lie–Trotter splitting scheme has been analyzed for low-regularity data on the periodic problem

$c=-1$39

After rewriting the PDE as a first-order Schrödinger-type system and introducing the spectral cut-off

$c=-1$40

the analysis in discrete Bourgain spaces $c=-1$41 yields, for $c=-1$42,

$c=-1$43

The paper states that this extends low-regularity convergence down to $c=-1$44, overcoming the classical $c=-1$45 bilinear barrier in smooth Sobolev spaces (Ji et al., 2024).

The numerical literature therefore reflects the same structural themes present in the analysis: dispersive stiffness, low-regularity sensitivity, and the special role of Hamiltonian and integrable structure in designing accurate long-time algorithms.